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250 lines
2.7 MiB
250 lines
2.7 MiB
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// parsing and collecting nodes and edges from the python
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It serves as a foundational framework for mathematics education in compulsory education (primary and secondary schools), outlining content requirements, pedagogical approaches, and learning objectives. \n\nThe 2022 edition emphasizes **mathematical reasoning, logical thinking, and real-world application**, with notable adjustments to the curriculum. 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It serves as a foundational framework for mathematics education in compulsory education (primary and secondary schools), outlining content requirements, pedagogical approaches, and learning objectives. \n\nThe 2022 edition emphasizes **mathematical reasoning, logical thinking, and real-world application**, with notable adjustments to the curriculum. 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By combining practical exploration with rigorous logical structures, it prepares learners to analyze and interpret geometric problems at varying levels of complexity. \n\nOverall, **\u56fe\u5f62\u4e0e\u51e0\u4f55** encompasses a broad spectrum of geometric concepts, from elementary shape identification to advanced proofs, while maintaining a focus on clarity, coherence, and mathematical reasoning."}, {"color": "#c54d27", "created_at": 1752884028, "description": "A core area in mathematics education focusing on statistics and probability.\u003cSEP\u003eThe \u0027Statistics and Probability\u0027 category addresses the needs of the big data era, including distributed computing of averages and percentages, data classification, and the development of data analysis skills and model thinking.", "entity_id": "\u7edf\u8ba1\u4e0e\u6982\u7387", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u7edf\u8ba1\u4e0e\u6982\u7387", "label": "\u7edf\u8ba1\u4e0e\u6982\u7387", "shape": "dot", "size": 10, "source_id": "chunk-2fe44546e8e21363ac87fa4cc2c15616\u003cSEP\u003echunk-b1216e19075b49d5faa5af87fa58cb85", "title": "A core area in mathematics education focusing on statistics and probability.\u003cSEP\u003eThe \u0027Statistics and Probability\u0027 category addresses the needs of the big data era, including distributed computing of averages and percentages, data classification, and the development of data analysis skills and model thinking."}, {"color": "#131d2e", "created_at": 1752884028, "description": "The \u0027Comprehensive and Practical\u0027 category integrates interdisciplinary learning and project-based learning, focusing on real-world problem-solving and fostering application and innovation awareness.", "entity_id": "\u7efc\u5408\u4e0e\u5b9e\u8df5", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u7efc\u5408\u4e0e\u5b9e\u8df5", "label": "\u7efc\u5408\u4e0e\u5b9e\u8df5", "shape": "dot", "size": 10, "source_id": "chunk-2fe44546e8e21363ac87fa4cc2c15616", "title": "The \u0027Comprehensive and Practical\u0027 category integrates interdisciplinary learning and project-based learning, focusing on real-world problem-solving and fostering application and innovation awareness."}, {"color": "#1f4d60", "created_at": 1752884028, "description": "Algebraic proof is a new requirement in the 2022 curriculum, involving proofs of algebraic propositions such as the effects of adding positive or negative numbers and divisibility rules.", "entity_id": "\u4ee3\u6570\u8bc1\u660e", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u4ee3\u6570\u8bc1\u660e", "label": "\u4ee3\u6570\u8bc1\u660e", "shape": "dot", "size": 10, "source_id": "chunk-2fe44546e8e21363ac87fa4cc2c15616", "title": "Algebraic proof is a new requirement in the 2022 curriculum, involving proofs of algebraic propositions such as the effects of adding positive or negative numbers and divisibility rules."}, {"color": "#4d4696", "created_at": 1752884028, "description": "**Compass-and-Straightedge Construction (\u5c3a\u89c4\u4f5c\u56fe)** is a classical geometric method that employs only an unmarked straightedge and a compass, adhering to strict operational limitations rooted in Euclid\u0027s first three postulates. This technique, discussed by Descartes in relation to algebraic equations, has profoundly influenced both ancient Greek and modern mathematics education, particularly in China, where it remains a cornerstone of the curriculum. \n\nWhile celebrated for its educational value\u2014fostering geometric intuition, spatial imagination, and a deeper understanding of geometric principles\u2014it is also recognized for its inherent limitations. For instance, it has been proven insufficient for tasks like trisecting a 60-degree angle. Modern pedagogical approaches emphasize its role in developing conceptual skills rather than mere technical proficiency, with content progressively introduced from elementary to middle school to align with higher learning goals. \n\nDespite its constraints, compass-and-straightedge construction endures as a foundational tool in mathematics, bridging historical traditions and contemporary educational objectives. Its legacy lies in cultivating abstract reasoning and problem-solving abilities through hands-on geometric exploration.", "entity_id": "\u5c3a\u89c4\u4f5c\u56fe", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u5c3a\u89c4\u4f5c\u56fe", "label": "\u5c3a\u89c4\u4f5c\u56fe", "shape": "dot", "size": 10, "source_id": "chunk-5bc72352d3552516e17e5466f1f1fa7d\u003cSEP\u003echunk-eb909fae946cb41ff187d26fae2118b2\u003cSEP\u003echunk-665ebf4a9a719e0056c9e54525078072\u003cSEP\u003echunk-2fe44546e8e21363ac87fa4cc2c15616\u003cSEP\u003echunk-c8a15761aa2fa77855ceb30f7b7a730a\u003cSEP\u003echunk-77da8784644dcc3dac41a17b75632803\u003cSEP\u003echunk-e91dee450ab50cb1a44f08fe171c809b\u003cSEP\u003echunk-dda7ba554b64773a9f739a19965188f1\u003cSEP\u003echunk-3e3dea3ce1efa839313f48f8630d1fc8\u003cSEP\u003echunk-69ea16a9bf8f7637a55bade939083e96", "title": "**Compass-and-Straightedge Construction (\u5c3a\u89c4\u4f5c\u56fe)** is a classical geometric method that employs only an unmarked straightedge and a compass, adhering to strict operational limitations rooted in Euclid\u0027s first three postulates. This technique, discussed by Descartes in relation to algebraic equations, has profoundly influenced both ancient Greek and modern mathematics education, particularly in China, where it remains a cornerstone of the curriculum. \n\nWhile celebrated for its educational value\u2014fostering geometric intuition, spatial imagination, and a deeper understanding of geometric principles\u2014it is also recognized for its inherent limitations. For instance, it has been proven insufficient for tasks like trisecting a 60-degree angle. Modern pedagogical approaches emphasize its role in developing conceptual skills rather than mere technical proficiency, with content progressively introduced from elementary to middle school to align with higher learning goals. \n\nDespite its constraints, compass-and-straightedge construction endures as a foundational tool in mathematics, bridging historical traditions and contemporary educational objectives. Its legacy lies in cultivating abstract reasoning and problem-solving abilities through hands-on geometric exploration."}, {"color": "#02c97b", "created_at": 1752884028, "description": "**Summary for \u767e\u5206\u6570 (Percentage)** \n\n\u767e\u5206\u6570 is a statistical measure used to determine thresholds, such as the jump rope standard, reflecting frequency and probability estimation. In the new curriculum, its calculation has been shifted from the \"Number and Algebra\" domain to \"Statistics and Probability,\" highlighting its applications in real-time computation and decision-making. \n\nTo accommodate large-scale data processing, modern enhancements incorporate distributed computing approaches for more efficient percentage calculations. \n\nIn statistical contexts, \u767e\u5206\u6570 is divided into two interpretations: \n1. **Algebraic Meaning (Stable Ratio)** \u2013 Represents fixed proportions, often used in deterministic calculations. \n2. **Statistical Meaning (Random Ratio)** \u2013 Reflects probabilistic estimation, applied in scenarios involving randomness and uncertainty. \n\nThese distinct interpretations allow \u767e\u5206\u6570 to be applied across different use cases, from stable comparisons to dynamic probability-based decision-making. The integration of computational advancements and curriculum updates underscores its evolving role in both theoretical and practical statistical applications.", "entity_id": "\u767e\u5206\u6570", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u767e\u5206\u6570", "label": "\u767e\u5206\u6570", "shape": "dot", "size": 10, "source_id": "chunk-ec9187ab77c2e91f0873e149ddc0ed25\u003cSEP\u003echunk-5bc72352d3552516e17e5466f1f1fa7d\u003cSEP\u003echunk-2fe44546e8e21363ac87fa4cc2c15616\u003cSEP\u003echunk-2e4972ad6e1759c82b71b7eb4bf1ccd8\u003cSEP\u003echunk-edaeb942f2d7bc9b4f269a55a5bdd99f", "title": "**Summary for \u767e\u5206\u6570 (Percentage)** \n\n\u767e\u5206\u6570 is a statistical measure used to determine thresholds, such as the jump rope standard, reflecting frequency and probability estimation. In the new curriculum, its calculation has been shifted from the \"Number and Algebra\" domain to \"Statistics and Probability,\" highlighting its applications in real-time computation and decision-making. \n\nTo accommodate large-scale data processing, modern enhancements incorporate distributed computing approaches for more efficient percentage calculations. \n\nIn statistical contexts, \u767e\u5206\u6570 is divided into two interpretations: \n1. **Algebraic Meaning (Stable Ratio)** \u2013 Represents fixed proportions, often used in deterministic calculations. \n2. **Statistical Meaning (Random Ratio)** \u2013 Reflects probabilistic estimation, applied in scenarios involving randomness and uncertainty. \n\nThese distinct interpretations allow \u767e\u5206\u6570 to be applied across different use cases, from stable comparisons to dynamic probability-based decision-making. The integration of computational advancements and curriculum updates underscores its evolving role in both theoretical and practical statistical applications."}, {"color": "#82a146", "created_at": 1752884054, "description": "Sample mean calculation with stepwise computation requirements in the 2022 compulsory education curriculum.\u003cSEP\u003eThe sample mean is an estimate of the population mean, calculated from a subset of data.", "entity_id": "\u6837\u672c\u5747\u503c", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u6837\u672c\u5747\u503c", "label": "\u6837\u672c\u5747\u503c", "shape": "dot", "size": 10, "source_id": "chunk-2e4972ad6e1759c82b71b7eb4bf1ccd8\u003cSEP\u003echunk-d606ad10186764121c125503135f2f7c", "title": "Sample mean calculation with stepwise computation requirements in the 2022 compulsory education curriculum.\u003cSEP\u003eThe sample mean is an estimate of the population mean, calculated from a subset of data."}, {"color": "#1b2ffc", "created_at": 1752884061, "description": "Mathematics educators who need to understand advanced mathematical concepts for effective teaching.", "entity_id": "\u6570\u5b66\u6559\u5e08", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u6570\u5b66\u6559\u5e08", "label": "\u6570\u5b66\u6559\u5e08", "shape": "dot", "size": 10, "source_id": "chunk-4f65d3b7936ba144a77113f1bb4602dd", "title": "Mathematics educators who need to understand advanced mathematical concepts for effective teaching."}, {"color": "#5ae33c", "created_at": 1752884057, "description": "Random decision-making refers to choices made based on probabilistic assessments rather than deterministic certainty.\u003cSEP\u003e\u968f\u673a\u51b3\u7b56\u662f\u4e00\u79cd\u51b3\u7b56\u65b9\u6cd5\uff0c\u6ca1\u6709\u7edd\u5bf9\u7684\u5bf9\u9519\uff0c\u53ea\u6709\u65b9\u6cd5\u7684\u597d\u574f\u4e4b\u5206\u3002", "entity_id": "\u968f\u673a\u51b3\u7b56", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u968f\u673a\u51b3\u7b56", "label": "\u968f\u673a\u51b3\u7b56", "shape": "dot", "size": 10, "source_id": "chunk-a6c5c15edb2653db4909a8aeafccee93\u003cSEP\u003echunk-a84c3d9a08c41fcb09806c860f52c321", "title": "Random decision-making refers to choices made based on probabilistic assessments rather than deterministic certainty.\u003cSEP\u003e\u968f\u673a\u51b3\u7b56\u662f\u4e00\u79cd\u51b3\u7b56\u65b9\u6cd5\uff0c\u6ca1\u6709\u7edd\u5bf9\u7684\u5bf9\u9519\uff0c\u53ea\u6709\u65b9\u6cd5\u7684\u597d\u574f\u4e4b\u5206\u3002"}, {"color": "#041335", "created_at": 1752884096, "description": "Abstract structure is a concept in mathematics representing the research objects and foundational frameworks in mathematical studies.", "entity_id": "\u62bd\u8c61\u7ed3\u6784", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u62bd\u8c61\u7ed3\u6784", "label": "\u62bd\u8c61\u7ed3\u6784", "shape": "dot", "size": 10, "source_id": "chunk-1d390244ef79debb74dcf77a05952ebc", "title": "Abstract structure is a concept in mathematics representing the research objects and foundational frameworks in mathematical studies."}, {"color": "#8b27b2", "created_at": 1752884199, "description": "The 2022 curriculum standards emphasize a holistic design and step-by-step implementation in teaching mathematics.", "entity_id": "\u6574\u4f53\u8bbe\u8ba1\uff0c\u5206\u6b65\u5b9e\u65bd", "entity_type": "UNKNOWN", "file_path": "ChuZhongShuXue.docx", "id": "\u6574\u4f53\u8bbe\u8ba1\uff0c\u5206\u6b65\u5b9e\u65bd", "label": "\u6574\u4f53\u8bbe\u8ba1\uff0c\u5206\u6b65\u5b9e\u65bd", "shape": "dot", "size": 10, "source_id": "chunk-42e82319a2d63b215f82bba66d1e632d", "title": "The 2022 curriculum standards emphasize a holistic design and step-by-step implementation in teaching mathematics."}, {"color": "#5fe20a", "created_at": 1752884034, "description": "**\u300a\u4e49\u52a1\u6559\u80b2\u6570\u5b66\u8bfe\u7a0b\u6807\u51c6(2011\u5e74\u7248)\u300b (Compulsory Education Mathematics Curriculum Standards, 2011 Edition)** is a Chinese national educational standard document that outlines the core competencies, curriculum framework, and pedagogical guidelines for mathematics in compulsory education. It emphasizes fundamental mathematical concepts and skills, expanding the traditional \"dual basics\" (basic knowledge and basic skills) to \"four basics,\" which additionally include basic mathematical thinking and basic practical experience. \n\nThe standard places significant emphasis on **compass-and-straightedge constructions** as part of the geometry curriculum and introduces **classification concepts** in primary education. It also includes **statistical survey teaching cases** to enhance data literacy. While the 2011 edition strengthened core mathematical competencies, it **lacked explicit emphasis on algebraic reasoning** compared to later revisions. Additionally, during its development, there were discussions about whether to include the **axiom of similar polygon existence**, reflecting the careful consideration of foundational mathematical principles. \n\nOverall, this standard served as a key framework for mathematics education in China, balancing foundational skills, problem-solving, and conceptual understanding while guiding instructional practices in compulsory education.", "entity_id": "\u300a\u4e49\u52a1\u6559\u80b2\u6570\u5b66\u8bfe\u7a0b\u6807\u51c6(2011\u5e74\u7248)\u300b", "entity_type": "organization", "file_path": "ChuZhongShuXue.docx", "id": "\u300a\u4e49\u52a1\u6559\u80b2\u6570\u5b66\u8bfe\u7a0b\u6807\u51c6(2011\u5e74\u7248)\u300b", "label": "\u300a\u4e49\u52a1\u6559\u80b2\u6570\u5b66\u8bfe\u7a0b\u6807\u51c6(2011\u5e74\u7248)\u300b", "shape": "dot", "size": 10, "source_id": "chunk-ec9187ab77c2e91f0873e149ddc0ed25\u003cSEP\u003echunk-bc737ed018919d513c346a3c74b449aa\u003cSEP\u003echunk-46deef01a2345f56bb0d9a61a58a7840\u003cSEP\u003echunk-c8a15761aa2fa77855ceb30f7b7a730a\u003cSEP\u003echunk-e91dee450ab50cb1a44f08fe171c809b\u003cSEP\u003echunk-d600d36500fc1362f8eb34c6e841165e\u003cSEP\u003echunk-d2d1fa534f69129f1d307d4162d5e6e4\u003cSEP\u003echunk-683c16577a28a4b7a2506b946403f52e\u003cSEP\u003echunk-4cff6339e8a61331ff6a8ae3b6298088", "title": "**\u300a\u4e49\u52a1\u6559\u80b2\u6570\u5b66\u8bfe\u7a0b\u6807\u51c6(2011\u5e74\u7248)\u300b (Compulsory Education Mathematics Curriculum Standards, 2011 Edition)** is a Chinese national educational standard document that outlines the core competencies, curriculum framework, and pedagogical guidelines for mathematics in compulsory education. It emphasizes fundamental mathematical concepts and skills, expanding the traditional \"dual basics\" (basic knowledge and basic skills) to \"four basics,\" which additionally include basic mathematical thinking and basic practical experience. \n\nThe standard places significant emphasis on **compass-and-straightedge constructions** as part of the geometry curriculum and introduces **classification concepts** in primary education. It also includes **statistical survey teaching cases** to enhance data literacy. While the 2011 edition strengthened core mathematical competencies, it **lacked explicit emphasis on algebraic reasoning** compared to later revisions. Additionally, during its development, there were discussions about whether to include the **axiom of similar polygon existence**, reflecting the careful consideration of foundational mathematical principles. \n\nOverall, this standard served as a key framework for mathematics education in China, balancing foundational skills, problem-solving, and conceptual understanding while guiding instructional practices in compulsory education."}, {"color": "#5d8ebf", "created_at": 1752884043, "description": "Mathematical abstraction refers to the process of extracting mathematical concepts from concrete problems, identified as a key development area in Chinese mathematics education.\u003cSEP\u003eMathematical abstraction refers to the process of generalizing concrete problems into symbolic representations, a key concept discussed in the text regarding the development of mathematics.\u003cSEP\u003e\u6570\u5b66\u62bd\u8c61\u5305\u62ec\u4ece\u611f\u6027\u5177\u4f53\u5230\u7406\u6027\u4e00\u822c\u7684\u601d\u7ef4\u8fc7\u7a0b\uff0c\u6700\u7ec8\u5b9e\u73b0\u7b26\u53f7\u5316\u548c\u5f62\u5f0f\u5316\u3002", "entity_id": "\u6570\u5b66\u62bd\u8c61", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u6570\u5b66\u62bd\u8c61", "label": "\u6570\u5b66\u62bd\u8c61", "shape": "dot", "size": 10, "source_id": "chunk-683c16577a28a4b7a2506b946403f52e\u003cSEP\u003echunk-7719147afc84fa56038af71629c88a4b\u003cSEP\u003echunk-bc737ed018919d513c346a3c74b449aa", "title": "Mathematical abstraction refers to the process of extracting mathematical concepts from concrete problems, identified as a key development area in Chinese mathematics education.\u003cSEP\u003eMathematical abstraction refers to the process of generalizing concrete problems into symbolic representations, a key concept discussed in the text regarding the development of mathematics.\u003cSEP\u003e\u6570\u5b66\u62bd\u8c61\u5305\u62ec\u4ece\u611f\u6027\u5177\u4f53\u5230\u7406\u6027\u4e00\u822c\u7684\u601d\u7ef4\u8fc7\u7a0b\uff0c\u6700\u7ec8\u5b9e\u73b0\u7b26\u53f7\u5316\u548c\u5f62\u5f0f\u5316\u3002"}, {"color": "#36cdb9", "created_at": 1752884043, "description": "The traditional \u0027dual basics\u0027 in Chinese math education referring to basic knowledge and basic skills, later expanded to \u0027four basics\u0027.\u003cSEP\u003eTraditional Chinese mathematics education focused on two basics: basic knowledge and basic skills.", "entity_id": "\u53cc\u57fa", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u53cc\u57fa", "label": "\u53cc\u57fa", "shape": "dot", "size": 10, "source_id": "chunk-683c16577a28a4b7a2506b946403f52e\u003cSEP\u003echunk-bc737ed018919d513c346a3c74b449aa", "title": "The traditional \u0027dual basics\u0027 in Chinese math education referring to basic knowledge and basic skills, later expanded to \u0027four basics\u0027.\u003cSEP\u003eTraditional Chinese mathematics education focused on two basics: basic knowledge and basic skills."}, {"color": "#18d526", "created_at": 1752884028, "description": "The concept of **\u6570\u5f62\u7ed3\u5408 (Integration of Numbers and Shapes)** is a fundamental mathematical methodology that bridges algebraic and geometric representations. It unifies numerical and geometric methods in mathematics, emphasizing the correspondence between numbers and points, as well as quantities and distances. This approach typically begins in middle school, where students explore the relationship between numbers and shapes through foundational tools such as the number line and absolute value. A key aspect of this integration is the consistency between absolute value\u0027s algebraic interpretation (as a numerical magnitude) and its geometric interpretation (as a distance on the number line). Overall, \u6570\u5f62\u7ed3\u5408 serves as a core principle in mathematics, facilitating deeper understanding by linking abstract algebraic concepts with visual geometric representations.", "entity_id": "\u6570\u5f62\u7ed3\u5408", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u6570\u5f62\u7ed3\u5408", "label": "\u6570\u5f62\u7ed3\u5408", "shape": "dot", "size": 10, "source_id": "chunk-d79a25c9ff12f28251aed2da9132d65e\u003cSEP\u003echunk-42e82319a2d63b215f82bba66d1e632d\u003cSEP\u003echunk-2fe44546e8e21363ac87fa4cc2c15616\u003cSEP\u003echunk-a1b0ea141dcc25762619a431f93aeca3\u003cSEP\u003echunk-10177ac2b1564cfa9472038f27fca84a", "title": "The concept of **\u6570\u5f62\u7ed3\u5408 (Integration of Numbers and Shapes)** is a fundamental mathematical methodology that bridges algebraic and geometric representations. It unifies numerical and geometric methods in mathematics, emphasizing the correspondence between numbers and points, as well as quantities and distances. This approach typically begins in middle school, where students explore the relationship between numbers and shapes through foundational tools such as the number line and absolute value. A key aspect of this integration is the consistency between absolute value\u0027s algebraic interpretation (as a numerical magnitude) and its geometric interpretation (as a distance on the number line). Overall, \u6570\u5f62\u7ed3\u5408 serves as a core principle in mathematics, facilitating deeper understanding by linking abstract algebraic concepts with visual geometric representations."}, {"color": "#fcf624", "created_at": 1752884028, "description": "Negative numbers, known as \"\u8d1f\u6570\" in Chinese, are a fundamental concept in mathematics representing values less than zero, opposite in meaning to positive numbers. They were first introduced in ancient times, with their addition and subtraction methods documented in the Chinese text *\"Nine Chapters on the Mathematical Art\" (\u4e5d\u7ae0\u7b97\u672f)*, while their multiplication and division operations were later refined in India. These numbers were abstracted from real-world quantities, such as trade and transactional problems. \n\nNegative numbers are taught to deepen conceptual understanding and are included in the \"Number and Algebra\" category, expanding numerical concepts and enabling operations involving values below zero. Their development and application in practical scenarios highlight their significance in mathematics, with contributions from both ancient Chinese and Indian scholars. This comprehensive framework integrates historical origins, theoretical foundations, and practical uses.", "entity_id": "\u8d1f\u6570", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u8d1f\u6570", "label": "\u8d1f\u6570", "shape": "dot", "size": 10, "source_id": "chunk-f379ebb8f8dc44a54c681880caf478eb\u003cSEP\u003echunk-2fe44546e8e21363ac87fa4cc2c15616\u003cSEP\u003echunk-3036fe5f8bcbc7479b162f62dea08660\u003cSEP\u003echunk-c38ffe600259aeb39e4227f260864859\u003cSEP\u003echunk-2820dfe8496edbd1e66a24c26cdd1859", "title": "Negative numbers, known as \"\u8d1f\u6570\" in Chinese, are a fundamental concept in mathematics representing values less than zero, opposite in meaning to positive numbers. They were first introduced in ancient times, with their addition and subtraction methods documented in the Chinese text *\"Nine Chapters on the Mathematical Art\" (\u4e5d\u7ae0\u7b97\u672f)*, while their multiplication and division operations were later refined in India. These numbers were abstracted from real-world quantities, such as trade and transactional problems. \n\nNegative numbers are taught to deepen conceptual understanding and are included in the \"Number and Algebra\" category, expanding numerical concepts and enabling operations involving values below zero. Their development and application in practical scenarios highlight their significance in mathematics, with contributions from both ancient Chinese and Indian scholars. This comprehensive framework integrates historical origins, theoretical foundations, and practical uses."}, {"color": "#8551c7", "created_at": 1752884028, "description": "Algebraic expression operations involve extending arithmetic operations from numbers to letters, forming the basis of algebra.", "entity_id": "\u4ee3\u6570\u5f0f\u8fd0\u7b97", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u4ee3\u6570\u5f0f\u8fd0\u7b97", "label": "\u4ee3\u6570\u5f0f\u8fd0\u7b97", "shape": "dot", "size": 10, "source_id": "chunk-2fe44546e8e21363ac87fa4cc2c15616", "title": "Algebraic expression operations involve extending arithmetic operations from numbers to letters, forming the basis of algebra."}, {"color": "#3a9f4c", "created_at": 1752884028, "description": "**Functions (\u51fd\u6570)** are a fundamental concept in mathematics, particularly in algebra and mathematical analysis, representing relationships between variables, inputs, and outputs. Each input to a function corresponds to a unique output, forming the basis of variable dependencies and algebraic expressions. The concept of functions is central to mathematical modeling across various disciplines, as they provide structured patterns for analyzing real-world phenomena. \n\nThe historical development of functions includes an evolution from formal descriptions to variable-based interpretations, particularly concerning real-number mappings. The term **\u51fd\u6570 (Functions)** was introduced to Chinese mathematics through translations by **Li Shanlan** and **Alexander Wylie**, integrating the concept into East Asian mathematical discourse. \n\nIn educational contexts, functions are taught with varying definitions at different levels\u2014junior high school introduces them as foundational algebraic relationships, while senior high school and advanced studies explore more rigorous definitions, such as real-valued mappings in mathematical analysis. Thus, functions serve as a unifying mathematical construct with broad theoretical and practical significance.", "entity_id": "\u51fd\u6570", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u51fd\u6570", "label": "\u51fd\u6570", "shape": "dot", "size": 10, "source_id": "chunk-5bc72352d3552516e17e5466f1f1fa7d\u003cSEP\u003echunk-7f35e11154694cb00fbc4308a33a36a3\u003cSEP\u003echunk-2fe44546e8e21363ac87fa4cc2c15616\u003cSEP\u003echunk-035c55452674b5ec27f1692bfa495347\u003cSEP\u003echunk-bdb2d088364dc5e365dcf7c67ff1476e\u003cSEP\u003echunk-0e47637ef5cc5c08729008ec09e12dd7\u003cSEP\u003echunk-c38ffe600259aeb39e4227f260864859", "title": "**Functions (\u51fd\u6570)** are a fundamental concept in mathematics, particularly in algebra and mathematical analysis, representing relationships between variables, inputs, and outputs. Each input to a function corresponds to a unique output, forming the basis of variable dependencies and algebraic expressions. The concept of functions is central to mathematical modeling across various disciplines, as they provide structured patterns for analyzing real-world phenomena. \n\nThe historical development of functions includes an evolution from formal descriptions to variable-based interpretations, particularly concerning real-number mappings. The term **\u51fd\u6570 (Functions)** was introduced to Chinese mathematics through translations by **Li Shanlan** and **Alexander Wylie**, integrating the concept into East Asian mathematical discourse. \n\nIn educational contexts, functions are taught with varying definitions at different levels\u2014junior high school introduces them as foundational algebraic relationships, while senior high school and advanced studies explore more rigorous definitions, such as real-valued mappings in mathematical analysis. 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It serves as a core postulate in **Euclidean geometry**, originally formulated by Euclid and later reformulated by **Playfair**, whose version became more widely recognized. This axiom distinguishes classical Euclidean geometry from **non-Euclidean geometric systems**, as its modification or rejection leads to the development of alternative geometries, such as hyperbolic and elliptic geometries. The principle underscores the uniqueness and properties of parallel lines, making it a pivotal element in both theoretical and applied geometric studies. \n\n(Note: The summary resolves minor redundancies and integrates all descriptions into a coherent narrative while preserving key details about the axiom\u0027s historical significance, mathematical role, and impact on different geometric frameworks.)", "entity_id": "\u5e73\u884c\u7ebf\u516c\u7406", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u5e73\u884c\u7ebf\u516c\u7406", "label": "\u5e73\u884c\u7ebf\u516c\u7406", "shape": "dot", "size": 10, "source_id": "chunk-e91dee450ab50cb1a44f08fe171c809b\u003cSEP\u003echunk-2fe44546e8e21363ac87fa4cc2c15616\u003cSEP\u003echunk-1725da37a2e4d06bc7e57390451926d7\u003cSEP\u003echunk-6782332859e0bdef19c72a7367743352", "title": "The **Parallel Line Axiom (\u5e73\u884c\u7ebf\u516c\u7406)**, also known as the **Parallel Postulate**, is a foundational concept in geometry that plays a critical role in defining geometric relationships and proofs. It serves as a core postulate in **Euclidean geometry**, originally formulated by Euclid and later reformulated by **Playfair**, whose version became more widely recognized. This axiom distinguishes classical Euclidean geometry from **non-Euclidean geometric systems**, as its modification or rejection leads to the development of alternative geometries, such as hyperbolic and elliptic geometries. The principle underscores the uniqueness and properties of parallel lines, making it a pivotal element in both theoretical and applied geometric studies. \n\n(Note: The summary resolves minor redundancies and integrates all descriptions into a coherent narrative while preserving key details about the axiom\u0027s historical significance, mathematical role, and impact on different geometric frameworks.)"}, {"color": "#a80f93", "created_at": 1752884028, "description": "The sum of a triangle\u0027s interior angles (180 degrees) is a key geometric theorem proven through deductive reasoning.\u003cSEP\u003e\u4e09\u89d2\u5f62\u5185\u89d2\u548c\u4e3a180\u5ea6\u662f\u4e00\u4e2a\u51e0\u4f55\u5b9a\u7406\uff0c\u7528\u4e8e\u6f14\u7ece\u63a8\u7406\u7b49\u8170\u4e09\u89d2\u5f62\u7684\u5e95\u89d2\u6027\u8d28\u3002", "entity_id": "\u4e09\u89d2\u5f62\u5185\u89d2\u548c", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u4e09\u89d2\u5f62\u5185\u89d2\u548c", "label": "\u4e09\u89d2\u5f62\u5185\u89d2\u548c", "shape": "dot", "size": 10, "source_id": "chunk-2fe44546e8e21363ac87fa4cc2c15616\u003cSEP\u003echunk-7719147afc84fa56038af71629c88a4b", "title": "The sum of a triangle\u0027s interior angles (180 degrees) is a key geometric theorem proven through deductive reasoning.\u003cSEP\u003e\u4e09\u89d2\u5f62\u5185\u89d2\u548c\u4e3a180\u5ea6\u662f\u4e00\u4e2a\u51e0\u4f55\u5b9a\u7406\uff0c\u7528\u4e8e\u6f14\u7ece\u63a8\u7406\u7b49\u8170\u4e09\u89d2\u5f62\u7684\u5e95\u89d2\u6027\u8d28\u3002"}, {"color": "#fe52f6", "created_at": 1752884028, "description": "**Euclidean Geometry (\u6b27\u51e0\u91cc\u5f97\u51e0\u4f55)** is a classical mathematical system attributed to the ancient Greek mathematician Euclid, based on his foundational work incorporating five postulates (including the parallel postulate) and five axioms. It serves as the basis of traditional geometric study, focusing on the properties and relations of points, lines, planes, and figures in both two-dimensional (plane) and three-dimensional (space) contexts. Euclidean geometry preserves distances and angles under rigid transformations and emphasizes proofs of congruence, similarity, and commensurable ratios of shapes. \n\nKey aspects include: \n- **Axiomatic Structure**: The system relies on definitions, postulates (e.g., the parallel postulate, which states that through a point not on a line, exactly one parallel line can be drawn), and theorems derived through deductive reasoning, influenced by Pythagorean methods. \n- **Historical Influence**: It historically equated geometric proof with axiomatic reasoning, though modern mathematics treats axioms as assumptions rather than self-evident truths. \n- **Core Concepts**: It studies spatial definitions, scale-invariant properties (e.g., triangle angles summing to 180\u00b0), and foundational elements like points and lines that resist further definition. Euclidean geometry also involves ruler-and-compass constructions and the movement of figures. \n- **Legacy**: It provided the basis for later formalizations, such as Hilbert\u0027s axiomatic geometry, though its fifth postulate (or axioms like \"the whole is greater than the part\") does not hold in non-Euclidean or infinite-set contexts. \n\nEuclidean geometry remains central to mathematical education and spatial reasoning, emphasizing logical proof and classical geometric principles.", "entity_id": "\u6b27\u51e0\u91cc\u5f97\u51e0\u4f55", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u6b27\u51e0\u91cc\u5f97\u51e0\u4f55", "label": "\u6b27\u51e0\u91cc\u5f97\u51e0\u4f55", "shape": "dot", "size": 10, "source_id": "chunk-d79a25c9ff12f28251aed2da9132d65e\u003cSEP\u003echunk-5bc72352d3552516e17e5466f1f1fa7d\u003cSEP\u003echunk-a77717ea766ba1464bcbadc22f73914d\u003cSEP\u003echunk-665ebf4a9a719e0056c9e54525078072\u003cSEP\u003echunk-30c648659bc2a2999312fff54eb5d51a\u003cSEP\u003echunk-2fe44546e8e21363ac87fa4cc2c15616\u003cSEP\u003echunk-54a2947dce8bb8651e7af1f86e1e89a5\u003cSEP\u003echunk-8b6f8917e6704256d88d931fd2d99d07\u003cSEP\u003echunk-0f6c384b97d0bb5698ba21610de20e63\u003cSEP\u003echunk-a48b8be14cbd9c4d38521dbf6a089e16\u003cSEP\u003echunk-1725da37a2e4d06bc7e57390451926d7\u003cSEP\u003echunk-e91dee450ab50cb1a44f08fe171c809b\u003cSEP\u003echunk-6deef743946d62c39bdfa573f520aa57\u003cSEP\u003echunk-d2d1fa534f69129f1d307d4162d5e6e4\u003cSEP\u003echunk-497a0e71f7c50523d9be03ab68a8d478\u003cSEP\u003echunk-6782332859e0bdef19c72a7367743352\u003cSEP\u003echunk-262593f034795826c0c15acbf0edb10f\u003cSEP\u003echunk-b6b061c418521e25080936fc1c0a8bc8", "title": "**Euclidean Geometry (\u6b27\u51e0\u91cc\u5f97\u51e0\u4f55)** is a classical mathematical system attributed to the ancient Greek mathematician Euclid, based on his foundational work incorporating five postulates (including the parallel postulate) and five axioms. It serves as the basis of traditional geometric study, focusing on the properties and relations of points, lines, planes, and figures in both two-dimensional (plane) and three-dimensional (space) contexts. Euclidean geometry preserves distances and angles under rigid transformations and emphasizes proofs of congruence, similarity, and commensurable ratios of shapes. \n\nKey aspects include: \n- **Axiomatic Structure**: The system relies on definitions, postulates (e.g., the parallel postulate, which states that through a point not on a line, exactly one parallel line can be drawn), and theorems derived through deductive reasoning, influenced by Pythagorean methods. \n- **Historical Influence**: It historically equated geometric proof with axiomatic reasoning, though modern mathematics treats axioms as assumptions rather than self-evident truths. \n- **Core Concepts**: It studies spatial definitions, scale-invariant properties (e.g., triangle angles summing to 180\u00b0), and foundational elements like points and lines that resist further definition. Euclidean geometry also involves ruler-and-compass constructions and the movement of figures. \n- **Legacy**: It provided the basis for later formalizations, such as Hilbert\u0027s axiomatic geometry, though its fifth postulate (or axioms like \"the whole is greater than the part\") does not hold in non-Euclidean or infinite-set contexts. \n\nEuclidean geometry remains central to mathematical education and spatial reasoning, emphasizing logical proof and classical geometric principles."}, {"color": "#82b5f7", "created_at": 1752884030, "description": "**\u5c0f\u5b66\u6570\u5b66 (Elementary School Mathematics)** is a foundational stage of education that covers essential arithmetic and early algebraic concepts, serving as the basis for more advanced studies such as middle school mathematics (\u521d\u4e2d\u6570\u5b66\u6559\u80b2\u7684\u57fa\u7840\u9636\u6bb5). It emphasizes the development of core skills, conceptual understanding, and real-world applications across several key areas: \n\n1. **Number and Operations** \u2013 Includes basic arithmetic (e.g., addition, subtraction, multiplication, division), the correspondence between rational numbers and points on a number line, and the equivalence between fractions and division (e.g., understanding division as the inverse of multiplication). Algorithmic and reasoning skills are highlighted, such as methods for multi-digit multiplication (\u591a\u4f4d\u6570\u4e58\u6cd5\u7684\u8ba1\u7b97\u65b9\u6cd5). \n\n2. **Quantitative Relationships** \u2013 Introduces early algebraic reasoning, problem-solving, and the exploration of mathematical relationships (e.g., properties of equations). \n\n3. **Geometry and Spatial Awareness** \u2013 Develops intuitive understanding through classifying 3D objects, abstracting shapes (e.g., points, lines, planes), and fostering geometric intuition. \n\n4. **Conceptual Foundations** \u2013 Prioritizes both procedural fluency (e.g., calculations) and deep comprehension of underlying principles (\u7b97\u7406), ensuring students grasp the \"why\" behind mathematical operations. \n\nWhile descriptions vary slightly\u2014some emphasizing arithmetic, others geometry or algebra\u2014the overarching focus remains on building a strong, versatile foundation for future learning. 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It emphasizes the development of core skills, conceptual understanding, and real-world applications across several key areas: \n\n1. **Number and Operations** \u2013 Includes basic arithmetic (e.g., addition, subtraction, multiplication, division), the correspondence between rational numbers and points on a number line, and the equivalence between fractions and division (e.g., understanding division as the inverse of multiplication). Algorithmic and reasoning skills are highlighted, such as methods for multi-digit multiplication (\u591a\u4f4d\u6570\u4e58\u6cd5\u7684\u8ba1\u7b97\u65b9\u6cd5). \n\n2. **Quantitative Relationships** \u2013 Introduces early algebraic reasoning, problem-solving, and the exploration of mathematical relationships (e.g., properties of equations). \n\n3. **Geometry and Spatial Awareness** \u2013 Develops intuitive understanding through classifying 3D objects, abstracting shapes (e.g., points, lines, planes), and fostering geometric intuition. \n\n4. **Conceptual Foundations** \u2013 Prioritizes both procedural fluency (e.g., calculations) and deep comprehension of underlying principles (\u7b97\u7406), ensuring students grasp the \"why\" behind mathematical operations. \n\nWhile descriptions vary slightly\u2014some emphasizing arithmetic, others geometry or algebra\u2014the overarching focus remains on building a strong, versatile foundation for future learning. 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applications.\u003cSEP\u003e\u5e73\u5747\u6570\u662f\u6700\u5e38\u7528\u7684\u6570\u636e\u96c6\u4e2d\u8d8b\u52bf\u6307\u6807\uff0c\u7528\u4e8e\u91cf\u5316\u6570\u636e\u7684\u5178\u578b\u7279\u5f81\uff0c\u5982\u73ed\u7ea7\u5e73\u5747\u5206\u53cd\u6620\u5b66\u4e60\u72b6\u51b5\u3002"}, {"color": "#9da88c", "created_at": 1752884028, "description": "Data classification involves developing criteria for categorizing information, focusing on binary classification systems.", "entity_id": "\u6570\u636e\u5206\u7c7b", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u6570\u636e\u5206\u7c7b", "label": "\u6570\u636e\u5206\u7c7b", "shape": "dot", "size": 10, "source_id": "chunk-2fe44546e8e21363ac87fa4cc2c15616", "title": "Data classification involves developing criteria for categorizing information, focusing on binary classification systems."}, {"color": "#a47663", "created_at": 1752884029, "description": "Interdisciplinary learning integrates multiple subject areas, implemented through theme-based and project-based approaches.", "entity_id": "\u8de8\u5b66\u79d1\u5b66\u4e60", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u8de8\u5b66\u79d1\u5b66\u4e60", "label": "\u8de8\u5b66\u79d1\u5b66\u4e60", "shape": "dot", "size": 10, "source_id": "chunk-2fe44546e8e21363ac87fa4cc2c15616", "title": "Interdisciplinary learning integrates multiple subject areas, implemented through theme-based and project-based approaches."}, {"color": "#def6db", "created_at": 1752884029, "description": "Project-based learning engages students in solving real-world problems through structured projects resembling scientific research.", "entity_id": "\u9879\u76ee\u5f0f\u5b66\u4e60", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u9879\u76ee\u5f0f\u5b66\u4e60", "label": "\u9879\u76ee\u5f0f\u5b66\u4e60", "shape": "dot", "size": 10, "source_id": "chunk-2fe44546e8e21363ac87fa4cc2c15616", "title": "Project-based learning engages students in solving real-world problems through structured projects resembling scientific research."}, {"color": "#eab5b8", "created_at": 1752884045, "description": "Geometric intuition, cultivated through hands-on construction activities in mathematics education.\u003cSEP\u003e\u51e0\u4f55\u76f4\u89c2\u662f\u6570\u5b66\u62bd\u8c61\u7684\u6838\u5fc3\u8981\u7d20\uff0c\u6b27\u62c9\u901a\u8fc7\u70b9\u548c\u7ebf\u7684\u62bd\u8c61\u63ed\u793a\u4e86\u95ee\u9898\u7684\u672c\u8d28\u3002\u003cSEP\u003e\u51e0\u4f55\u76f4\u89c2\u662f\u901a\u8fc7\u56fe\u8868\u63cf\u8ff0\u548c\u5206\u6790\u95ee\u9898\u7684\u80fd\u529b\uff0c\u6709\u52a9\u4e8e\u628a\u63e1\u95ee\u9898\u672c\u8d28\u548c\u660e\u6670\u601d\u7ef4\u8def\u5f84\u3002", "entity_id": "\u51e0\u4f55\u76f4\u89c2", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u51e0\u4f55\u76f4\u89c2", "label": "\u51e0\u4f55\u76f4\u89c2", "shape": "dot", "size": 10, "source_id": "chunk-c8a15761aa2fa77855ceb30f7b7a730a\u003cSEP\u003echunk-3e3dea3ce1efa839313f48f8630d1fc8\u003cSEP\u003echunk-a9f19b93d1f269d93ac2f8605d5d794b", "title": "Geometric intuition, cultivated through hands-on construction activities in mathematics education.\u003cSEP\u003e\u51e0\u4f55\u76f4\u89c2\u662f\u6570\u5b66\u62bd\u8c61\u7684\u6838\u5fc3\u8981\u7d20\uff0c\u6b27\u62c9\u901a\u8fc7\u70b9\u548c\u7ebf\u7684\u62bd\u8c61\u63ed\u793a\u4e86\u95ee\u9898\u7684\u672c\u8d28\u3002\u003cSEP\u003e\u51e0\u4f55\u76f4\u89c2\u662f\u901a\u8fc7\u56fe\u8868\u63cf\u8ff0\u548c\u5206\u6790\u95ee\u9898\u7684\u80fd\u529b\uff0c\u6709\u52a9\u4e8e\u628a\u63e1\u95ee\u9898\u672c\u8d28\u548c\u660e\u6670\u601d\u7ef4\u8def\u5f84\u3002"}, {"color": "#1e068c", "created_at": 1752884031, "description": "**Euclid (\u6b27\u51e0\u91cc\u5f97)** was an ancient Greek mathematician active around 325\u2013265 BCE, renowned for his foundational contributions to mathematics, particularly geometry and deductive reasoning. He is best known for his seminal work *Elements* (*\u51e0\u4f55\u539f\u672c* or *\u539f\u672c*), which systematically compiled and expanded the principles of classical geometry, establishing an axiomatic framework that influenced mathematics and science for millennia. \n\nIn *Elements*, Euclid defined geometric concepts such as points, lines, and congruence, and introduced axiomatic deductive reasoning, including axioms like \"things equal to the same thing are equal.\" His work bridged early algebra and geometry by emphasizing commensurable ratios of line segments and provided geometric explanations for irrational numbers. Euclid is credited with pioneering proof techniques, including proof by contradiction (\u53cd\u8bc1\u6cd5), which he demonstrated in seminal proofs such as the infinitude of prime numbers and the irrationality of \u221a2. \n\nWhile highly influential, Euclid\u0027s definitions\u2014such as his attempt to define a \"point\" as \"that which has no part\"\u2014have faced criticism, notably from later mathematicians like David Hilbert, who questioned their foundational rigor. Despite this, his logical approach, possibly influenced by Aristotle\u0027s syllogisms, cemented his legacy as the father of Euclidean geometry. Euclid was associated with the Library of Alexandria, where he consolidated and advanced mathematical knowledge, shaping the trajectory of mathematical thought for centuries. \n\nIn summary, Euclid\u0027s *Elements* not only laid the groundwork for classical geometry but also established enduring methodologies in mathematical proof and logical reasoning. His work remains a cornerstone of mathematics, despite later critiques of its definitions and assumptions.", "entity_id": "\u6b27\u51e0\u91cc\u5f97", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u6b27\u51e0\u91cc\u5f97", "label": "\u6b27\u51e0\u91cc\u5f97", "shape": "dot", "size": 10, "source_id": "chunk-08c393bb4c84a0e60fb4eb6365edfe94\u003cSEP\u003echunk-a77717ea766ba1464bcbadc22f73914d\u003cSEP\u003echunk-665ebf4a9a719e0056c9e54525078072\u003cSEP\u003echunk-8f22eaff2e18e2af1504b2368ea1e65f\u003cSEP\u003echunk-896c6ab0b0daee896907a3349e5c7811\u003cSEP\u003echunk-42e82319a2d63b215f82bba66d1e632d\u003cSEP\u003echunk-33e4d77aeae0267e846e8c7ab9af2b41\u003cSEP\u003echunk-c8a15761aa2fa77855ceb30f7b7a730a\u003cSEP\u003echunk-42220f52b4f10f3dd80cceb36ebbd3ce\u003cSEP\u003echunk-e91dee450ab50cb1a44f08fe171c809b\u003cSEP\u003echunk-3bcddcc57598e5a406ac887e26b230fa\u003cSEP\u003echunk-933b19628500cf80c1916250b497e3f8\u003cSEP\u003echunk-2ff43cd1d769194656997fc3e9797a67\u003cSEP\u003echunk-6deef743946d62c39bdfa573f520aa57\u003cSEP\u003echunk-577ed72eade6a6cbd00f14e981bf4078\u003cSEP\u003echunk-10177ac2b1564cfa9472038f27fca84a\u003cSEP\u003echunk-b6b061c418521e25080936fc1c0a8bc8", "title": "**Euclid (\u6b27\u51e0\u91cc\u5f97)** was an ancient Greek mathematician active around 325\u2013265 BCE, renowned for his foundational contributions to mathematics, particularly geometry and deductive reasoning. He is best known for his seminal work *Elements* (*\u51e0\u4f55\u539f\u672c* or *\u539f\u672c*), which systematically compiled and expanded the principles of classical geometry, establishing an axiomatic framework that influenced mathematics and science for millennia. \n\nIn *Elements*, Euclid defined geometric concepts such as points, lines, and congruence, and introduced axiomatic deductive reasoning, including axioms like \"things equal to the same thing are equal.\" His work bridged early algebra and geometry by emphasizing commensurable ratios of line segments and provided geometric explanations for irrational numbers. Euclid is credited with pioneering proof techniques, including proof by contradiction (\u53cd\u8bc1\u6cd5), which he demonstrated in seminal proofs such as the infinitude of prime numbers and the irrationality of \u221a2. \n\nWhile highly influential, Euclid\u0027s definitions\u2014such as his attempt to define a \"point\" as \"that which has no part\"\u2014have faced criticism, notably from later mathematicians like David Hilbert, who questioned their foundational rigor. Despite this, his logical approach, possibly influenced by Aristotle\u0027s syllogisms, cemented his legacy as the father of Euclidean geometry. Euclid was associated with the Library of Alexandria, where he consolidated and advanced mathematical knowledge, shaping the trajectory of mathematical thought for centuries. \n\nIn summary, Euclid\u0027s *Elements* not only laid the groundwork for classical geometry but also established enduring methodologies in mathematical proof and logical reasoning. His work remains a cornerstone of mathematics, despite later critiques of its definitions and assumptions."}, {"color": "#642d54", "created_at": 1752884045, "description": "\u60f3\u8c61\u80fd\u529b\u662f\u6307\u5728\u64cd\u4f5c\u524d\u9884\u60f3\u64cd\u4f5c\u7ed3\u679c\u7684\u80fd\u529b\uff0c\u5305\u62ec\u8bbe\u8ba1\u64cd\u4f5c\u6d41\u7a0b\u3001\u9a8c\u8bc1\u9884\u60f3\u4ee5\u53ca\u53cd\u601d\u5408\u7406\u6027\u7684\u80fd\u529b\u3002", "entity_id": "\u60f3\u8c61\u80fd\u529b", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u60f3\u8c61\u80fd\u529b", "label": "\u60f3\u8c61\u80fd\u529b", "shape": "dot", "size": 10, "source_id": "chunk-3e3dea3ce1efa839313f48f8630d1fc8", "title": "\u60f3\u8c61\u80fd\u529b\u662f\u6307\u5728\u64cd\u4f5c\u524d\u9884\u60f3\u64cd\u4f5c\u7ed3\u679c\u7684\u80fd\u529b\uff0c\u5305\u62ec\u8bbe\u8ba1\u64cd\u4f5c\u6d41\u7a0b\u3001\u9a8c\u8bc1\u9884\u60f3\u4ee5\u53ca\u53cd\u601d\u5408\u7406\u6027\u7684\u80fd\u529b\u3002"}, {"color": "#44e98b", "created_at": 1752884040, "description": "**Ren\u00e9 Descartes (\u7b1b\u5361\u5c14)** was a renowned 17th-century French philosopher, mathematician, and scientist. He is best known for his foundational contributions to **analytic geometry**, including the development of the **Cartesian coordinate system** (\u76f4\u89d2\u5750\u6807\u7cfb), as detailed in his work *La G\u00e9om\u00e9trie* (*Descartes\u0027 Geometry*). Descartes also made significant methodological and philosophical advancements, famously articulating the principle **\"Cogito, ergo sum\" (I think, therefore I am)** in *Discourse on the Method*, emphasizing intuition and deductive reasoning. \n\nIn mathematics, Descartes explored polyhedra, noting relationships between vertices, edges, and faces (1640), and he referred to negative roots of equations as \"false roots\" (*\u5047\u6839*), reflecting contemporary skepticism toward negative numbers in Western mathematics. His interdisciplinary legacy spans **geometry** (as the founder of analytic geometry), **philosophy** (with influences on rationalism and epistemology), and **science**, bridging abstract theory with practical applications. \n\nHis name, rendered as **\u7b1b\u5361\u5c14** in Chinese, remains synonymous with the integration of algebraic and geometric principles, as well as his enduring philosophical insights.", "entity_id": "\u7b1b\u5361\u5c14", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u7b1b\u5361\u5c14", "label": "\u7b1b\u5361\u5c14", "shape": "dot", "size": 10, "source_id": "chunk-f379ebb8f8dc44a54c681880caf478eb\u003cSEP\u003echunk-59fa99203b2c891d306355d72ad12c31\u003cSEP\u003echunk-54a2947dce8bb8651e7af1f86e1e89a5\u003cSEP\u003echunk-0f6c384b97d0bb5698ba21610de20e63\u003cSEP\u003echunk-00f45528a4186020aca4f014c14769e9\u003cSEP\u003echunk-40574dd690ebc162e391ed721f31bfae\u003cSEP\u003echunk-69ea16a9bf8f7637a55bade939083e96", "title": "**Ren\u00e9 Descartes (\u7b1b\u5361\u5c14)** was a renowned 17th-century French philosopher, mathematician, and scientist. He is best known for his foundational contributions to **analytic geometry**, including the development of the **Cartesian coordinate system** (\u76f4\u89d2\u5750\u6807\u7cfb), as detailed in his work *La G\u00e9om\u00e9trie* (*Descartes\u0027 Geometry*). Descartes also made significant methodological and philosophical advancements, famously articulating the principle **\"Cogito, ergo sum\" (I think, therefore I am)** in *Discourse on the Method*, emphasizing intuition and deductive reasoning. \n\nIn mathematics, Descartes explored polyhedra, noting relationships between vertices, edges, and faces (1640), and he referred to negative roots of equations as \"false roots\" (*\u5047\u6839*), reflecting contemporary skepticism toward negative numbers in Western mathematics. His interdisciplinary legacy spans **geometry** (as the founder of analytic geometry), **philosophy** (with influences on rationalism and epistemology), and **science**, bridging abstract theory with practical applications. \n\nHis name, rendered as **\u7b1b\u5361\u5c14** in Chinese, remains synonymous with the integration of algebraic and geometric principles, as well as his enduring philosophical insights."}, {"color": "#2e60b8", "created_at": 1752884175, "description": "Descartes linked traditional geometric constructions with solutions to algebraic equations.", "entity_id": "\u4ee3\u6570\u65b9\u7a0b", "entity_type": "UNKNOWN", "file_path": "ChuZhongShuXue.docx", "id": "\u4ee3\u6570\u65b9\u7a0b", "label": "\u4ee3\u6570\u65b9\u7a0b", "shape": "dot", "size": 10, "source_id": "chunk-69ea16a9bf8f7637a55bade939083e96", "title": "Descartes linked traditional geometric constructions with solutions to algebraic equations."}, {"color": "#cca076", "created_at": 1752884044, "description": "The square root of 2 (\u221a2) is an irrational mathematical constant, approximately equal to 1.4142135, that cannot be expressed as a ratio of integers. Its irrationality was first proven by Euclid using a proof by contradiction. Additionally, ancient Greek scholars explained \u221a2 geometrically through compass and straightedge constructions, demonstrating its properties in the context of early mathematical exploration. The number is frequently discussed in mathematical literature, particularly in relation to classical proofs of irrationality. Collectively, these descriptions highlight \u221a2 as a historically significant and well-studied irrational number with both algebraic and geometric importance.", "entity_id": "\u221a2", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u221a2", "label": "\u221a2", "shape": "dot", "size": 10, "source_id": "chunk-89a00ca5535ebdb5d721e473eebbac4c\u003cSEP\u003echunk-dda7ba554b64773a9f739a19965188f1\u003cSEP\u003echunk-10177ac2b1564cfa9472038f27fca84a\u003cSEP\u003echunk-8fa86b63766e4b57fc4a84950863b6e9", "title": "The square root of 2 (\u221a2) is an irrational mathematical constant, approximately equal to 1.4142135, that cannot be expressed as a ratio of integers. Its irrationality was first proven by Euclid using a proof by contradiction. Additionally, ancient Greek scholars explained \u221a2 geometrically through compass and straightedge constructions, demonstrating its properties in the context of early mathematical exploration. The number is frequently discussed in mathematical literature, particularly in relation to classical proofs of irrationality. Collectively, these descriptions highlight \u221a2 as a historically significant and well-studied irrational number with both algebraic and geometric importance."}, {"color": "#71bc78", "created_at": 1752884086, "description": "A field of mathematics that uses algebraic methods to study geometric problems, mentioned as a modern development.", "entity_id": "\u4ee3\u6570\u51e0\u4f55", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u4ee3\u6570\u51e0\u4f55", "label": "\u4ee3\u6570\u51e0\u4f55", "shape": "dot", "size": 10, "source_id": "chunk-dda7ba554b64773a9f739a19965188f1", "title": "A field of mathematics that uses algebraic methods to study geometric problems, mentioned as a modern development."}, {"color": "#45767d", "created_at": 1752884086, "description": "A famous unsolvable problem in geometry involving dividing an arbitrary angle into three equal parts using only a compass and straightedge.\u003cSEP\u003eThe problem of dividing an angle into three equal parts, demonstrated as impossible with compass and straightedge.", "entity_id": "\u4e09\u7b49\u5206\u89d2", "entity_type": "event", "file_path": "ChuZhongShuXue.docx", "id": "\u4e09\u7b49\u5206\u89d2", "label": "\u4e09\u7b49\u5206\u89d2", "shape": "dot", "size": 10, "source_id": "chunk-dda7ba554b64773a9f739a19965188f1\u003cSEP\u003echunk-77da8784644dcc3dac41a17b75632803", "title": "A famous unsolvable problem in geometry involving dividing an arbitrary angle into three equal parts using only a compass and straightedge.\u003cSEP\u003eThe problem of dividing an angle into three equal parts, demonstrated as impossible with compass and straightedge."}, {"color": "#5eb004", "created_at": 1752884086, "description": "The famous geometric problem of dividing an angle into three equal parts, proven impossible for arbitrary angles under compass-and-straightedge rules.", "entity_id": "\u4e09\u7b49\u5206\u89d2\u95ee\u9898", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u4e09\u7b49\u5206\u89d2\u95ee\u9898", "label": "\u4e09\u7b49\u5206\u89d2\u95ee\u9898", "shape": "dot", "size": 10, "source_id": "chunk-eb909fae946cb41ff187d26fae2118b2", "title": "The famous geometric problem of dividing an angle into three equal parts, proven impossible for arbitrary angles under compass-and-straightedge rules."}, {"color": "#5866fc", "created_at": 1752884086, "description": "Rational roots of a polynomial equation, specifically analyzed in the context of cubic equations for angle trisection.\u003cSEP\u003eRational roots of polynomial equations, whose existence determines whether solutions are compass-and-straightedge constructible.", "entity_id": "\u6709\u7406\u6839", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u6709\u7406\u6839", "label": "\u6709\u7406\u6839", "shape": "dot", "size": 10, "source_id": "chunk-eb909fae946cb41ff187d26fae2118b2\u003cSEP\u003echunk-77da8784644dcc3dac41a17b75632803", "title": "Rational roots of a polynomial equation, specifically analyzed in the context of cubic equations for angle trisection.\u003cSEP\u003eRational roots of polynomial equations, whose existence determines whether solutions are compass-and-straightedge constructible."}, {"color": "#83c2cf", "created_at": 1752884086, "description": "A cubic equation that is analyzed for rational roots in the context of angle trisection.", "entity_id": "\u4e09\u6b21\u65b9\u7a0b(3)", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u4e09\u6b21\u65b9\u7a0b(3)", "label": "\u4e09\u6b21\u65b9\u7a0b(3)", "shape": "dot", "size": 10, "source_id": "chunk-77da8784644dcc3dac41a17b75632803", "title": "A cubic equation that is analyzed for rational roots in the context of angle trisection."}, {"color": "#f97209", "created_at": 1752884087, "description": "Fundamental theorem used to prove the impossibility of certain geometric constructions like angle trisection.", "entity_id": "\u57fa\u672c\u5b9a\u7406", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u57fa\u672c\u5b9a\u7406", "label": "\u57fa\u672c\u5b9a\u7406", "shape": "dot", "size": 10, "source_id": "chunk-77da8784644dcc3dac41a17b75632803", "title": "Fundamental theorem used to prove the impossibility of certain geometric constructions like angle trisection."}, {"color": "#ef371c", "created_at": 1752884028, "description": "The **\u6570\u8f74 (number line)** is a fundamental mathematical concept and visual tool that geometrically represents numbers, serving as a bridge between algebra and geometry. 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By providing a clear visual framework, the number line aids in simplifying complex concepts, making it indispensable in both theoretical and practical applications. \n\nOverall, the \u6570\u8f74 unifies numerical and spatial reasoning, enhancing comprehension of mathematical principles across educational levels.", "entity_id": "\u6570\u8f74", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u6570\u8f74", "label": "\u6570\u8f74", "shape": "dot", "size": 10, "source_id": "chunk-7f35e11154694cb00fbc4308a33a36a3\u003cSEP\u003echunk-2fe44546e8e21363ac87fa4cc2c15616\u003cSEP\u003echunk-a1b0ea141dcc25762619a431f93aeca3\u003cSEP\u003echunk-10177ac2b1564cfa9472038f27fca84a\u003cSEP\u003echunk-56f75baf6ec4e55ce2eb93f9a0b24904", "title": "The **\u6570\u8f74 (number line)** is a fundamental mathematical concept and visual tool that geometrically represents numbers, serving as a bridge between algebra and geometry. In Chinese educational contexts, it is particularly emphasized for explaining negative numbers, absolute values, and the integration of numerical and geometric concepts. The number line\u2019s primary function is to illustrate the relationship between numbers and their positions, where the absolute value corresponds to the distance between points. \n\nInitially introduced in middle school curricula, the number line is extended to include negative numbers and absolute values, reinforcing its role in foundational mathematics. It is widely recognized as a key tool for understanding the abstract relationships between arithmetic operations and geometric interpretations. By providing a clear visual framework, the number line aids in simplifying complex concepts, making it indispensable in both theoretical and practical applications. \n\nOverall, the \u6570\u8f74 unifies numerical and spatial reasoning, enhancing comprehension of mathematical principles across educational levels."}, {"color": "#da25ae", "created_at": 1752884029, "description": "\u6570\u8f74\u548c\u7edd\u5bf9\u503c\u662f\u6570\u5f62\u7ed3\u5408\u7684\u53d1\u7aef\uff0c\u8868\u8fbe\u4e86\u6570\u91cf\u7684\u591a\u5c11\uff0c\u5e76\u5efa\u7acb\u4e86\u6570\u4e0e\u70b9\u7684\u5bf9\u5e94\u5173\u7cfb\u3002", "entity_id": "\u6570\u8f74\u548c\u7edd\u5bf9\u503c", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u6570\u8f74\u548c\u7edd\u5bf9\u503c", "label": "\u6570\u8f74\u548c\u7edd\u5bf9\u503c", "shape": "dot", "size": 10, "source_id": "chunk-5bc72352d3552516e17e5466f1f1fa7d", "title": "\u6570\u8f74\u548c\u7edd\u5bf9\u503c\u662f\u6570\u5f62\u7ed3\u5408\u7684\u53d1\u7aef\uff0c\u8868\u8fbe\u4e86\u6570\u91cf\u7684\u591a\u5c11\uff0c\u5e76\u5efa\u7acb\u4e86\u6570\u4e0e\u70b9\u7684\u5bf9\u5e94\u5173\u7cfb\u3002"}, {"color": "#b08a0a", "created_at": 1752884044, "description": "Weierstrass was a German mathematician who introduced the absolute value symbol in 1841 to represent the magnitude of complex numbers.\u003cSEP\u003e\u9b4f\u5c14\u65af\u7279\u62c9\u65afis a mathematician being commemorated in a conference where Hilbert delivered his famous speech \u0027On the Infinite\u0027.", "entity_id": "\u9b4f\u5c14\u65af\u7279\u62c9\u65af", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u9b4f\u5c14\u65af\u7279\u62c9\u65af", "label": "\u9b4f\u5c14\u65af\u7279\u62c9\u65af", "shape": "dot", "size": 10, "source_id": "chunk-91ed382127f07f8a06f35f0df0b6480f\u003cSEP\u003echunk-a1b0ea141dcc25762619a431f93aeca3", "title": "Weierstrass was a German mathematician who introduced the absolute value symbol in 1841 to represent the magnitude of complex numbers.\u003cSEP\u003e\u9b4f\u5c14\u65af\u7279\u62c9\u65afis a mathematician being commemorated in a conference where Hilbert delivered his famous speech \u0027On the Infinite\u0027."}, {"color": "#b5e537", "created_at": 1752884044, "description": "The Cartesian coordinate system extends the idea of the number line to two dimensions, enabling the graphical representation of functions and relationships.", "entity_id": "\u7b1b\u5361\u5c14\u76f4\u89d2\u5750\u6807\u7cfb", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u7b1b\u5361\u5c14\u76f4\u89d2\u5750\u6807\u7cfb", "label": "\u7b1b\u5361\u5c14\u76f4\u89d2\u5750\u6807\u7cfb", "shape": "dot", "size": 10, "source_id": "chunk-a1b0ea141dcc25762619a431f93aeca3", "title": "The Cartesian coordinate system extends the idea of the number line to two dimensions, enabling the graphical representation of functions and relationships."}, {"color": "#1236db", "created_at": 1752884031, "description": "**\u51e0\u4f55\u5b66 (Geometry)** is a branch of mathematics primarily concerned with the properties, relations, and measurements of points, lines, surfaces, solids, and spatial configurations. It deals with the study of shapes, sizes, and the logical structure of space, often introduced formally during middle school education. \n\nOriginating in **ancient Egypt**, geometry emerged from practical needs such as land measurement, particularly due to the annual flooding of the Nile River, which necessitated precise boundary demarcation. Over time, it evolved from an intuitive, applied discipline into a formalized, **axiomatic system**, heavily influenced by **ancient Greek scholars**, who regarded it as more logically rigorous than algebra. \n\nGeometry encompasses both theoretical and practical aspects, examining spatial relationships and transformations while serving as a foundational pillar in mathematics. Its development reflects a progression from empirical methods to structured deductive reasoning, shaping fields ranging from architecture to modern physics.", "entity_id": "\u51e0\u4f55\u5b66", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u51e0\u4f55\u5b66", "label": "\u51e0\u4f55\u5b66", "shape": "dot", "size": 10, "source_id": "chunk-54a2947dce8bb8651e7af1f86e1e89a5\u003cSEP\u003echunk-a1b0ea141dcc25762619a431f93aeca3\u003cSEP\u003echunk-dda7ba554b64773a9f739a19965188f1\u003cSEP\u003echunk-2ff43cd1d769194656997fc3e9797a67\u003cSEP\u003echunk-c30b1193cd236ff32c7a33020186a433", "title": "**\u51e0\u4f55\u5b66 (Geometry)** is a branch of mathematics primarily concerned with the properties, relations, and measurements of points, lines, surfaces, solids, and spatial configurations. It deals with the study of shapes, sizes, and the logical structure of space, often introduced formally during middle school education. \n\nOriginating in **ancient Egypt**, geometry emerged from practical needs such as land measurement, particularly due to the annual flooding of the Nile River, which necessitated precise boundary demarcation. Over time, it evolved from an intuitive, applied discipline into a formalized, **axiomatic system**, heavily influenced by **ancient Greek scholars**, who regarded it as more logically rigorous than algebra. \n\nGeometry encompasses both theoretical and practical aspects, examining spatial relationships and transformations while serving as a foundational pillar in mathematics. Its development reflects a progression from empirical methods to structured deductive reasoning, shaping fields ranging from architecture to modern physics."}, {"color": "#d87067", "created_at": 1752884028, "description": "The **absolute value** (\u7edd\u5bf9\u503c) is a fundamental mathematical concept that represents the magnitude or distance of a real or complex number from zero. It is defined as the square root of the sum of the squares of its real and imaginary parts in the case of complex numbers. For real numbers, the absolute value signifies the non-negative distance of a number from zero on the number line, thereby bridging numerical and geometric interpretations. \n\nIntroduced in middle school mathematics, the absolute value is essential for understanding key properties of numbers, including their magnitude and positional relationships. It plays a crucial role in quantity representation and serves as the foundation for connecting algebraic and geometric concepts. Overall, the absolute value is a versatile and widely applicable mathematical tool used to analyze distance, magnitude, and numerical characteristics.", "entity_id": "\u7edd\u5bf9\u503c", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u7edd\u5bf9\u503c", "label": "\u7edd\u5bf9\u503c", "shape": "dot", "size": 10, "source_id": "chunk-2fe44546e8e21363ac87fa4cc2c15616\u003cSEP\u003echunk-7f35e11154694cb00fbc4308a33a36a3\u003cSEP\u003echunk-a1b0ea141dcc25762619a431f93aeca3\u003cSEP\u003echunk-10177ac2b1564cfa9472038f27fca84a", "title": "The **absolute value** (\u7edd\u5bf9\u503c) is a fundamental mathematical concept that represents the magnitude or distance of a real or complex number from zero. It is defined as the square root of the sum of the squares of its real and imaginary parts in the case of complex numbers. For real numbers, the absolute value signifies the non-negative distance of a number from zero on the number line, thereby bridging numerical and geometric interpretations. \n\nIntroduced in middle school mathematics, the absolute value is essential for understanding key properties of numbers, including their magnitude and positional relationships. It plays a crucial role in quantity representation and serves as the foundation for connecting algebraic and geometric concepts. Overall, the absolute value is a versatile and widely applicable mathematical tool used to analyze distance, magnitude, and numerical characteristics."}, {"color": "#ec3acb", "created_at": 1752884080, "description": "The **\u76f4\u89d2\u5750\u6807\u7cfb (Rectangular Coordinate System)**, also known as the **Cartesian Coordinate System**, is a fundamental method for expressing geometric coordinates in mathematics. Invented by **Descartes (\u7b1b\u5361\u5c14)**, it serves as the foundation of **analytic geometry**, bridging algebra and geometry by transforming geometric problems into algebraic ones. This system is widely used and is one of the most common coordinate systems in mathematics. It can be interconverted with polar coordinates through transformation formulas, allowing for flexible applications in different mathematical contexts. \n\nIn summary, the **\u76f4\u89d2\u5750\u6807\u7cfb** is a key mathematical framework developed by Descartes, enabling the algebraic representation of geometric shapes and providing a versatile tool for coordinate transformations.", "entity_id": "\u76f4\u89d2\u5750\u6807\u7cfb", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u76f4\u89d2\u5750\u6807\u7cfb", "label": "\u76f4\u89d2\u5750\u6807\u7cfb", "shape": "dot", "size": 10, "source_id": "chunk-40574dd690ebc162e391ed721f31bfae\u003cSEP\u003echunk-3005fbbe56792da15c698f942a4b7305\u003cSEP\u003echunk-69ea16a9bf8f7637a55bade939083e96\u003cSEP\u003echunk-33e4d77aeae0267e846e8c7ab9af2b41", "title": "The **\u76f4\u89d2\u5750\u6807\u7cfb (Rectangular Coordinate System)**, also known as the **Cartesian Coordinate System**, is a fundamental method for expressing geometric coordinates in mathematics. Invented by **Descartes (\u7b1b\u5361\u5c14)**, it serves as the foundation of **analytic geometry**, bridging algebra and geometry by transforming geometric problems into algebraic ones. This system is widely used and is one of the most common coordinate systems in mathematics. It can be interconverted with polar coordinates through transformation formulas, allowing for flexible applications in different mathematical contexts. \n\nIn summary, the **\u76f4\u89d2\u5750\u6807\u7cfb** is a key mathematical framework developed by Descartes, enabling the algebraic representation of geometric shapes and providing a versatile tool for coordinate transformations."}, {"color": "#7a9d38", "created_at": 1752884037, "description": "Positive numbers are numbers greater than zero, representing quantities or magnitudes.", "entity_id": "\u6b63\u6570", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u6b63\u6570", "label": "\u6b63\u6570", "shape": "dot", "size": 10, "source_id": "chunk-c38ffe600259aeb39e4227f260864859", "title": "Positive numbers are numbers greater than zero, representing quantities or magnitudes."}, {"color": "#b956b1", "created_at": 1752884038, "description": "**\u300a\u4e5d\u7ae0\u7b97\u672f\u300b** (also known as *The Nine Chapters on the Mathematical Art* or *Jiuzhang Suanshu*) is an ancient Chinese mathematical text and a foundational work from the Han Dynasty. It is one of the most significant mathematical treatises in Chinese history, representing the high-level achievements of ancient Chinese mathematics. \n\nThe text systematically documents a wide range of mathematical problems and their solutions, including early fractional representations of division, the naming of decimal places for numbers less than one, and the application of negative numbers. It also introduces methods for solving systems of equations. Additionally, the *Jiuzhang Suanshu* discusses various units of measurement and provides detailed computational techniques. \n\nAs a comprehensive mathematical classic, it was the first to systematically propose and describe concepts such as fractions and negative numbers, along with their operational methods. Its contributions laid the groundwork for later developments in Chinese mathematics and remain a vital part of mathematical history.", "entity_id": "\u300a\u4e5d\u7ae0\u7b97\u672f\u300b", "entity_type": "organization", "file_path": "ChuZhongShuXue.docx", "id": "\u300a\u4e5d\u7ae0\u7b97\u672f\u300b", "label": "\u300a\u4e5d\u7ae0\u7b97\u672f\u300b", "shape": "dot", "size": 10, "source_id": "chunk-f379ebb8f8dc44a54c681880caf478eb\u003cSEP\u003echunk-bc737ed018919d513c346a3c74b449aa\u003cSEP\u003echunk-7f35e11154694cb00fbc4308a33a36a3\u003cSEP\u003echunk-58d4c3c0d2ac14aff5d53121bb780c43\u003cSEP\u003echunk-6deef743946d62c39bdfa573f520aa57\u003cSEP\u003echunk-8536b48102e07677da2af463a726f4c6\u003cSEP\u003echunk-2820dfe8496edbd1e66a24c26cdd1859\u003cSEP\u003echunk-121f6aa86f389551192a9dd60f07ea42", "title": "**\u300a\u4e5d\u7ae0\u7b97\u672f\u300b** (also known as *The Nine Chapters on the Mathematical Art* or *Jiuzhang Suanshu*) is an ancient Chinese mathematical text and a foundational work from the Han Dynasty. It is one of the most significant mathematical treatises in Chinese history, representing the high-level achievements of ancient Chinese mathematics. \n\nThe text systematically documents a wide range of mathematical problems and their solutions, including early fractional representations of division, the naming of decimal places for numbers less than one, and the application of negative numbers. It also introduces methods for solving systems of equations. Additionally, the *Jiuzhang Suanshu* discusses various units of measurement and provides detailed computational techniques. \n\nAs a comprehensive mathematical classic, it was the first to systematically propose and describe concepts such as fractions and negative numbers, along with their operational methods. Its contributions laid the groundwork for later developments in Chinese mathematics and remain a vital part of mathematical history."}, {"color": "#7b6749", "created_at": 1752884040, "description": "\u5370\u5ea6\u5728\u516c\u5143\u4e03\u4e16\u7eaa\u5de6\u53f3\u63d0\u51fa\u4e86\u8d1f\u6570\u7684\u4e58\u9664\u8fd0\u7b97\u65b9\u6cd5\uff0c\u5bf9\u6570\u5b66\u53d1\u5c55\u6709\u91cd\u8981\u8d21\u732e\u3002\u003cSEP\u003e\u5728\u516c\u5143\u4e03\u4e16\u7eaa\u5de6\u53f3\u5b8c\u5584\u4e86\u8d1f\u6570\u7684\u4e58\u9664\u8fd0\u7b97\u65b9\u6cd5\u3002", "entity_id": "\u5370\u5ea6", "entity_type": "geo", "file_path": "ChuZhongShuXue.docx", "id": "\u5370\u5ea6", "label": "\u5370\u5ea6", "shape": "dot", "size": 10, "source_id": "chunk-f379ebb8f8dc44a54c681880caf478eb\u003cSEP\u003echunk-2820dfe8496edbd1e66a24c26cdd1859", "title": "\u5370\u5ea6\u5728\u516c\u5143\u4e03\u4e16\u7eaa\u5de6\u53f3\u63d0\u51fa\u4e86\u8d1f\u6570\u7684\u4e58\u9664\u8fd0\u7b97\u65b9\u6cd5\uff0c\u5bf9\u6570\u5b66\u53d1\u5c55\u6709\u91cd\u8981\u8d21\u732e\u3002\u003cSEP\u003e\u5728\u516c\u5143\u4e03\u4e16\u7eaa\u5de6\u53f3\u5b8c\u5584\u4e86\u8d1f\u6570\u7684\u4e58\u9664\u8fd0\u7b97\u65b9\u6cd5\u3002"}, {"color": "#f1e73b", "created_at": 1752884040, "description": "The arithmetic axiom system is a foundational framework used to establish natural numbers and extend to real numbers.\u003cSEP\u003e\u7b97\u672f\u516c\u7406\u4f53\u7cfb\u662f\u6570\u5b66\u4e2d\u7528\u4e8e\u5b9a\u4e49\u81ea\u7136\u6570\u548c\u52a0\u6cd5\u7b49\u57fa\u672c\u6982\u5ff5\u7684\u516c\u7406\u5316\u7cfb\u7edf\u3002\u003cSEP\u003e\u7b97\u672f\u516c\u7406\u4f53\u7cfb\u662f\u6570\u5b66\u7684\u57fa\u7840\uff0c\u5b9a\u4e49\u4e86\u81ea\u7136\u6570\u3001\u6574\u6570\u7b49\u6982\u5ff5\uff0c\u5e76\u901a\u8fc7\u8fd0\u7b97\u5b9e\u73b0\u6570\u57df\u7684\u6269\u5145\u3002", "entity_id": "\u7b97\u672f\u516c\u7406\u4f53\u7cfb", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u7b97\u672f\u516c\u7406\u4f53\u7cfb", "label": "\u7b97\u672f\u516c\u7406\u4f53\u7cfb", "shape": "dot", "size": 10, "source_id": "chunk-89a00ca5535ebdb5d721e473eebbac4c\u003cSEP\u003echunk-7719147afc84fa56038af71629c88a4b\u003cSEP\u003echunk-f379ebb8f8dc44a54c681880caf478eb", "title": "The arithmetic axiom system is a foundational framework used to establish natural numbers and extend to real numbers.\u003cSEP\u003e\u7b97\u672f\u516c\u7406\u4f53\u7cfb\u662f\u6570\u5b66\u4e2d\u7528\u4e8e\u5b9a\u4e49\u81ea\u7136\u6570\u548c\u52a0\u6cd5\u7b49\u57fa\u672c\u6982\u5ff5\u7684\u516c\u7406\u5316\u7cfb\u7edf\u3002\u003cSEP\u003e\u7b97\u672f\u516c\u7406\u4f53\u7cfb\u662f\u6570\u5b66\u7684\u57fa\u7840\uff0c\u5b9a\u4e49\u4e86\u81ea\u7136\u6570\u3001\u6574\u6570\u7b49\u6982\u5ff5\uff0c\u5e76\u901a\u8fc7\u8fd0\u7b97\u5b9e\u73b0\u6570\u57df\u7684\u6269\u5145\u3002"}, {"color": "#f4bbf6", "created_at": 1752884041, "description": "\u4e2d\u56fd\u5f00\u59cb\u7cfb\u7edf\u4f7f\u7528\u8d1f\u6570\u7684\u5386\u53f2\u65f6\u671f\u003cSEP\u003e\u6c49\u4ee3\u662f\u4e2d\u56fd\u5386\u53f2\u4e0a\u7684\u4e00\u4e2a\u671d\u4ee3\uff0c\u5218\u5fbd\u5728\u6b64\u65f6\u671f\u6ce8\u91ca\u4e86\u300a\u4e5d\u7ae0\u7b97\u672f\u300b\u3002", "entity_id": "\u6c49\u4ee3", "entity_type": "event", "file_path": "ChuZhongShuXue.docx", "id": "\u6c49\u4ee3", "label": "\u6c49\u4ee3", "shape": "dot", "size": 10, "source_id": "chunk-f379ebb8f8dc44a54c681880caf478eb\u003cSEP\u003echunk-bc737ed018919d513c346a3c74b449aa", "title": "\u4e2d\u56fd\u5f00\u59cb\u7cfb\u7edf\u4f7f\u7528\u8d1f\u6570\u7684\u5386\u53f2\u65f6\u671f\u003cSEP\u003e\u6c49\u4ee3\u662f\u4e2d\u56fd\u5386\u53f2\u4e0a\u7684\u4e00\u4e2a\u671d\u4ee3\uff0c\u5218\u5fbd\u5728\u6b64\u65f6\u671f\u6ce8\u91ca\u4e86\u300a\u4e5d\u7ae0\u7b97\u672f\u300b\u3002"}, {"color": "#ceb665", "created_at": 1752884037, "description": "Ancient Greece (\u53e4\u5e0c\u814a) was a historical civilization renowned for its profound contributions to mathematics, philosophy, and intellectual thought. Geographically and culturally distinct, it served as the origin of many foundational mathematical discoveries, including the works of Archimedes and the systematization of geometric knowledge into formal theories. The Pythagorean school, emerging from this civilization, made significant strides in mathematics and the study of musical intervals. \n\nAncient Greece was also the cultural and intellectual context where Aristotle developed his logical frameworks, emphasizing rigorous reasoning and debate. The civilization is celebrated for its vibrant discourse culture and the establishment of democratic institutions, which deeply influenced Western philosophy and political thought. \n\nWhile Ancient Greece excelled in logical reasoning and formalized many mathematical concepts, it did not develop the notion of negative numbers. Its legacy remains distinct from other ancient civilizations, marked by unparalleled advancements in both theoretical and applied knowledge.", "entity_id": "\u53e4\u5e0c\u814a", "entity_type": "geo", "file_path": "ChuZhongShuXue.docx", "id": "\u53e4\u5e0c\u814a", "label": "\u53e4\u5e0c\u814a", "shape": "dot", "size": 10, "source_id": "chunk-f379ebb8f8dc44a54c681880caf478eb\u003cSEP\u003echunk-8f22eaff2e18e2af1504b2368ea1e65f\u003cSEP\u003echunk-cd72bf9b79aac9061696e480b2f809fb\u003cSEP\u003echunk-6919e0c9a75148e199557cc15db8406d\u003cSEP\u003echunk-fa7a7a68a2388eb9ade79559ac5334d0\u003cSEP\u003echunk-c38ffe600259aeb39e4227f260864859\u003cSEP\u003echunk-75f67b5b9db86524b3fadd3f9f5a2715", "title": "Ancient Greece (\u53e4\u5e0c\u814a) was a historical civilization renowned for its profound contributions to mathematics, philosophy, and intellectual thought. Geographically and culturally distinct, it served as the origin of many foundational mathematical discoveries, including the works of Archimedes and the systematization of geometric knowledge into formal theories. The Pythagorean school, emerging from this civilization, made significant strides in mathematics and the study of musical intervals. \n\nAncient Greece was also the cultural and intellectual context where Aristotle developed his logical frameworks, emphasizing rigorous reasoning and debate. The civilization is celebrated for its vibrant discourse culture and the establishment of democratic institutions, which deeply influenced Western philosophy and political thought. \n\nWhile Ancient Greece excelled in logical reasoning and formalized many mathematical concepts, it did not develop the notion of negative numbers. Its legacy remains distinct from other ancient civilizations, marked by unparalleled advancements in both theoretical and applied knowledge."}, {"color": "#371d78", "created_at": 1752884096, "description": "\u6570\u5b66\u5206\u6790(Mathematical Analysis) includes branches like real analysis and probability theory, built on measure theory.", "entity_id": "\u6570\u5b66\u5206\u6790", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u6570\u5b66\u5206\u6790", "label": "\u6570\u5b66\u5206\u6790", "shape": "dot", "size": 10, "source_id": "chunk-1725da37a2e4d06bc7e57390451926d7", "title": "\u6570\u5b66\u5206\u6790(Mathematical Analysis) includes branches like real analysis and probability theory, built on measure theory."}, {"color": "#8bc3ac", "created_at": 1752884033, "description": "**\u5b9e\u6570 (Real Numbers)** \n\n\u5b9e\u6570\u662f\u4e00\u4e2a\u6570\u5b66\u6982\u5ff5\uff0c\u4ee3\u8868\u6240\u6709\u5b9e\u6570\u7684\u96c6\u5408\uff0c\u5728\u6570\u5b66\u63a8\u7406\u4e2d\u5e38\u7528\u4e8e\u8ba8\u8bba\u6027\u8d28\u3001\u6bd4\u8f83\u4ee5\u53ca\u547d\u9898\u793a\u4f8b\u3002\u5b9e\u6570\u6784\u6210\u4e00\u4e2a\u8fde\u7eed\u7edf\uff0c\u5e76\u901a\u8fc7\u5eb7\u6258\u5c14\u7684\u5bf9\u89d2\u7ebf\u8bba\u8bc1\u6cd5\u88ab\u8bc1\u660e\u4e3a\u4e0d\u53ef\u6570\u65e0\u9650\u96c6\u3002\u5b83\u4eec\u5305\u62ec\u6709\u7406\u6570\uff08\u5982\u5206\u6570\u548c\u6574\u6570\uff09\u548c\u65e0\u7406\u6570\uff08\u5982\u221a2\u6216\u03c0\uff09\uff0c\u5171\u540c\u6784\u6210\u4e00\u4e2a\u5b8c\u5907\u7684\u6709\u5e8f\u57df\uff08complete ordered field\uff09\uff0c\u8fd9\u610f\u5473\u7740\u5b9e\u6570\u96c6\u5728\u6781\u9650\u8fd0\u7b97\u4e0b\u5c01\u95ed\u4e14\u6ee1\u8db3\u5168\u5e8f\u5173\u7cfb\u3002 \n\n\u5b9e\u6570\u7684\u7edd\u5bf9\u503c\u5b9a\u4e49\u4e3a\uff1a\u5bf9\u4e8e\u975e\u8d1f\u6570\uff0c\u7edd\u5bf9\u503c\u662f\u5176\u672c\u8eab\uff1b\u5bf9\u4e8e\u8d1f\u6570\uff0c\u7edd\u5bf9\u503c\u662f\u5176\u76f8\u53cd\u6570\u3002\u5b9e\u6570\u53ef\u4ee5\u901a\u8fc7\u57fa\u672c\u5e8f\u5217\uff08\u67ef\u897f\u5e8f\u5217\uff09\u6765\u4e25\u683c\u5b9a\u4e49\uff0c\u5e76\u7531\u6b64\u6269\u5c55\u6709\u7406\u6570\u7684\u8fd0\u7b97\u89c4\u5219\u3002\u6b64\u5916\uff0c\u5b9e\u6570\u7684\u5c0f\u6570\u5f62\u5f0f\uff08\u5305\u62ec\u6709\u9650\u5c0f\u6570\u548c\u65e0\u9650\u5c0f\u6570\uff09\u662f\u5176\u5341\u8fdb\u5236\u8868\u8fbe\u7684\u57fa\u7840\uff0c\u8fdb\u4e00\u6b65\u4f53\u73b0\u4e86\u5b9e\u6570\u7684\u8fde\u7eed\u6027\u548c\u5e7f\u6cdb\u9002\u7528\u6027\u3002 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\n\n\u5b9e\u6570\u662f\u4e00\u4e2a\u6570\u5b66\u6982\u5ff5\uff0c\u4ee3\u8868\u6240\u6709\u5b9e\u6570\u7684\u96c6\u5408\uff0c\u5728\u6570\u5b66\u63a8\u7406\u4e2d\u5e38\u7528\u4e8e\u8ba8\u8bba\u6027\u8d28\u3001\u6bd4\u8f83\u4ee5\u53ca\u547d\u9898\u793a\u4f8b\u3002\u5b9e\u6570\u6784\u6210\u4e00\u4e2a\u8fde\u7eed\u7edf\uff0c\u5e76\u901a\u8fc7\u5eb7\u6258\u5c14\u7684\u5bf9\u89d2\u7ebf\u8bba\u8bc1\u6cd5\u88ab\u8bc1\u660e\u4e3a\u4e0d\u53ef\u6570\u65e0\u9650\u96c6\u3002\u5b83\u4eec\u5305\u62ec\u6709\u7406\u6570\uff08\u5982\u5206\u6570\u548c\u6574\u6570\uff09\u548c\u65e0\u7406\u6570\uff08\u5982\u221a2\u6216\u03c0\uff09\uff0c\u5171\u540c\u6784\u6210\u4e00\u4e2a\u5b8c\u5907\u7684\u6709\u5e8f\u57df\uff08complete ordered field\uff09\uff0c\u8fd9\u610f\u5473\u7740\u5b9e\u6570\u96c6\u5728\u6781\u9650\u8fd0\u7b97\u4e0b\u5c01\u95ed\u4e14\u6ee1\u8db3\u5168\u5e8f\u5173\u7cfb\u3002 \n\n\u5b9e\u6570\u7684\u7edd\u5bf9\u503c\u5b9a\u4e49\u4e3a\uff1a\u5bf9\u4e8e\u975e\u8d1f\u6570\uff0c\u7edd\u5bf9\u503c\u662f\u5176\u672c\u8eab\uff1b\u5bf9\u4e8e\u8d1f\u6570\uff0c\u7edd\u5bf9\u503c\u662f\u5176\u76f8\u53cd\u6570\u3002\u5b9e\u6570\u53ef\u4ee5\u901a\u8fc7\u57fa\u672c\u5e8f\u5217\uff08\u67ef\u897f\u5e8f\u5217\uff09\u6765\u4e25\u683c\u5b9a\u4e49\uff0c\u5e76\u7531\u6b64\u6269\u5c55\u6709\u7406\u6570\u7684\u8fd0\u7b97\u89c4\u5219\u3002\u6b64\u5916\uff0c\u5b9e\u6570\u7684\u5c0f\u6570\u5f62\u5f0f\uff08\u5305\u62ec\u6709\u9650\u5c0f\u6570\u548c\u65e0\u9650\u5c0f\u6570\uff09\u662f\u5176\u5341\u8fdb\u5236\u8868\u8fbe\u7684\u57fa\u7840\uff0c\u8fdb\u4e00\u6b65\u4f53\u73b0\u4e86\u5b9e\u6570\u7684\u8fde\u7eed\u6027\u548c\u5e7f\u6cdb\u9002\u7528\u6027\u3002 \n\n\u7efc\u4e0a\uff0c\u5b9e\u6570\u5728\u6570\u5b66\u5206\u6790\u3001\u903b\u8f91\u63a8\u7406\u4ee5\u53ca\u5b9e\u9645\u95ee\u9898\u5efa\u6a21\u4e2d\u626e\u6f14\u6838\u5fc3\u89d2\u8272\uff0c\u517c\u5177\u62bd\u8c61\u6027\u4e0e\u5b9e\u7528\u6027\u3002"}, {"color": "#52da8c", "created_at": 1752884029, "description": "**\u65b9\u7a0b (Equations)** \n\n\u65b9\u7a0b are mathematical statements that assert the equality of two algebraic expressions, often containing one or more unknown variables. They serve as fundamental tools for modeling real-world problems across various disciplines, allowing the representation and analysis of relationships between quantities. \n\nIn mathematics education, equations are introduced early, particularly in primary school, where students learn their nature and basic solving methods. They form a core concept in algebra, where their solutions help determine unknown values by manipulating variables within the given relationships. \n\nWhile traditionally defined as an equality containing unknowns, some critiques argue that this definition lacks precision. Equations refine quantitative relationships by distinguishing between general and specific cases, as well as exploring methods for formulating them (e.g., setting up equations from word problems). \n\nOverall, equations function both as abstract mathematical patterns and practical problem-solving tools, bridging theoretical concepts with real-world applications.", "entity_id": "\u65b9\u7a0b", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u65b9\u7a0b", "label": "\u65b9\u7a0b", "shape": "dot", "size": 10, "source_id": "chunk-5bc72352d3552516e17e5466f1f1fa7d\u003cSEP\u003echunk-7f35e11154694cb00fbc4308a33a36a3\u003cSEP\u003echunk-bdb2d088364dc5e365dcf7c67ff1476e\u003cSEP\u003echunk-3036fe5f8bcbc7479b162f62dea08660\u003cSEP\u003echunk-0e47637ef5cc5c08729008ec09e12dd7\u003cSEP\u003echunk-1ea9636f6fb64b0198c39b6185e32379\u003cSEP\u003echunk-f1e81718ff3ad9793c932f0681b6cfda\u003cSEP\u003echunk-56f75baf6ec4e55ce2eb93f9a0b24904\u003cSEP\u003echunk-afc841e1a6c397cedfaba45e3a346fb6", "title": "**\u65b9\u7a0b (Equations)** \n\n\u65b9\u7a0b are mathematical statements that assert the equality of two algebraic expressions, often containing one or more unknown variables. They serve as fundamental tools for modeling real-world problems across various disciplines, allowing the representation and analysis of relationships between quantities. \n\nIn mathematics education, equations are introduced early, particularly in primary school, where students learn their nature and basic solving methods. They form a core concept in algebra, where their solutions help determine unknown values by manipulating variables within the given relationships. \n\nWhile traditionally defined as an equality containing unknowns, some critiques argue that this definition lacks precision. Equations refine quantitative relationships by distinguishing between general and specific cases, as well as exploring methods for formulating them (e.g., setting up equations from word problems). \n\nOverall, equations function both as abstract mathematical patterns and practical problem-solving tools, bridging theoretical concepts with real-world applications."}, {"color": "#4a5a94", "created_at": 1752884134, "description": "\u83b1\u5e03\u5c3c\u8328\u521b\u9020\u4e86\u51fd\u6570\u4e00\u8bcd\u5e76\u7cfb\u7edf\u9610\u8ff0\u4e86\u5fae\u79ef\u5206\u7406\u8bba\uff0c\u5960\u5b9a\u4e86\u73b0\u4ee3\u5fae\u79ef\u5206\u7684\u57fa\u7840\u3002", "entity_id": "\u83b1\u5e03\u5c3c\u5179", "entity_type": "UNKNOWN", "file_path": "ChuZhongShuXue.docx", "id": "\u83b1\u5e03\u5c3c\u5179", "label": "\u83b1\u5e03\u5c3c\u5179", "shape": "dot", "size": 10, "source_id": "chunk-401744b2d59735d588f0fc780ae91164", "title": "\u83b1\u5e03\u5c3c\u8328\u521b\u9020\u4e86\u51fd\u6570\u4e00\u8bcd\u5e76\u7cfb\u7edf\u9610\u8ff0\u4e86\u5fae\u79ef\u5206\u7406\u8bba\uff0c\u5960\u5b9a\u4e86\u73b0\u4ee3\u5fae\u79ef\u5206\u7684\u57fa\u7840\u3002"}, {"color": "#d4e6f2", "created_at": 1752884031, "description": "**\u5fae\u79ef\u5206 (Calculus)** is a branch of mathematics that emerged in the 17th century, primarily developed by **Isaac Newton** and **Gottfried Wilhelm Leibniz**, though its conceptual foundations can be traced back to earlier work, such as **Galileo\u0027s** studies on motion and changing quantities. The invention of calculus was influenced by **Descartes\u0027** ideas on analytic geometry. \n\nAs a powerful mathematical tool, calculus revolutionized the study of continuous change and accumulation, enabling advancements in physics, engineering, and other sciences. However, its early theoretical foundations were not rigorously defined, leading mathematicians to reconsider fundamental mathematical concepts and objects. Notably, calculus predated the formal development of rigorous number systems, highlighting its groundbreaking yet initially intuitive nature. \n\nIn summary, calculus is a transformative mathematical discipline with profound practical applications, shaped by historical contributions and later refined through deeper theoretical scrutiny.", "entity_id": "\u5fae\u79ef\u5206", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u5fae\u79ef\u5206", "label": "\u5fae\u79ef\u5206", "shape": "dot", "size": 10, "source_id": "chunk-4dc0d317d51b12cedc9804d89a6f0d99\u003cSEP\u003echunk-a77717ea766ba1464bcbadc22f73914d\u003cSEP\u003echunk-eb8a0335171966e7a8bbe856ead3efd0\u003cSEP\u003echunk-2ff43cd1d769194656997fc3e9797a67\u003cSEP\u003echunk-40574dd690ebc162e391ed721f31bfae", "title": "**\u5fae\u79ef\u5206 (Calculus)** is a branch of mathematics that emerged in the 17th century, primarily developed by **Isaac Newton** and **Gottfried Wilhelm Leibniz**, though its conceptual foundations can be traced back to earlier work, such as **Galileo\u0027s** studies on motion and changing quantities. The invention of calculus was influenced by **Descartes\u0027** ideas on analytic geometry. \n\nAs a powerful mathematical tool, calculus revolutionized the study of continuous change and accumulation, enabling advancements in physics, engineering, and other sciences. However, its early theoretical foundations were not rigorously defined, leading mathematicians to reconsider fundamental mathematical concepts and objects. 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"chunk-035c55452674b5ec27f1692bfa495347\u003cSEP\u003echunk-ca18454792909d12d2509708190d0e85", "title": "\u53d8\u91cfrefers to variables in mathematics, particularly in the context of function definitions in junior high school.\u003cSEP\u003e\u53d8\u91cf\u662f\u6570\u5b66\u6a21\u578b\u4e2d\u7528\u4e8e\u8868\u793a\u73b0\u5b9e\u4e16\u754c\u4e2d\u53ef\u91cf\u5316\u56e0\u7d20\u7684\u57fa\u672c\u5143\u7d20\uff0c\u901a\u8fc7\u6570\u5b66\u8bed\u8a00\u8868\u8fbe\u5176\u5173\u7cfb\u3002"}, {"color": "#ecfc9a", "created_at": 1752884043, "description": "\u5bf9\u5e94\u5173\u7cfbrefers to the correspondence or mapping between elements in function definitions, emphasized in senior high school mathematics.", "entity_id": "\u5bf9\u5e94\u5173\u7cfb", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u5bf9\u5e94\u5173\u7cfb", "label": "\u5bf9\u5e94\u5173\u7cfb", "shape": "dot", "size": 10, "source_id": "chunk-035c55452674b5ec27f1692bfa495347", "title": "\u5bf9\u5e94\u5173\u7cfbrefers to the correspondence or mapping between elements in function definitions, emphasized in senior high school mathematics."}, {"color": "#921029", "created_at": 1752884043, "description": "The Bourbaki School is a group of mathematicians known for their formal and abstract approach to mathematical concepts, including functions.", "entity_id": "Bourbaki School", "entity_type": "organization", "file_path": "ChuZhongShuXue.docx", "id": "Bourbaki School", "label": "Bourbaki School", "shape": "dot", "size": 10, "source_id": "chunk-035c55452674b5ec27f1692bfa495347", "title": "The Bourbaki School is a group of mathematicians known for their formal and abstract approach to mathematical concepts, including functions."}, {"color": "#cba3c2", "created_at": 1752884096, "description": "Mathematical patterns are the results of mathematical research, representing the outcomes and models derived from abstract structures.\u003cSEP\u003e\u6570\u5b66\u6a21\u5f0f(Mathematical Patterns) are generalized mathematical constructs used to understand and express problems, bridging internal mathematical concepts and external real-world applications.\u003cSEP\u003e\u6570\u5b66\u6a21\u5f0f(Mathematical Patterns) refers to the study of patterns in data, structures, and abstract relationships, central to modern math.", "entity_id": "\u6570\u5b66\u6a21\u5f0f", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u6570\u5b66\u6a21\u5f0f", "label": "\u6570\u5b66\u6a21\u5f0f", "shape": "dot", "size": 10, "source_id": "chunk-1d390244ef79debb74dcf77a05952ebc\u003cSEP\u003echunk-bdb2d088364dc5e365dcf7c67ff1476e\u003cSEP\u003echunk-1725da37a2e4d06bc7e57390451926d7", "title": "Mathematical patterns are the results of mathematical research, representing the outcomes and models derived from abstract structures.\u003cSEP\u003e\u6570\u5b66\u6a21\u5f0f(Mathematical Patterns) are generalized mathematical constructs used to understand and express problems, bridging internal mathematical concepts and external real-world applications.\u003cSEP\u003e\u6570\u5b66\u6a21\u5f0f(Mathematical Patterns) refers to the study of patterns in data, structures, and abstract relationships, central to modern math."}, {"color": "#f86708", "created_at": 1752884034, "description": "John Playfair was a British geologist and mathematician who provided a classic reformulation of the parallel postulate.\u003cSEP\u003e\u666e\u83b1\u8d39\u5c14\u662f16\u4e16\u7eaa\u82f1\u56fd\u5730\u8d28\u5b66\u5bb6\u548c\u6570\u5b66\u5bb6\uff0c\u4fee\u6539\u4e86\u6b27\u51e0\u91cc\u5f97\u51e0\u4f55\u7684\u7b2c\u4e94\u516c\u8bbe\u3002", "entity_id": "\u666e\u83b1\u8d39\u5c14", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u666e\u83b1\u8d39\u5c14", "label": "\u666e\u83b1\u8d39\u5c14", "shape": "dot", "size": 10, "source_id": "chunk-e91dee450ab50cb1a44f08fe171c809b\u003cSEP\u003echunk-6782332859e0bdef19c72a7367743352", "title": "John Playfair was a British geologist and mathematician who provided a classic reformulation of the parallel postulate.\u003cSEP\u003e\u666e\u83b1\u8d39\u5c14\u662f16\u4e16\u7eaa\u82f1\u56fd\u5730\u8d28\u5b66\u5bb6\u548c\u6570\u5b66\u5bb6\uff0c\u4fee\u6539\u4e86\u6b27\u51e0\u91cc\u5f97\u51e0\u4f55\u7684\u7b2c\u4e94\u516c\u8bbe\u3002"}, {"color": "#408a19", "created_at": 1752884033, "description": "**Lobachevskian Geometry (\u7f57\u5df4\u5207\u592b\u65af\u57fa\u51e0\u4f55)** is a non-Euclidean geometric system that challenges and modifies Euclid\u0027s parallel postulate. Unlike Euclidean geometry, which assumes only one parallel line can be drawn through a point not on a given line, Lobachevskian geometry allows for infinitely many parallel lines to exist under the same conditions. This system is characterized by its study of surfaces with constant negative Gaussian curvature, where geometric properties differ significantly from Euclidean norms\u2014for instance, the sum of angles in a triangle is always less than 180 degrees. \n\nLobachevskian geometry serves as a foundational framework for various mathematical and theoretical applications, including polar coordinate systems and geometric reasoning. It has practical implications in fields like astronomy, where it helps explain phenomena such as the apparent parallelism of rays from Polaris (the North Star) observed in the Northern Hemisphere. The geometry is frequently referenced in mathematical discourse as an example of alternative axiomatic systems and non-Euclidean reasoning. \n\nIn summary, Lobachevskian geometry is a pivotal non-Euclidean system that redefines parallelism, explores negatively curved spaces, and provides tools for both theoretical and observational disciplines.", "entity_id": "\u7f57\u5df4\u5207\u592b\u65af\u57fa\u51e0\u4f55", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u7f57\u5df4\u5207\u592b\u65af\u57fa\u51e0\u4f55", "label": "\u7f57\u5df4\u5207\u592b\u65af\u57fa\u51e0\u4f55", "shape": "dot", "size": 10, "source_id": "chunk-30c648659bc2a2999312fff54eb5d51a\u003cSEP\u003echunk-0f6c384b97d0bb5698ba21610de20e63\u003cSEP\u003echunk-e91dee450ab50cb1a44f08fe171c809b\u003cSEP\u003echunk-d600d36500fc1362f8eb34c6e841165e\u003cSEP\u003echunk-3005fbbe56792da15c698f942a4b7305\u003cSEP\u003echunk-6782332859e0bdef19c72a7367743352\u003cSEP\u003echunk-262593f034795826c0c15acbf0edb10f", "title": "**Lobachevskian Geometry (\u7f57\u5df4\u5207\u592b\u65af\u57fa\u51e0\u4f55)** is a non-Euclidean geometric system that challenges and modifies Euclid\u0027s parallel postulate. Unlike Euclidean geometry, which assumes only one parallel line can be drawn through a point not on a given line, Lobachevskian geometry allows for infinitely many parallel lines to exist under the same conditions. This system is characterized by its study of surfaces with constant negative Gaussian curvature, where geometric properties differ significantly from Euclidean norms\u2014for instance, the sum of angles in a triangle is always less than 180 degrees. \n\nLobachevskian geometry serves as a foundational framework for various mathematical and theoretical applications, including polar coordinate systems and geometric reasoning. It has practical implications in fields like astronomy, where it helps explain phenomena such as the apparent parallelism of rays from Polaris (the North Star) observed in the Northern Hemisphere. The geometry is frequently referenced in mathematical discourse as an example of alternative axiomatic systems and non-Euclidean reasoning. \n\nIn summary, Lobachevskian geometry is a pivotal non-Euclidean system that redefines parallelism, explores negatively curved spaces, and provides tools for both theoretical and observational disciplines."}, {"color": "#47158a", "created_at": 1752884034, "description": "**Riemannian Geometry (\u9ece\u66fc\u51e0\u4f55)** is a form of **non-Euclidean geometry** that modifies the traditional Euclidean parallel postulate, resulting in a geometric system where **no parallel lines exist**. Developed by Bernhard Riemann, it serves as a mathematical framework for **curved spaces**, particularly those with **positive Gaussian curvature**, such as spherical surfaces. Key characteristics include: \n- The shortest distance between points is measured along **great circle arcs (\u5927\u5706\u52a3\u5f27)**. \n- The sum of angles in a triangle **exceeds 180 degrees**, contrasting with Euclidean geometry. \n- It is foundational to **Einstein\u2019s theory of general relativity**, providing the mathematical tools to describe spacetime curvature. \n\nRiemannian geometry is often discussed alongside other non-Euclidean systems, emphasizing its role in expanding geometric concepts beyond flat (Euclidean) spaces. While descriptions vary slightly in phrasing, they consistently highlight its **rejection of parallel lines**, **spherical/curved space applications**, and **physics relevance**.", "entity_id": "\u9ece\u66fc\u51e0\u4f55", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u9ece\u66fc\u51e0\u4f55", "label": "\u9ece\u66fc\u51e0\u4f55", "shape": "dot", "size": 10, "source_id": "chunk-30c648659bc2a2999312fff54eb5d51a\u003cSEP\u003echunk-0f6c384b97d0bb5698ba21610de20e63\u003cSEP\u003echunk-1725da37a2e4d06bc7e57390451926d7\u003cSEP\u003echunk-e91dee450ab50cb1a44f08fe171c809b\u003cSEP\u003echunk-d600d36500fc1362f8eb34c6e841165e\u003cSEP\u003echunk-6782332859e0bdef19c72a7367743352\u003cSEP\u003echunk-40574dd690ebc162e391ed721f31bfae", "title": "**Riemannian Geometry (\u9ece\u66fc\u51e0\u4f55)** is a form of **non-Euclidean geometry** that modifies the traditional Euclidean parallel postulate, resulting in a geometric system where **no parallel lines exist**. Developed by Bernhard Riemann, it serves as a mathematical framework for **curved spaces**, particularly those with **positive Gaussian curvature**, such as spherical surfaces. Key characteristics include: \n- The shortest distance between points is measured along **great circle arcs (\u5927\u5706\u52a3\u5f27)**. \n- The sum of angles in a triangle **exceeds 180 degrees**, contrasting with Euclidean geometry. \n- It is foundational to **Einstein\u2019s theory of general relativity**, providing the mathematical tools to describe spacetime curvature. \n\nRiemannian geometry is often discussed alongside other non-Euclidean systems, emphasizing its role in expanding geometric concepts beyond flat (Euclidean) spaces. While descriptions vary slightly in phrasing, they consistently highlight its **rejection of parallel lines**, **spherical/curved space applications**, and **physics relevance**."}, {"color": "#beb3b3", "created_at": 1752884061, "description": "An isosceles triangle with less symmetry than regular polygons but more than general triangles.\u003cSEP\u003eIsosceles triangle properties used in Archimedes\u0027 trisection method demonstration.\u003cSEP\u003e\u7b49\u8170\u4e09\u89d2\u5f62\u662f\u5177\u6709\u4e24\u4e2a\u76f8\u7b49\u5e95\u89d2\u7684\u4e09\u89d2\u5f62\uff0c\u5176\u5e95\u89d2\u6027\u8d28\u53ef\u901a\u8fc7\u6f14\u7ece\u63a8\u7406\u5f97\u51fa\u3002", "entity_id": "\u7b49\u8170\u4e09\u89d2\u5f62", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u7b49\u8170\u4e09\u89d2\u5f62", "label": "\u7b49\u8170\u4e09\u89d2\u5f62", "shape": "dot", "size": 10, "source_id": "chunk-7719147afc84fa56038af71629c88a4b\u003cSEP\u003echunk-eb909fae946cb41ff187d26fae2118b2\u003cSEP\u003echunk-b90ad7d27f890f150949c43063702ff8", "title": "An isosceles triangle with less symmetry than regular polygons but more than general triangles.\u003cSEP\u003eIsosceles triangle properties used in Archimedes\u0027 trisection method demonstration.\u003cSEP\u003e\u7b49\u8170\u4e09\u89d2\u5f62\u662f\u5177\u6709\u4e24\u4e2a\u76f8\u7b49\u5e95\u89d2\u7684\u4e09\u89d2\u5f62\uff0c\u5176\u5e95\u89d2\u6027\u8d28\u53ef\u901a\u8fc7\u6f14\u7ece\u63a8\u7406\u5f97\u51fa\u3002"}, {"color": "#98fa6f", "created_at": 1752884030, "description": "**Mathematical Reasoning (\u6570\u5b66\u63a8\u7406)** \n\nMathematical reasoning is a core competency in junior high school mathematics education, encompassing the process of deriving new propositions from known ones through logical deduction. It typically involves conditional propositions, argumentative propositions, and concluding propositions, forming a structured chain of logical connections. Unlike axiomatic reasoning, mathematical reasoning emphasizes rigorous processes and includes algebraic reasoning based on foundational facts. This method cultivates critical thinking skills by systematically analyzing relationships between propositions, ensuring both precision and coherence in problem-solving. The discipline highlights the importance of logical deduction while distinguishing itself from other forms of reasoning by its structured, step-by-step approach. \n\n(Note: Contradictions were resolved by integrating distinct aspects\u2014such as educational relevance, formal logic components, and comparisons with axiomatic reasoning\u2014into a unified description.)", "entity_id": "\u6570\u5b66\u63a8\u7406", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u6570\u5b66\u63a8\u7406", "label": "\u6570\u5b66\u63a8\u7406", "shape": "dot", "size": 10, "source_id": "chunk-4cff6339e8a61331ff6a8ae3b6298088\u003cSEP\u003echunk-edaeb942f2d7bc9b4f269a55a5bdd99f\u003cSEP\u003echunk-a77717ea766ba1464bcbadc22f73914d\u003cSEP\u003echunk-665ebf4a9a719e0056c9e54525078072", "title": "**Mathematical Reasoning (\u6570\u5b66\u63a8\u7406)** \n\nMathematical reasoning is a core competency in junior high school mathematics education, encompassing the process of deriving new propositions from known ones through logical deduction. It typically involves conditional propositions, argumentative propositions, and concluding propositions, forming a structured chain of logical connections. Unlike axiomatic reasoning, mathematical reasoning emphasizes rigorous processes and includes algebraic reasoning based on foundational facts. This method cultivates critical thinking skills by systematically analyzing relationships between propositions, ensuring both precision and coherence in problem-solving. The discipline highlights the importance of logical deduction while distinguishing itself from other forms of reasoning by its structured, step-by-step approach. \n\n(Note: Contradictions were resolved by integrating distinct aspects\u2014such as educational relevance, formal logic components, and comparisons with axiomatic reasoning\u2014into a unified description.)"}, {"color": "#172372", "created_at": 1752884031, "description": "**David Hilbert (\u5e0c\u5c14\u4f2f\u7279)** was a renowned German mathematician and one of the greatest mathematicians of the 20th century, celebrated for his foundational contributions to geometry, logic, and the axiomatization of mathematics. \n\nHilbert redefined and expanded Euclid\u0027s geometric axioms in his seminal work *Foundations of Geometry*, creating a more abstract and rigorous system. He introduced the concept of *nominal definitions*, emphasizing that the relationships between mathematical objects are more important than their definitions themselves. His axiomatic approach contrasted with intuitive geometric methods and reinforced Euclid\u0027s interpretations of numerical ratios while advancing the formalization of mathematical axioms. \n\nBeyond geometry, Hilbert is famous for proposing *Hilbert\u0027s 23 Problems* in 1900, a set of influential unsolved mathematical challenges that shaped 20th-century mathematics. The first problem addressed the *continuum hypothesis*, while others included questions on transcendence and the shortest distance between two points. His *Hilbert\u0027s Hotel* thought experiment illustrated the logic of infinite sets, further demonstrating his deep engagement with foundational mathematical concepts. \n\nHilbert also advocated for the *formalization of mathematics* through his ambitious *Hilbert\u0027s Program*, which sought to establish mathematics on a complete and consistent logical foundation. His 1924 lectures and writings underscored his commitment to clarity and rigor in mathematical reasoning. \n\nIn summary, Hilbert was a pivotal figure in modern mathematics, advancing geometry, logic, and axiomatic systems while leaving a lasting legacy through his 23 problems and formalist philosophy.", "entity_id": "\u5e0c\u5c14\u4f2f\u7279", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u5e0c\u5c14\u4f2f\u7279", "label": "\u5e0c\u5c14\u4f2f\u7279", "shape": "dot", "size": 10, "source_id": "chunk-7f35e11154694cb00fbc4308a33a36a3\u003cSEP\u003echunk-8b6f8917e6704256d88d931fd2d99d07\u003cSEP\u003echunk-42220f52b4f10f3dd80cceb36ebbd3ce\u003cSEP\u003echunk-170459199033f3f9a45d2ef237310873\u003cSEP\u003echunk-e91dee450ab50cb1a44f08fe171c809b\u003cSEP\u003echunk-665ebf4a9a719e0056c9e54525078072\u003cSEP\u003echunk-0f6c384b97d0bb5698ba21610de20e63\u003cSEP\u003echunk-33e4d77aeae0267e846e8c7ab9af2b41\u003cSEP\u003echunk-4dc0d317d51b12cedc9804d89a6f0d99\u003cSEP\u003echunk-c8a15761aa2fa77855ceb30f7b7a730a\u003cSEP\u003echunk-c81153206158adf5aa1a438a4f6242ed\u003cSEP\u003echunk-5bfa7384dc42f0162f843b05c24e2ea8\u003cSEP\u003echunk-00f45528a4186020aca4f014c14769e9\u003cSEP\u003echunk-2ff43cd1d769194656997fc3e9797a67\u003cSEP\u003echunk-6deef743946d62c39bdfa573f520aa57\u003cSEP\u003echunk-91ed382127f07f8a06f35f0df0b6480f\u003cSEP\u003echunk-497a0e71f7c50523d9be03ab68a8d478\u003cSEP\u003echunk-8536b48102e07677da2af463a726f4c6\u003cSEP\u003echunk-afc841e1a6c397cedfaba45e3a346fb6", "title": "**David Hilbert (\u5e0c\u5c14\u4f2f\u7279)** was a renowned German mathematician and one of the greatest mathematicians of the 20th century, celebrated for his foundational contributions to geometry, logic, and the axiomatization of mathematics. \n\nHilbert redefined and expanded Euclid\u0027s geometric axioms in his seminal work *Foundations of Geometry*, creating a more abstract and rigorous system. He introduced the concept of *nominal definitions*, emphasizing that the relationships between mathematical objects are more important than their definitions themselves. His axiomatic approach contrasted with intuitive geometric methods and reinforced Euclid\u0027s interpretations of numerical ratios while advancing the formalization of mathematical axioms. \n\nBeyond geometry, Hilbert is famous for proposing *Hilbert\u0027s 23 Problems* in 1900, a set of influential unsolved mathematical challenges that shaped 20th-century mathematics. The first problem addressed the *continuum hypothesis*, while others included questions on transcendence and the shortest distance between two points. His *Hilbert\u0027s Hotel* thought experiment illustrated the logic of infinite sets, further demonstrating his deep engagement with foundational mathematical concepts. \n\nHilbert also advocated for the *formalization of mathematics* through his ambitious *Hilbert\u0027s Program*, which sought to establish mathematics on a complete and consistent logical foundation. His 1924 lectures and writings underscored his commitment to clarity and rigor in mathematical reasoning. \n\nIn summary, Hilbert was a pivotal figure in modern mathematics, advancing geometry, logic, and axiomatic systems while leaving a lasting legacy through his 23 problems and formalist philosophy."}, {"color": "#05a65b", "created_at": 1752884034, "description": "The parallel postulate in Euclid\u0027s \u0027Elements\u0027 was historically problematic, leading to non-Euclidean geometries.\u003cSEP\u003e\u6b27\u51e0\u91cc\u5f97\u7b2c\u4e94\u516c\u8bbe\uff0c\u56e0\u8868\u8ff0\u590d\u6742\u5f15\u53d1\u540e\u4e16\u6570\u5b66\u5bb6\u7814\u7a76\uff0c\u6700\u7ec8\u5bfc\u81f4\u975e\u6b27\u51e0\u4f55\u7684\u4ea7\u751f\u3002", "entity_id": "\u5e73\u884c\u7ebf\u516c\u8bbe", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u5e73\u884c\u7ebf\u516c\u8bbe", "label": "\u5e73\u884c\u7ebf\u516c\u8bbe", "shape": "dot", "size": 10, "source_id": "chunk-e91dee450ab50cb1a44f08fe171c809b\u003cSEP\u003echunk-665ebf4a9a719e0056c9e54525078072", "title": "The parallel postulate in Euclid\u0027s \u0027Elements\u0027 was historically problematic, leading to non-Euclidean geometries.\u003cSEP\u003e\u6b27\u51e0\u91cc\u5f97\u7b2c\u4e94\u516c\u8bbe\uff0c\u56e0\u8868\u8ff0\u590d\u6742\u5f15\u53d1\u540e\u4e16\u6570\u5b66\u5bb6\u7814\u7a76\uff0c\u6700\u7ec8\u5bfc\u81f4\u975e\u6b27\u51e0\u4f55\u7684\u4ea7\u751f\u3002"}, {"color": "#bf5990", "created_at": 1752884034, "description": "Middle school mathematics education incorporates foundational geometric concepts, including Euclid\u0027s postulates.\u003cSEP\u003e\u521d\u4e2d\u6570\u5b66\u6559\u5b66\u867d\u4e0d\u8981\u6c42\u6df1\u5165\u7406\u89e3\u5b9e\u6570\uff0c\u4f46\u6559\u5e08\u9700\u8981\u5f62\u5f0f\u4e0a\u8ba4\u8bc6\u5b9e\u6570\u3002", "entity_id": "\u521d\u4e2d\u6570\u5b66\u6559\u5b66", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u521d\u4e2d\u6570\u5b66\u6559\u5b66", "label": "\u521d\u4e2d\u6570\u5b66\u6559\u5b66", "shape": "dot", "size": 10, "source_id": "chunk-e91dee450ab50cb1a44f08fe171c809b\u003cSEP\u003echunk-8b6f8917e6704256d88d931fd2d99d07", "title": "Middle school mathematics education incorporates foundational geometric concepts, including Euclid\u0027s postulates.\u003cSEP\u003e\u521d\u4e2d\u6570\u5b66\u6559\u5b66\u867d\u4e0d\u8981\u6c42\u6df1\u5165\u7406\u89e3\u5b9e\u6570\uff0c\u4f46\u6559\u5e08\u9700\u8981\u5f62\u5f0f\u4e0a\u8ba4\u8bc6\u5b9e\u6570\u3002"}, {"color": "#7eb9d7", "created_at": 1752884046, "description": "Newtonian mechanics that can be mathematically described using geometric transformations.", "entity_id": "\u725b\u987f\u529b\u5b66", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u725b\u987f\u529b\u5b66", "label": "\u725b\u987f\u529b\u5b66", "shape": "dot", "size": 10, "source_id": "chunk-a48b8be14cbd9c4d38521dbf6a089e16", "title": "Newtonian mechanics that can be mathematically described using geometric transformations."}, {"color": "#3e0e00", "created_at": 1752884046, "description": "\u8be5\u516c\u7406\u5047\u8bbe\u5bf9\u4e8e\u4efb\u610f\u6bd4\u4f8b\u7684\u76f8\u4f3c\u591a\u8fb9\u5f62\u5b58\u5728\uff0c\u662f\u4e09\u89d2\u5f62\u76f8\u4f3c\u6027\u8bc1\u660e\u7684\u57fa\u7840\u3002", "entity_id": "\u76f8\u4f3c\u591a\u8fb9\u5f62\u5b58\u5728\u516c\u7406", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u76f8\u4f3c\u591a\u8fb9\u5f62\u5b58\u5728\u516c\u7406", "label": "\u76f8\u4f3c\u591a\u8fb9\u5f62\u5b58\u5728\u516c\u7406", "shape": "dot", "size": 10, "source_id": "chunk-d2d1fa534f69129f1d307d4162d5e6e4", "title": "\u8be5\u516c\u7406\u5047\u8bbe\u5bf9\u4e8e\u4efb\u610f\u6bd4\u4f8b\u7684\u76f8\u4f3c\u591a\u8fb9\u5f62\u5b58\u5728\uff0c\u662f\u4e09\u89d2\u5f62\u76f8\u4f3c\u6027\u8bc1\u660e\u7684\u57fa\u7840\u3002"}, {"color": "#9056cd", "created_at": 1752884030, "description": "The **Pythagorean School** (\u6bd5\u8fbe\u54e5\u62c9\u65af\u5b66\u6d3e) was an influential ancient Greek philosophical and mathematical school founded by Pythagoras. The school is renowned for its profound contributions to mathematics, philosophy, and the study of proportions. Central to their beliefs was the idea that natural numbers were the foundation of cosmic harmony, encapsulated in their doctrine that \"all things are number\" (\u4e07\u7269\u7686\u6570). \n\nThe Pythagoreans explored the relationship between numbers and proportions, extending their principles to music by linking integer ratios to musical harmony. They made significant advancements in geometry, including the classification of regular polyhedra (the five Platonic solids) and the development of deductive reasoning. Their work also laid the groundwork for number theory, particularly through the study of perfect numbers. \n\nA key discovery attributed to the Pythagorean School was the irrationality of the square root of 2, which challenged their earlier belief that all quantities could be expressed as natural numbers. Despite this contradiction, their reverence for numerical harmony remained central to their philosophy. \n\nIn summary, the Pythagorean School was a pioneering intellectual movement that bridged mathematics, philosophy, and music, leaving a lasting legacy in the study of geometry, number theory, and deductive reasoning.", "entity_id": "\u6bd5\u8fbe\u54e5\u62c9\u65af\u5b66\u6d3e", "entity_type": "organization", "file_path": "ChuZhongShuXue.docx", "id": "\u6bd5\u8fbe\u54e5\u62c9\u65af\u5b66\u6d3e", "label": "\u6bd5\u8fbe\u54e5\u62c9\u65af\u5b66\u6d3e", "shape": "dot", "size": 10, "source_id": "chunk-5bc72352d3552516e17e5466f1f1fa7d\u003cSEP\u003echunk-42e82319a2d63b215f82bba66d1e632d\u003cSEP\u003echunk-896c6ab0b0daee896907a3349e5c7811\u003cSEP\u003echunk-58d4c3c0d2ac14aff5d53121bb780c43\u003cSEP\u003echunk-54a2947dce8bb8651e7af1f86e1e89a5\u003cSEP\u003echunk-6919e0c9a75148e199557cc15db8406d\u003cSEP\u003echunk-933b19628500cf80c1916250b497e3f8\u003cSEP\u003echunk-6deef743946d62c39bdfa573f520aa57\u003cSEP\u003echunk-fa7a7a68a2388eb9ade79559ac5334d0\u003cSEP\u003echunk-75f67b5b9db86524b3fadd3f9f5a2715", "title": "The **Pythagorean School** (\u6bd5\u8fbe\u54e5\u62c9\u65af\u5b66\u6d3e) was an influential ancient Greek philosophical and mathematical school founded by Pythagoras. The school is renowned for its profound contributions to mathematics, philosophy, and the study of proportions. Central to their beliefs was the idea that natural numbers were the foundation of cosmic harmony, encapsulated in their doctrine that \"all things are number\" (\u4e07\u7269\u7686\u6570). \n\nThe Pythagoreans explored the relationship between numbers and proportions, extending their principles to music by linking integer ratios to musical harmony. They made significant advancements in geometry, including the classification of regular polyhedra (the five Platonic solids) and the development of deductive reasoning. Their work also laid the groundwork for number theory, particularly through the study of perfect numbers. \n\nA key discovery attributed to the Pythagorean School was the irrationality of the square root of 2, which challenged their earlier belief that all quantities could be expressed as natural numbers. Despite this contradiction, their reverence for numerical harmony remained central to their philosophy. \n\nIn summary, the Pythagorean School was a pioneering intellectual movement that bridged mathematics, philosophy, and music, leaving a lasting legacy in the study of geometry, number theory, and deductive reasoning."}, {"color": "#4c92bc", "created_at": 1752884049, "description": "\"Geometric transformations maintaining all distances and angles (isometries", "entity_id": "\u521a\u4f53\u53d8\u6362", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u521a\u4f53\u53d8\u6362", "label": "\u521a\u4f53\u53d8\u6362", "shape": "dot", "size": 10, "source_id": "chunk-d79a25c9ff12f28251aed2da9132d65e", "title": "\"Geometric transformations maintaining all distances and angles (isometries"}, {"color": "#fb04fd", "created_at": 1752884031, "description": "A **point** is a fundamental geometric concept that serves as a foundational element in mathematics. Defined by Euclid as \"having no part,\" it is a zero-dimensional mathematical abstraction representing a precise position in space without any length, area, or volume. While it lacks a substantial definition due to its primitive nature and cannot be distinguished from other geometric primitives by inherent properties, it is nominally described as a marker of location. Points are used to define more complex geometric objects, such as lines, shapes, and higher-dimensional constructs, by serving as their basic building blocks. Despite differing phrasings across descriptions, all agree on its role as an abstract, dimensionless entity that denotes spatial coordinates.", "entity_id": "\u70b9", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u70b9", "label": "\u70b9", "shape": "dot", "size": 10, "source_id": "chunk-d79a25c9ff12f28251aed2da9132d65e\u003cSEP\u003echunk-c8a15761aa2fa77855ceb30f7b7a730a\u003cSEP\u003echunk-a77717ea766ba1464bcbadc22f73914d\u003cSEP\u003echunk-b6b061c418521e25080936fc1c0a8bc8", "title": "A **point** is a fundamental geometric concept that serves as a foundational element in mathematics. Defined by Euclid as \"having no part,\" it is a zero-dimensional mathematical abstraction representing a precise position in space without any length, area, or volume. While it lacks a substantial definition due to its primitive nature and cannot be distinguished from other geometric primitives by inherent properties, it is nominally described as a marker of location. Points are used to define more complex geometric objects, such as lines, shapes, and higher-dimensional constructs, by serving as their basic building blocks. Despite differing phrasings across descriptions, all agree on its role as an abstract, dimensionless entity that denotes spatial coordinates."}, {"color": "#419973", "created_at": 1752884160, "description": "\u5eb7\u6258\u5b9a\u4e49\u4e86\u65e0\u7a77\u96c6\u5408\uff0c\u5e76\u8bc1\u660e\u4e86\u81ea\u7136\u6570\u548c\u6709\u7406\u6570\u7684\u4e2a\u6570\u4e00\u6837\u591a\u3002", "entity_id": "\u65e0\u7a77\u96c6\u5408", "entity_type": "UNKNOWN", "file_path": "ChuZhongShuXue.docx", "id": "\u65e0\u7a77\u96c6\u5408", "label": "\u65e0\u7a77\u96c6\u5408", "shape": "dot", "size": 10, "source_id": "chunk-8b6f8917e6704256d88d931fd2d99d07", "title": "\u5eb7\u6258\u5b9a\u4e49\u4e86\u65e0\u7a77\u96c6\u5408\uff0c\u5e76\u8bc1\u660e\u4e86\u81ea\u7136\u6570\u548c\u6709\u7406\u6570\u7684\u4e2a\u6570\u4e00\u6837\u591a\u3002"}, {"color": "#871e5d", "created_at": 1752884033, "description": "The Pythagorean theorem, known as **\u52fe\u80a1\u5b9a\u7406 (G\u014dug\u01d4 D\u00ecngl\u01d0)** in Chinese, is a fundamental principle in Euclidean geometry that describes the relationship between the sides of a right-angled triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem is a cornerstone of geometric theory and has widespread applications in mathematics, astronomy, and geography. \n\nIn ancient China, **\u52fe\u80a1\u5b9a\u7406** was employed for astronomical and geographical measurements, as well as for solving geometric problems, as documented in classical texts like the **\u300a\u4e5d\u7ae0\u7b97\u672f\u300b (Ji\u01d4 Zh\u0101ng Su\u00e0nsh\u00f9, *The Nine Chapters on the Mathematical Art*)**. The theorem not only serves as a foundational proposition in Euclidean geometry for calculating line segment lengths but also works in conjunction with trigonometric principles, forming a core tool in geometric analysis. \n\nThus, **\u52fe\u80a1\u5b9a\u7406 (Pythagorean theorem)** remains an essential mathematical concept, bridging theoretical geometry and practical applications across history.", "entity_id": "\u52fe\u80a1\u5b9a\u7406", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u52fe\u80a1\u5b9a\u7406", "label": "\u52fe\u80a1\u5b9a\u7406", "shape": "dot", "size": 10, "source_id": "chunk-bc737ed018919d513c346a3c74b449aa\u003cSEP\u003echunk-c64e6f44879da2fe257177d9e60e88f1\u003cSEP\u003echunk-d2d1fa534f69129f1d307d4162d5e6e4\u003cSEP\u003echunk-6782332859e0bdef19c72a7367743352\u003cSEP\u003echunk-f65f3e755944ffb1fd1280feebd5aa52\u003cSEP\u003echunk-69ea16a9bf8f7637a55bade939083e96\u003cSEP\u003echunk-b86ecc17467c79a88d8dfa681dcdb3ec\u003cSEP\u003echunk-afc841e1a6c397cedfaba45e3a346fb6", "title": "The Pythagorean theorem, known as **\u52fe\u80a1\u5b9a\u7406 (G\u014dug\u01d4 D\u00ecngl\u01d0)** in Chinese, is a fundamental principle in Euclidean geometry that describes the relationship between the sides of a right-angled triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem is a cornerstone of geometric theory and has widespread applications in mathematics, astronomy, and geography. \n\nIn ancient China, **\u52fe\u80a1\u5b9a\u7406** was employed for astronomical and geographical measurements, as well as for solving geometric problems, as documented in classical texts like the **\u300a\u4e5d\u7ae0\u7b97\u672f\u300b (Ji\u01d4 Zh\u0101ng Su\u00e0nsh\u00f9, *The Nine Chapters on the Mathematical Art*)**. The theorem not only serves as a foundational proposition in Euclidean geometry for calculating line segment lengths but also works in conjunction with trigonometric principles, forming a core tool in geometric analysis. \n\nThus, **\u52fe\u80a1\u5b9a\u7406 (Pythagorean theorem)** remains an essential mathematical concept, bridging theoretical geometry and practical applications across history."}, {"color": "#902829", "created_at": 1752884031, "description": "**\u300a\u51e0\u4f55\u57fa\u7840\u300b** (*Foundations of Geometry*) is a foundational mathematical text written by the renowned mathematician David Hilbert. Originally published in 1899, the work is a seminal contribution to the field of geometry, formalizing and redefining geometric axioms while addressing the limitations of Euclid\u0027s classical system. Hilbert\u0027s approach introduced a rigorous axiomatic framework, organizing geometric principles into five groups of axioms and employing symbolic notation for fundamental geometric objects. This modernized, formalized treatment laid the groundwork for the axiomatic method in mathematics and significantly influenced subsequent developments in the discipline. \n\nThe text has been translated into Chinese under the title **\u300a\u51e0\u4f55\u57fa\u7840\u300b**, with notable translations contributed by Jiang Zehan (\u6c5f\u6cfd\u6db5) and Zhu Dingxun (\u6731\u9f0e\u52cb). Hilbert\u0027s work is often regarded as an attempt to establish a strict, logical foundation for geometry through axiomatization, mirroring the structure of Euclid\u0027s *Elements* while advancing beyond it in clarity and generality. It remains a cornerstone of mathematical literature, bridging classical and modern geometric thought. \n\n(*Note: The English title *Foundations of Geometry* is occasionally referenced in Chinese contexts, but the work is predominantly recognized by its Chinese title, \u300a\u51e0\u4f55\u57fa\u7840\u300b.*)", "entity_id": "\u300a\u51e0\u4f55\u57fa\u7840\u300b", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u300a\u51e0\u4f55\u57fa\u7840\u300b", "label": "\u300a\u51e0\u4f55\u57fa\u7840\u300b", "shape": "dot", "size": 10, "source_id": "chunk-7f35e11154694cb00fbc4308a33a36a3\u003cSEP\u003echunk-0f6c384b97d0bb5698ba21610de20e63\u003cSEP\u003echunk-33e4d77aeae0267e846e8c7ab9af2b41\u003cSEP\u003echunk-e91dee450ab50cb1a44f08fe171c809b\u003cSEP\u003echunk-c81153206158adf5aa1a438a4f6242ed\u003cSEP\u003echunk-2ff43cd1d769194656997fc3e9797a67\u003cSEP\u003echunk-836dc146c06f0d9243867645657906ed", "title": "**\u300a\u51e0\u4f55\u57fa\u7840\u300b** (*Foundations of Geometry*) is a foundational mathematical text written by the renowned mathematician David Hilbert. Originally published in 1899, the work is a seminal contribution to the field of geometry, formalizing and redefining geometric axioms while addressing the limitations of Euclid\u0027s classical system. Hilbert\u0027s approach introduced a rigorous axiomatic framework, organizing geometric principles into five groups of axioms and employing symbolic notation for fundamental geometric objects. This modernized, formalized treatment laid the groundwork for the axiomatic method in mathematics and significantly influenced subsequent developments in the discipline. \n\nThe text has been translated into Chinese under the title **\u300a\u51e0\u4f55\u57fa\u7840\u300b**, with notable translations contributed by Jiang Zehan (\u6c5f\u6cfd\u6db5) and Zhu Dingxun (\u6731\u9f0e\u52cb). Hilbert\u0027s work is often regarded as an attempt to establish a strict, logical foundation for geometry through axiomatization, mirroring the structure of Euclid\u0027s *Elements* while advancing beyond it in clarity and generality. It remains a cornerstone of mathematical literature, bridging classical and modern geometric thought. \n\n(*Note: The English title *Foundations of Geometry* is occasionally referenced in Chinese contexts, but the work is predominantly recognized by its Chinese title, \u300a\u51e0\u4f55\u57fa\u7840\u300b.*)"}, {"color": "#52358b", "created_at": 1752884032, "description": "The shortest path between two points on a surface, with defined endpoints and length.\u003cSEP\u003e\u76f4\u7ebf\u6bb5is an abstract concept representing geometric shapes, used to demonstrate mathematical abstraction.", "entity_id": "\u76f4\u7ebf\u6bb5", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u76f4\u7ebf\u6bb5", "label": "\u76f4\u7ebf\u6bb5", "shape": "dot", "size": 10, "source_id": "chunk-56f75baf6ec4e55ce2eb93f9a0b24904\u003cSEP\u003echunk-c8a15761aa2fa77855ceb30f7b7a730a", "title": "The shortest path between two points on a surface, with defined endpoints and length.\u003cSEP\u003e\u76f4\u7ebf\u6bb5is an abstract concept representing geometric shapes, used to demonstrate mathematical abstraction."}, {"color": "#71bbf5", "created_at": 1752884032, "description": "Property of sequential computation where each step depends on the previous one.\u003cSEP\u003e\u4f20\u9012\u6027is a fundamental property in mathematics, serving as the logical foundation for reasoning and inference.\u003cSEP\u003e\u4f20\u9012\u6027refers to the transitive property, a fundamental logical structure in human cognition.", "entity_id": "\u4f20\u9012\u6027", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u4f20\u9012\u6027", "label": "\u4f20\u9012\u6027", "shape": "dot", "size": 10, "source_id": "chunk-4d700c37f93e40f9422446b4fe6a019c\u003cSEP\u003echunk-1d53a8b79c459a7736aa43124180db58\u003cSEP\u003echunk-56f75baf6ec4e55ce2eb93f9a0b24904", "title": "Property of sequential computation where each step depends on the previous one.\u003cSEP\u003e\u4f20\u9012\u6027is a fundamental property in mathematics, serving as the logical foundation for reasoning and inference.\u003cSEP\u003e\u4f20\u9012\u6027refers to the transitive property, a fundamental logical structure in human cognition."}, {"color": "#1b1b56", "created_at": 1752884040, "description": "**\u6234\u5fb7\u91d1 (Richard Dedekind)** \u662f\u4e00\u4f4d\u5fb7\u56fd\u6570\u5b66\u5bb6\uff0c\u4ee5\u5176\u5bf9\u6570\u8bba\u548c\u5b9e\u6570\u7406\u8bba\u7684\u91cd\u5927\u8d21\u732e\u800c\u95fb\u540d\u3002\u4ed6\u63d0\u51fa\u4e86**\u6234\u5fb7\u91d1\u5206\u5272 (Dedekind Cut)** \u7684\u6982\u5ff5\uff0c\u5e76\u4e8e1872\u5e74\u901a\u8fc7\u8fd9\u4e00\u65b9\u6cd5\u6784\u9020\u6027\u5730\u5b9a\u4e49\u4e86\u5b9e\u6570\uff0c\u89e3\u51b3\u4e86\u8fde\u7eed\u6027\u7684\u95ee\u9898\u3002\u6b64\u5916\uff0c\u4ed6\u8fd8\u5229\u7528\u7edd\u5bf9\u503c\u7684\u51e0\u4f55\u89e3\u91ca\uff0c\u63d0\u51fa\u4e86**\u6234\u5fb7\u91d1\u8fde\u7eed\u6027\u516c\u7406 (Dedekind continuity axiom)**\uff0c\u8fd9\u4e00\u516c\u7406\u6210\u4e3a\u5b9e\u6570\u7406\u8bba\u7684\u57fa\u7840\u3002\u6234\u5fb7\u91d1\u7684\u5de5\u4f5c\u8fd8\u5305\u62ec\u5bf9**\u540e\u7ee7\u6570 (successor)** \u6982\u5ff5\u7684\u63a2\u8ba8\uff0c\u8fd9\u4e00\u601d\u60f3\u5bf9**\u76ae\u4e9a\u8bfa (Peano)** \u7684\u7b97\u672f\u516c\u7406\u4f53\u7cfb\u4ea7\u751f\u4e86\u6df1\u8fdc\u5f71\u54cd\u3002\u603b\u4f53\u800c\u8a00\uff0c\u6234\u5fb7\u91d1\u5728\u6570\u8bba\u3001\u5b9e\u6570\u8fde\u7eed\u6027\u4ee5\u53ca\u516c\u7406\u5316\u65b9\u6cd5\u7b49\u9886\u57df\u5747\u4f5c\u51fa\u4e86\u5f00\u521b\u6027\u7684\u8d21\u732e\uff0c\u662f19\u4e16\u7eaa\u6570\u5b66\u754c\u7684\u91cd\u8981\u4eba\u7269\u4e4b\u4e00\u3002", "entity_id": "\u6234\u5fb7\u91d1", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u6234\u5fb7\u91d1", "label": "\u6234\u5fb7\u91d1", "shape": "dot", "size": 10, "source_id": "chunk-f379ebb8f8dc44a54c681880caf478eb\u003cSEP\u003echunk-42e82319a2d63b215f82bba66d1e632d\u003cSEP\u003echunk-4dc0d317d51b12cedc9804d89a6f0d99\u003cSEP\u003echunk-c81153206158adf5aa1a438a4f6242ed\u003cSEP\u003echunk-10177ac2b1564cfa9472038f27fca84a", "title": "**\u6234\u5fb7\u91d1 (Richard Dedekind)** \u662f\u4e00\u4f4d\u5fb7\u56fd\u6570\u5b66\u5bb6\uff0c\u4ee5\u5176\u5bf9\u6570\u8bba\u548c\u5b9e\u6570\u7406\u8bba\u7684\u91cd\u5927\u8d21\u732e\u800c\u95fb\u540d\u3002\u4ed6\u63d0\u51fa\u4e86**\u6234\u5fb7\u91d1\u5206\u5272 (Dedekind Cut)** \u7684\u6982\u5ff5\uff0c\u5e76\u4e8e1872\u5e74\u901a\u8fc7\u8fd9\u4e00\u65b9\u6cd5\u6784\u9020\u6027\u5730\u5b9a\u4e49\u4e86\u5b9e\u6570\uff0c\u89e3\u51b3\u4e86\u8fde\u7eed\u6027\u7684\u95ee\u9898\u3002\u6b64\u5916\uff0c\u4ed6\u8fd8\u5229\u7528\u7edd\u5bf9\u503c\u7684\u51e0\u4f55\u89e3\u91ca\uff0c\u63d0\u51fa\u4e86**\u6234\u5fb7\u91d1\u8fde\u7eed\u6027\u516c\u7406 (Dedekind continuity axiom)**\uff0c\u8fd9\u4e00\u516c\u7406\u6210\u4e3a\u5b9e\u6570\u7406\u8bba\u7684\u57fa\u7840\u3002\u6234\u5fb7\u91d1\u7684\u5de5\u4f5c\u8fd8\u5305\u62ec\u5bf9**\u540e\u7ee7\u6570 (successor)** \u6982\u5ff5\u7684\u63a2\u8ba8\uff0c\u8fd9\u4e00\u601d\u60f3\u5bf9**\u76ae\u4e9a\u8bfa (Peano)** \u7684\u7b97\u672f\u516c\u7406\u4f53\u7cfb\u4ea7\u751f\u4e86\u6df1\u8fdc\u5f71\u54cd\u3002\u603b\u4f53\u800c\u8a00\uff0c\u6234\u5fb7\u91d1\u5728\u6570\u8bba\u3001\u5b9e\u6570\u8fde\u7eed\u6027\u4ee5\u53ca\u516c\u7406\u5316\u65b9\u6cd5\u7b49\u9886\u57df\u5747\u4f5c\u51fa\u4e86\u5f00\u521b\u6027\u7684\u8d21\u732e\uff0c\u662f19\u4e16\u7eaa\u6570\u5b66\u754c\u7684\u91cd\u8981\u4eba\u7269\u4e4b\u4e00\u3002"}, {"color": "#aea671", "created_at": 1752884037, "description": "**Summary of Rational Numbers (\u6709\u7406\u6570):** \n\nRational numbers (\u6709\u7406\u6570) are numbers that can be expressed as a ratio (or fraction) of two integers, where the denominator is not zero. They form a fundamental concept in mathematics, serving as a cornerstone in geometric constructions, real number theory, and historical mathematical studies. Rational numbers include integers, proper and improper fractions, and can also be represented in decimal form, either as terminating or repeating decimals. \n\nKey properties of rational numbers include: \n- **Representation:** They can be written as fractions \\( \\frac{a}{b} \\) where \\( a \\) and \\( b \\) are integers, or equivalently as finite or repeating decimals. \n- **Cardinality:** The set of rational numbers has the same cardinality (size) as the set of natural numbers, meaning they are countably infinite. \n- **Geometric Interpretation:** Rational numbers correspond to points on a number line, and their properties are further explored using concepts like absolute value. \n- **Historical and Educational Significance:** They were foundational in ancient mathematics and remain a crucial topic in early mathematics education, particularly in middle school (\u521d\u4e2d\u6570\u5b66), where their decimal representations are studied. \n\nRational numbers are a subset of real numbers and play a key role in extending numerical systems from integers to more complex structures in mathematics.", "entity_id": "\u6709\u7406\u6570", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u6709\u7406\u6570", "label": "\u6709\u7406\u6570", "shape": "dot", "size": 10, "source_id": "chunk-aca1faae06ff2fb9d92a04ee4d07e4f7\u003cSEP\u003echunk-8b6f8917e6704256d88d931fd2d99d07\u003cSEP\u003echunk-89a00ca5535ebdb5d721e473eebbac4c\u003cSEP\u003echunk-dda7ba554b64773a9f739a19965188f1\u003cSEP\u003echunk-a2266049e857fa018842eec4ce9a2e9d\u003cSEP\u003echunk-c38ffe600259aeb39e4227f260864859\u003cSEP\u003echunk-2820dfe8496edbd1e66a24c26cdd1859\u003cSEP\u003echunk-10177ac2b1564cfa9472038f27fca84a", "title": "**Summary of Rational Numbers (\u6709\u7406\u6570):** \n\nRational numbers (\u6709\u7406\u6570) are numbers that can be expressed as a ratio (or fraction) of two integers, where the denominator is not zero. They form a fundamental concept in mathematics, serving as a cornerstone in geometric constructions, real number theory, and historical mathematical studies. Rational numbers include integers, proper and improper fractions, and can also be represented in decimal form, either as terminating or repeating decimals. \n\nKey properties of rational numbers include: \n- **Representation:** They can be written as fractions \\( \\frac{a}{b} \\) where \\( a \\) and \\( b \\) are integers, or equivalently as finite or repeating decimals. \n- **Cardinality:** The set of rational numbers has the same cardinality (size) as the set of natural numbers, meaning they are countably infinite. \n- **Geometric Interpretation:** Rational numbers correspond to points on a number line, and their properties are further explored using concepts like absolute value. \n- **Historical and Educational Significance:** They were foundational in ancient mathematics and remain a crucial topic in early mathematics education, particularly in middle school (\u521d\u4e2d\u6570\u5b66), where their decimal representations are studied. \n\nRational numbers are a subset of real numbers and play a key role in extending numerical systems from integers to more complex structures in mathematics."}, {"color": "#3fd177", "created_at": 1752884044, "description": "Invented the absolute value symbol in 1841 to represent the magnitude of complex numbers, laying the foundation for its algebraic representation.", "entity_id": "\u65af", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u65af", "label": "\u65af", "shape": "dot", "size": 10, "source_id": "chunk-10177ac2b1564cfa9472038f27fca84a", "title": "Invented the absolute value symbol in 1841 to represent the magnitude of complex numbers, laying the foundation for its algebraic representation."}, {"color": "#8f9a14", "created_at": 1752884044, "description": "A German mathematician who in 1905 generalized the concept of absolute value to represent the length of vectors, providing a geometric interpretation.", "entity_id": "\u7518\u65af", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u7518\u65af", "label": "\u7518\u65af", "shape": "dot", "size": 10, "source_id": "chunk-10177ac2b1564cfa9472038f27fca84a", "title": "A German mathematician who in 1905 generalized the concept of absolute value to represent the length of vectors, providing a geometric interpretation."}, {"color": "#1cc9c1", "created_at": 1752884030, "description": "**Complex numbers (\u590d\u6570)** are a mathematical concept that extends the real numbers to allow for the complete solution of polynomial equations, including those involving square roots of negative numbers. They consist of a real part and an imaginary part, forming expressions like \\( a + bi \\), where \\( i \\) is the imaginary unit. Historically, the absolute value (or modulus) of a complex number was defined as the square root of the sum of the squares of its real and imaginary components. \n\nWhile complex numbers are widely used in mathematics and engineering, they differ from real numbers in that they do not possess a natural ordering (i.e., they lack a conventional \"greater than\" or \"less than\" relationship). Some perspectives argue that complex numbers are not \"true\" numbers in the same sense as real numbers due to their lack of comparability. Additionally, they serve as an illustrative example where improper syllogistic reasoning can lead to incorrect conclusions, highlighting the need for careful logical treatment in mathematical proofs. \n\nIn summary, complex numbers are a fundamental extension of real numbers with unique properties, enabling solutions to equations that would otherwise have no real roots, while also presenting conceptual challenges regarding their nature and ordering.", "entity_id": "\u590d\u6570", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u590d\u6570", "label": "\u590d\u6570", "shape": "dot", "size": 10, "source_id": "chunk-8b6f8917e6704256d88d931fd2d99d07\u003cSEP\u003echunk-cd72bf9b79aac9061696e480b2f809fb\u003cSEP\u003echunk-a2266049e857fa018842eec4ce9a2e9d\u003cSEP\u003echunk-edaeb942f2d7bc9b4f269a55a5bdd99f\u003cSEP\u003echunk-10177ac2b1564cfa9472038f27fca84a", "title": "**Complex numbers (\u590d\u6570)** are a mathematical concept that extends the real numbers to allow for the complete solution of polynomial equations, including those involving square roots of negative numbers. They consist of a real part and an imaginary part, forming expressions like \\( a + bi \\), where \\( i \\) is the imaginary unit. Historically, the absolute value (or modulus) of a complex number was defined as the square root of the sum of the squares of its real and imaginary components. \n\nWhile complex numbers are widely used in mathematics and engineering, they differ from real numbers in that they do not possess a natural ordering (i.e., they lack a conventional \"greater than\" or \"less than\" relationship). Some perspectives argue that complex numbers are not \"true\" numbers in the same sense as real numbers due to their lack of comparability. Additionally, they serve as an illustrative example where improper syllogistic reasoning can lead to incorrect conclusions, highlighting the need for careful logical treatment in mathematical proofs. \n\nIn summary, complex numbers are a fundamental extension of real numbers with unique properties, enabling solutions to equations that would otherwise have no real roots, while also presenting conceptual challenges regarding their nature and ordering."}, {"color": "#d25e3a", "created_at": 1752884044, "description": "The length between two points on a number line, which can be expressed using absolute value in geometric interpretation.", "entity_id": "\u7ebf\u6bb5\u957f\u5ea6", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u7ebf\u6bb5\u957f\u5ea6", "label": "\u7ebf\u6bb5\u957f\u5ea6", "shape": "dot", "size": 10, "source_id": "chunk-10177ac2b1564cfa9472038f27fca84a", "title": "The length between two points on a number line, which can be expressed using absolute value in geometric interpretation."}, {"color": "#0eb2ec", "created_at": 1752884055, "description": "A computational method where multiple calculations are carried out simultaneously, demonstrated through the example of average calculation.", "entity_id": "\u5e76\u884c\u8ba1\u7b97", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u5e76\u884c\u8ba1\u7b97", "label": "\u5e76\u884c\u8ba1\u7b97", "shape": "dot", "size": 10, "source_id": "chunk-2e4972ad6e1759c82b71b7eb4bf1ccd8", "title": "A computational method where multiple calculations are carried out simultaneously, demonstrated through the example of average calculation."}, {"color": "#5dab97", "created_at": 1752884055, "description": "\u6570\u636e\u7684\u96c6\u4e2d\u8d8b\u52bf\u662f\u6307\u901a\u8fc7\u5e73\u5747\u6570\u7b49\u6307\u6807\u53cd\u6620\u6570\u636e\u7684\u5178\u578b\u7279\u5f81\uff0c\u7528\u4e8e\u7406\u89e3\u548c\u6bd4\u8f83\u4e8b\u7269\u7684\u91cf\u5316\u8868\u73b0\u3002", "entity_id": 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"ChuZhongShuXue.docx", "id": "\u8bdd\u9898\u7bc7", "label": "\u8bdd\u9898\u7bc7", "shape": "dot", "size": 10, "source_id": "chunk-3036fe5f8bcbc7479b162f62dea08660", "title": "Section of the book discussing topics suitable for project-based learning"}, {"color": "#1fc2ae", "created_at": 1752884035, "description": "**\u4ee3\u6570\u63a8\u7406 (Algebraic Reasoning)** \n\n\u4ee3\u6570\u63a8\u7406 is a form of mathematical reasoning centered on algebraic structures, relationships, and logical problem-solving using algebraic concepts. It involves both inductive and deductive processes in mathematical proof, utilizing mathematical symbols and logical rules to analyze and solve problems. Unlike traditional geometric proofs, algebraic reasoning emphasizes symbolic manipulation and abstract thinking. \n\nIn recent years, \u4ee3\u6570\u63a8\u7406 has gained prominence in modern mathematics education, particularly in Chinese curriculum reforms, where it is increasingly emphasized to complement conventional geometric approaches. This shift highlights its importance in fostering logical thinking and structured problem-solving skills in students. \n\nOverall, \u4ee3\u6570\u63a8\u7406 is a systematic method for mathematical argumentation and problem-solving, blending symbolic representation with rigorous logical reasoning.", "entity_id": "\u4ee3\u6570\u63a8\u7406", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u4ee3\u6570\u63a8\u7406", "label": "\u4ee3\u6570\u63a8\u7406", "shape": "dot", "size": 10, "source_id": "chunk-0f93ea4d651934743a1c0c3eefd7ac60\u003cSEP\u003echunk-4d700c37f93e40f9422446b4fe6a019c\u003cSEP\u003echunk-46deef01a2345f56bb0d9a61a58a7840\u003cSEP\u003echunk-a1b0ea141dcc25762619a431f93aeca3", "title": "**\u4ee3\u6570\u63a8\u7406 (Algebraic Reasoning)** \n\n\u4ee3\u6570\u63a8\u7406 is a form of mathematical reasoning centered on algebraic structures, relationships, and logical problem-solving using algebraic concepts. It involves both inductive and deductive processes in mathematical proof, utilizing mathematical symbols and logical rules to analyze and solve problems. Unlike traditional geometric proofs, algebraic reasoning emphasizes symbolic manipulation and abstract thinking. \n\nIn recent years, \u4ee3\u6570\u63a8\u7406 has gained prominence in modern mathematics education, particularly in Chinese curriculum reforms, where it is increasingly emphasized to complement conventional geometric approaches. This shift highlights its importance in fostering logical thinking and structured problem-solving skills in students. \n\nOverall, \u4ee3\u6570\u63a8\u7406 is a systematic method for mathematical argumentation and problem-solving, blending symbolic representation with rigorous logical reasoning."}, {"color": "#139ed3", "created_at": 1752884034, "description": "\u300a\u666e\u901a\u9ad8\u4e2d\u6570\u5b66\u8bfe\u7a0b\u6807\u51c6(2017\u5e74\u7248)\u300b(Senior High School Mathematics Curriculum Standards, 2017 Edition) is a Chinese national educational standard issued by the Ministry of Education, defining the curriculum content and competencies for high school mathematics. It emphasizes algebraic reasoning, advocating the principle of \"building intuition through geometry and expressing it through algebra.\" The standard further refines the foundational framework, evolving the \"four basics\" into \"three abilities\" focused on observing, thinking, and expressing the real world mathematically. \n\nKey components include logical reasoning, computational methods (such as the bisection method), and a deepened understanding of geometric transformations, including isometric transformations and fundamental transformations. The curriculum integrates practical applications, ensuring students develop both theoretical knowledge and problem-solving skills. By balancing geometric intuition with algebraic precision, the standard aims to cultivate mathematical literacy and analytical thinking among high school students in China.", "entity_id": "\u300a\u666e\u901a\u9ad8\u4e2d\u6570\u5b66\u8bfe\u7a0b\u6807\u51c6(2017\u5e74\u7248)\u300b", "entity_type": "organization", "file_path": "ChuZhongShuXue.docx", "id": "\u300a\u666e\u901a\u9ad8\u4e2d\u6570\u5b66\u8bfe\u7a0b\u6807\u51c6(2017\u5e74\u7248)\u300b", "label": "\u300a\u666e\u901a\u9ad8\u4e2d\u6570\u5b66\u8bfe\u7a0b\u6807\u51c6(2017\u5e74\u7248)\u300b", "shape": "dot", "size": 10, "source_id": "chunk-46deef01a2345f56bb0d9a61a58a7840\u003cSEP\u003echunk-d600d36500fc1362f8eb34c6e841165e\u003cSEP\u003echunk-1d53a8b79c459a7736aa43124180db58\u003cSEP\u003echunk-3e3dea3ce1efa839313f48f8630d1fc8\u003cSEP\u003echunk-683c16577a28a4b7a2506b946403f52e", "title": "\u300a\u666e\u901a\u9ad8\u4e2d\u6570\u5b66\u8bfe\u7a0b\u6807\u51c6(2017\u5e74\u7248)\u300b(Senior High School Mathematics Curriculum Standards, 2017 Edition) is a Chinese national educational standard issued by the Ministry of Education, defining the curriculum content and competencies for high school mathematics. It emphasizes algebraic reasoning, advocating the principle of \"building intuition through geometry and expressing it through algebra.\" The standard further refines the foundational framework, evolving the \"four basics\" into \"three abilities\" focused on observing, thinking, and expressing the real world mathematically. \n\nKey components include logical reasoning, computational methods (such as the bisection method), and a deepened understanding of geometric transformations, including isometric transformations and fundamental transformations. The curriculum integrates practical applications, ensuring students develop both theoretical knowledge and problem-solving skills. By balancing geometric intuition with algebraic precision, the standard aims to cultivate mathematical literacy and analytical thinking among high school students in China."}, {"color": "#a63724", "created_at": 1752884035, "description": "**Deductive Reasoning (\u6f14\u7ece\u63a8\u7406)** is a fundamental form of logical reasoning in which conclusions are necessarily derived from true premises, making it essential for mathematical proofs and validations. Pioneered by the Pythagorean School and further developed in Aristotle\u0027s syllogistic forms, deductive reasoning serves as the foundation of Euclidean geometry and other structured logical systems. It operates by moving from general principles to specific conclusions, ensuring certainty in its outcomes\u2014unlike statistical inference, which relies on probability. \n\nIn mathematics, deductive reasoning is widely used to validate algebraic expressions, verify conjectures, and establish rigorous arguments. It encompasses various forms, including holistic (\u6574\u4f53\u6f14\u7ece\u63a8\u7406) and simple (\u7b80\u5355\u6f14\u7ece\u63a8\u7406) deductive methods, all characterized by their inevitability and logical consistency. 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Pioneered by the Pythagorean School and further developed in Aristotle\u0027s syllogistic forms, deductive reasoning serves as the foundation of Euclidean geometry and other structured logical systems. It operates by moving from general principles to specific conclusions, ensuring certainty in its outcomes\u2014unlike statistical inference, which relies on probability. \n\nIn mathematics, deductive reasoning is widely used to validate algebraic expressions, verify conjectures, and establish rigorous arguments. It encompasses various forms, including holistic (\u6574\u4f53\u6f14\u7ece\u63a8\u7406) and simple (\u7b80\u5355\u6f14\u7ece\u63a8\u7406) deductive methods, all characterized by their inevitability and logical consistency. 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They serve as foundational elements for equations and functions, utilizing algebraic notation to express quantitative relationships. These expressions involve the representation and manipulation of variables and constants through arithmetic operations, forming the basis for solving mathematical problems. By integrating these components, algebraic expressions provide a structured way to model and analyze relationships in mathematics. \n\nThis summary consolidates all provided descriptions, emphasizing their role in equations, their compositional elements (numbers, variables, and operations), and their broader mathematical significance.", "entity_id": "\u4ee3\u6570\u5f0f", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u4ee3\u6570\u5f0f", "label": "\u4ee3\u6570\u5f0f", "shape": "dot", "size": 10, "source_id": "chunk-3036fe5f8bcbc7479b162f62dea08660\u003cSEP\u003echunk-0e47637ef5cc5c08729008ec09e12dd7\u003cSEP\u003echunk-1ea9636f6fb64b0198c39b6185e32379\u003cSEP\u003echunk-c38ffe600259aeb39e4227f260864859\u003cSEP\u003echunk-f1e81718ff3ad9793c932f0681b6cfda", "title": "**Algebraic expressions (\u4ee3\u6570\u5f0f)** are mathematical phrases that combine numbers, variables, and operations to represent general or specific numerical relationships. They serve as foundational elements for equations and functions, utilizing algebraic notation to express quantitative relationships. These expressions involve the representation and manipulation of variables and constants through arithmetic operations, forming the basis for solving mathematical problems. By integrating these components, algebraic expressions provide a structured way to model and analyze relationships in mathematics. \n\nThis summary consolidates all provided descriptions, emphasizing their role in equations, their compositional elements (numbers, variables, and operations), and their broader mathematical significance."}, {"color": "#de91c2", "created_at": 1752884037, "description": "Algebra involves the study of mathematical symbols and rules for manipulating these symbols.\u003cSEP\u003eAlgebra is a mathematical discipline introduced to Chinese through translations by Li Shanlan and Alexander Wylie.", "entity_id": "\u4ee3\u6570", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u4ee3\u6570", "label": "\u4ee3\u6570", "shape": "dot", "size": 10, "source_id": "chunk-c38ffe600259aeb39e4227f260864859\u003cSEP\u003echunk-7f35e11154694cb00fbc4308a33a36a3", "title": "Algebra involves the study of mathematical symbols and rules for manipulating these symbols.\u003cSEP\u003eAlgebra is a mathematical discipline introduced to Chinese through translations by Li Shanlan and Alexander Wylie."}, {"color": "#6e759c", "created_at": 1752884041, "description": "A theorem in algebra that extends numerical comparisons to algebraic expressions, forming the basis for equations and inequalities.", "entity_id": "\u4e09\u5c90\u6027\u5b9a\u7406", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u4e09\u5c90\u6027\u5b9a\u7406", "label": "\u4e09\u5c90\u6027\u5b9a\u7406", "shape": "dot", "size": 10, "source_id": "chunk-0e47637ef5cc5c08729008ec09e12dd7", "title": "A theorem in algebra that extends numerical comparisons to algebraic expressions, forming the basis for equations and inequalities."}, {"color": "#89088f", "created_at": 1752884041, "description": "A formula in algebra expressing the difference of squares, derived through deductive reasoning or inductive methods.\u003cSEP\u003eA mathematical formula used to calculate the 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It marks a significant milestone in the development of algebra by introducing symbolic representations of coefficients and general solutions to equations, effectively serving as a cornerstone for modern algebraic theory. \n\nOriginally applied to quadratic equations, Vi\u00e8te\u0027s formulas establish clear connections between the roots and coefficients of a polynomial, enabling deeper analysis of constructible solutions. 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"\u4e3b\u8c13\u4e0d\u5bf9\u79f0\u7684\u6027\u8d28\u547d\u9898refers to propositions where the subject implies the predicate but not vice versa, indicating a one-directional relationship.", "entity_id": "\u4e3b\u8c13\u4e0d\u5bf9\u79f0\u7684\u6027\u8d28\u547d\u9898", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u4e3b\u8c13\u4e0d\u5bf9\u79f0\u7684\u6027\u8d28\u547d\u9898", "label": "\u4e3b\u8c13\u4e0d\u5bf9\u79f0\u7684\u6027\u8d28\u547d\u9898", "shape": "dot", "size": 10, "source_id": "chunk-afc841e1a6c397cedfaba45e3a346fb6", "title": "\u4e3b\u8c13\u4e0d\u5bf9\u79f0\u7684\u6027\u8d28\u547d\u9898refers to propositions where the subject implies the predicate but not vice versa, indicating a one-directional relationship."}, {"color": "#0fdd0d", "created_at": 1752884041, "description": "Mathematical inequalities comparing algebraic expressions, solved to determine ranges of possible values.", "entity_id": "\u4e0d\u7b49\u5f0f", "entity_type": "category", "file_path": 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"file_path": "ChuZhongShuXue.docx", "id": "\u9e21\u5154\u540c\u7b3c\u95ee\u9898", "label": "\u9e21\u5154\u540c\u7b3c\u95ee\u9898", "shape": "dot", "size": 10, "source_id": "chunk-1ea9636f6fb64b0198c39b6185e32379\u003cSEP\u003echunk-f1e81718ff3ad9793c932f0681b6cfda", "title": "A classic Chinese mathematical problem involving systems of equations to solve for unknown quantities.\u003cSEP\u003eA classic mathematical problem involving counting the number of chickens and rabbits in a cage based on the total number of heads and legs."}, {"color": "#55d3a1", "created_at": 1752884042, "description": "Distance-rate-time problems in mathematics, often solved using equation systems.\u003cSEP\u003eDistance-speed-time problems commonly used in mathematical education.", "entity_id": "\u884c\u7a0b\u95ee\u9898", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u884c\u7a0b\u95ee\u9898", "label": "\u884c\u7a0b\u95ee\u9898", "shape": "dot", "size": 10, "source_id": "chunk-1ea9636f6fb64b0198c39b6185e32379\u003cSEP\u003echunk-f1e81718ff3ad9793c932f0681b6cfda", "title": "Distance-rate-time problems in mathematics, often solved using equation systems.\u003cSEP\u003eDistance-speed-time problems commonly used in mathematical education."}, {"color": "#7b4154", "created_at": 1752884097, "description": "**\u6570\u5b66\u6a21\u578b (Mathematical Models)** are analytical tools that employ mathematical language and frameworks to represent, describe, and analyze real-world phenomena. These models serve as rational, structured representations of complex systems, emphasizing the relationships between variables and their quantitative interactions. Their primary value lies in their ability to simulate and interpret real-world scenarios, rather than in the mathematical constructs themselves. By translating observable patterns into mathematical terms, mathematical models enable users\u2014such as students or researchers\u2014to apply mathematical reasoning to practical problems, validate hypotheses through computational results, and gain deeper insights into systemic behaviors. Whether used for educational purposes or scientific research, these models provide a versatile foundation for understanding and predicting dynamic phenomena across various disciplines. \n\n*(Note: Contradictions were resolved by unifying the descriptions\u0027 core themes\u2014representation of reality, utility in analysis, and educational or scientific application\u2014while omitting redundant or overly narrow phrasing.)*", "entity_id": "\u6570\u5b66\u6a21\u578b", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u6570\u5b66\u6a21\u578b", "label": "\u6570\u5b66\u6a21\u578b", "shape": "dot", "size": 10, "source_id": "chunk-42e82319a2d63b215f82bba66d1e632d\u003cSEP\u003echunk-bdb2d088364dc5e365dcf7c67ff1476e\u003cSEP\u003echunk-42220f52b4f10f3dd80cceb36ebbd3ce\u003cSEP\u003echunk-ca18454792909d12d2509708190d0e85\u003cSEP\u003echunk-acdcbf2ecfb171615d68bd9be50f3470", "title": "**\u6570\u5b66\u6a21\u578b (Mathematical Models)** are analytical tools that employ mathematical language and frameworks to represent, describe, and analyze real-world phenomena. These models serve as rational, structured representations of complex systems, emphasizing the relationships between variables and their quantitative interactions. Their primary value lies in their ability to simulate and interpret real-world scenarios, rather than in the mathematical constructs themselves. By translating observable patterns into mathematical terms, mathematical models enable users\u2014such as students or researchers\u2014to apply mathematical reasoning to practical problems, validate hypotheses through computational results, and gain deeper insights into systemic behaviors. Whether used for educational purposes or scientific research, these models provide a versatile foundation for understanding and predicting dynamic phenomena across various disciplines. \n\n*(Note: Contradictions were resolved by unifying the descriptions\u0027 core themes\u2014representation of reality, utility in analysis, and educational or scientific application\u2014while omitting redundant or overly narrow phrasing.)*"}, {"color": "#7143b8", "created_at": 1752884043, "description": "Algebra, whose beginnings are discussed in relation to Vi\u00e8te\u0027s work on symbolic representation.\u003cSEP\u003eThe branch of mathematics focused on symbols and rules for manipulating them, contrasted with geometry in the text.", "entity_id": "\u4ee3\u6570\u5b66", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u4ee3\u6570\u5b66", "label": "\u4ee3\u6570\u5b66", "shape": "dot", "size": 10, "source_id": "chunk-683c16577a28a4b7a2506b946403f52e\u003cSEP\u003echunk-dda7ba554b64773a9f739a19965188f1", "title": "Algebra, whose beginnings are discussed in relation to Vi\u00e8te\u0027s work on symbolic representation.\u003cSEP\u003eThe branch of mathematics focused on symbols and rules for manipulating them, contrasted with geometry in the text."}, {"color": "#2a05b1", "created_at": 1752884037, "description": "A **\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b (quadratic equation with one unknown)** is a fundamental type of mathematical equation, specifically a polynomial equation of degree two with a single variable. It is typically expressed in the standard form **ax\u00b2 + bx + c = 0**, where *a*, *b*, and *c* are coefficients, and *x* represents the unknown variable. The solutions (roots) of a quadratic equation depend on the discriminant (\u0394 = b\u00b2 - 4ac), which determines whether the equation has two distinct real roots, one repeated real root, or complex roots. \n\nHistorically, quadratic equations have played a significant role in mathematics. As early as the Chinese text **\u300a\u4e5d\u7ae0\u7b97\u672f\u300b (The Nine Chapters on the Mathematical Art)**, methods for solving quadratic equations were well-established, demonstrating their long-standing importance in mathematical abstraction and problem-solving. \n\nIn summary, a **\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b** is a second-degree polynomial equation in one unknown, widely studied for its theoretical and practical applications, with roots tracing back to ancient mathematical traditions.", "entity_id": "\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b", "label": "\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b", "shape": "dot", "size": 10, "source_id": "chunk-c38ffe600259aeb39e4227f260864859\u003cSEP\u003echunk-bc737ed018919d513c346a3c74b449aa\u003cSEP\u003echunk-1ea9636f6fb64b0198c39b6185e32379\u003cSEP\u003echunk-a2266049e857fa018842eec4ce9a2e9d\u003cSEP\u003echunk-683c16577a28a4b7a2506b946403f52e", "title": "A **\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b (quadratic equation with one unknown)** is a fundamental type of mathematical equation, specifically a polynomial equation of degree two with a single variable. It is typically expressed in the standard form **ax\u00b2 + bx + c = 0**, where *a*, *b*, and *c* are coefficients, and *x* represents the unknown variable. The solutions (roots) of a quadratic equation depend on the discriminant (\u0394 = b\u00b2 - 4ac), which determines whether the equation has two distinct real roots, one repeated real root, or complex roots. \n\nHistorically, quadratic equations have played a significant role in mathematics. As early as the Chinese text **\u300a\u4e5d\u7ae0\u7b97\u672f\u300b (The Nine Chapters on the Mathematical Art)**, methods for solving quadratic equations were well-established, demonstrating their long-standing importance in mathematical abstraction and problem-solving. \n\nIn summary, a **\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b** is a second-degree polynomial equation in one unknown, widely studied for its theoretical and practical applications, with roots tracing back to ancient mathematical traditions."}, {"color": "#e36db6", "created_at": 1752884042, "description": "Fran\u00e7ois Vi\u00e8te who pioneered symbolic algebra and quadratic formula.\u003cSEP\u003eFran\u00e7ois Vi\u00e8te, a French mathematician who pioneered the use of letters for abstract representation in algebra, enabling general solutions to equations.\u003cSEP\u003e\u97e6\u8fbe\u662f\u897f\u65b9\u6570\u5b66\u5bb6\uff0c\u4ee5\u4ee3\u6570\u5b66\u7684\u7814\u7a76\u95fb\u540d\uff0c\u5176\u5de5\u4f5c\u6bd4\u5218\u5fbd\u6ce8\u91ca\u300a\u4e5d\u7ae0\u7b97\u672f\u300b\u665a\u4e86\u4e00\u5343\u591a\u5e74\u3002", "entity_id": "\u97e6\u8fbe", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u97e6\u8fbe", "label": "\u97e6\u8fbe", "shape": "dot", "size": 10, "source_id": "chunk-683c16577a28a4b7a2506b946403f52e\u003cSEP\u003echunk-a2266049e857fa018842eec4ce9a2e9d\u003cSEP\u003echunk-bc737ed018919d513c346a3c74b449aa", "title": "Fran\u00e7ois Vi\u00e8te who pioneered symbolic algebra and quadratic formula.\u003cSEP\u003eFran\u00e7ois Vi\u00e8te, a French mathematician who pioneered the use of letters for abstract representation in algebra, enabling general solutions to equations.\u003cSEP\u003e\u97e6\u8fbe\u662f\u897f\u65b9\u6570\u5b66\u5bb6\uff0c\u4ee5\u4ee3\u6570\u5b66\u7684\u7814\u7a76\u95fb\u540d\uff0c\u5176\u5de5\u4f5c\u6bd4\u5218\u5fbd\u6ce8\u91ca\u300a\u4e5d\u7ae0\u7b97\u672f\u300b\u665a\u4e86\u4e00\u5343\u591a\u5e74\u3002"}, {"color": "#63aec3", "created_at": 1752884134, "description": "\u97e6\u8fbe\u5b9a\u7406\u53ef\u4ee5\u63a8\u5e7f\u5230\u4e00\u5143\u4e09\u6b21\u65b9\u7a0b\uff0c\u63cf\u8ff0\u5176\u6839\u4e0e\u7cfb\u6570\u7684\u5173\u7cfb\u3002", "entity_id": "\u4e00\u5143\u4e09\u6b21\u65b9\u7a0b", "entity_type": "UNKNOWN", "file_path": "ChuZhongShuXue.docx", "id": "\u4e00\u5143\u4e09\u6b21\u65b9\u7a0b", "label": "\u4e00\u5143\u4e09\u6b21\u65b9\u7a0b", "shape": "dot", "size": 10, "source_id": "chunk-401744b2d59735d588f0fc780ae91164", "title": "\u97e6\u8fbe\u5b9a\u7406\u53ef\u4ee5\u63a8\u5e7f\u5230\u4e00\u5143\u4e09\u6b21\u65b9\u7a0b\uff0c\u63cf\u8ff0\u5176\u6839\u4e0e\u7cfb\u6570\u7684\u5173\u7cfb\u3002"}, {"color": "#4bad4b", "created_at": 1752884086, "description": "Cubic equations that emerge when translating the angle trisection problem into algebraic terms.", "entity_id": "\u4e09\u6b21\u65b9\u7a0b", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u4e09\u6b21\u65b9\u7a0b", "label": "\u4e09\u6b21\u65b9\u7a0b", "shape": "dot", "size": 10, "source_id": "chunk-eb909fae946cb41ff187d26fae2118b2", "title": "Cubic equations that emerge when translating the angle trisection problem into algebraic terms."}, {"color": "#2488e4", "created_at": 1752884119, "description": "\u53cd\u8bc1\u6cd5\u7684\u903b\u8f91\u57fa\u7840\u662f\u77db\u76fe\u5f8b\u548c\u6392\u4e2d\u5f8b\uff0c\u7528\u4e8e\u9a8c\u8bc1\u6570\u5b66\u547d\u9898\u3002", "entity_id": "\u77db\u76fe\u5f8b\u548c\u6392\u4e2d\u5f8b", "entity_type": "UNKNOWN", "file_path": "ChuZhongShuXue.docx", "id": "\u77db\u76fe\u5f8b\u548c\u6392\u4e2d\u5f8b", "label": "\u77db\u76fe\u5f8b\u548c\u6392\u4e2d\u5f8b", "shape": "dot", "size": 10, "source_id": "chunk-5bc72352d3552516e17e5466f1f1fa7d", "title": "\u53cd\u8bc1\u6cd5\u7684\u903b\u8f91\u57fa\u7840\u662f\u77db\u76fe\u5f8b\u548c\u6392\u4e2d\u5f8b\uff0c\u7528\u4e8e\u9a8c\u8bc1\u6570\u5b66\u547d\u9898\u3002"}, {"color": "#f5cd1c", "created_at": 1752884035, "description": "**Prime numbers (\u7d20\u6570)** are natural numbers greater than 1 that can only be divided by 1 and themselves, making them the fundamental building blocks of natural numbers. They are a central object of study in number theory (\u6570\u8bba\u7814\u7a76\u7684\u6838\u5fc3\u5bf9\u8c61), with their properties often used to illustrate consistent or inconsistent logical reasoning in mathematics. Euclid famously proved, through contradiction, that prime numbers are infinite in number. This foundational characteristic underscores their importance in both theoretical and applied mathematical contexts. \n\nThe summary integrates all provided descriptions, resolving any potential contradictions by presenting a unified overview of prime numbers, their definition, mathematical significance, and historical proof of their infinitude.", "entity_id": "\u7d20\u6570", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u7d20\u6570", "label": "\u7d20\u6570", "shape": "dot", "size": 10, "source_id": "chunk-933b19628500cf80c1916250b497e3f8\u003cSEP\u003echunk-46deef01a2345f56bb0d9a61a58a7840\u003cSEP\u003echunk-10177ac2b1564cfa9472038f27fca84a\u003cSEP\u003echunk-896c6ab0b0daee896907a3349e5c7811", "title": "**Prime numbers (\u7d20\u6570)** are natural numbers greater than 1 that can only be divided by 1 and themselves, making them the fundamental building blocks of natural numbers. They are a central object of study in number theory (\u6570\u8bba\u7814\u7a76\u7684\u6838\u5fc3\u5bf9\u8c61), with their properties often used to illustrate consistent or inconsistent logical reasoning in mathematics. Euclid famously proved, through contradiction, that prime numbers are infinite in number. This foundational characteristic underscores their importance in both theoretical and applied mathematical contexts. \n\nThe summary integrates all provided descriptions, resolving any potential contradictions by presenting a unified overview of prime numbers, their definition, mathematical significance, and historical proof of their infinitude."}, {"color": "#dd9710", "created_at": 1752884044, "description": "Parallel line properties theorem, where proof by contradiction is introduced in China\u0027s compulsory education math curriculum.", "entity_id": "\u5e73\u884c\u7ebf\u6027\u8d28\u5b9a\u7406", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u5e73\u884c\u7ebf\u6027\u8d28\u5b9a\u7406", "label": "\u5e73\u884c\u7ebf\u6027\u8d28\u5b9a\u7406", "shape": "dot", "size": 10, "source_id": "chunk-10177ac2b1564cfa9472038f27fca84a", "title": "Parallel line properties theorem, where proof by contradiction is introduced in China\u0027s compulsory education math curriculum."}, {"color": "#00f849", "created_at": 1752884060, "description": "The mathematical law of excluded middle states that for any proposition P, either P or not P must hold in a given set.", "entity_id": "\u6570\u5b66\u6392\u4e2d\u5f8b", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u6570\u5b66\u6392\u4e2d\u5f8b", "label": "\u6570\u5b66\u6392\u4e2d\u5f8b", "shape": "dot", "size": 10, "source_id": "chunk-08c393bb4c84a0e60fb4eb6365edfe94", "title": "The mathematical law of excluded middle states that for any proposition P, either P or not P must hold in a given set."}, {"color": "#064888", "created_at": 1752884082, "description": "Rigid motion refers to transformations that preserve the shape and size of geometric figures, relevant to discussions of congruence in Euclid\u0027s axioms.", "entity_id": "\u521a\u4f53\u8fd0\u52a8", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u521a\u4f53\u8fd0\u52a8", "label": "\u521a\u4f53\u8fd0\u52a8", "shape": "dot", "size": 10, "source_id": "chunk-33e4d77aeae0267e846e8c7ab9af2b41", "title": "Rigid motion refers to transformations that preserve the shape and size of geometric figures, relevant to discussions of congruence in Euclid\u0027s axioms."}, {"color": "#040c10", "created_at": 1752884029, "description": "\u76f8\u4f3c\u56fe\u5f62\u7684\u672c\u8d28\u5728\u4e8e\u5bf9\u5e94\u89d2\u76f8\u7b49\u548c\u5bf9\u5e94\u8fb9\u6210\u6bd4\u4f8b\uff0c\u5e73\u884c\u7ebf\u5207\u5272\u5b9a\u7406\u662f\u5176\u57fa\u672c\u4e8b\u5b9e\u3002", "entity_id": "\u76f8\u4f3c\u56fe\u5f62", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u76f8\u4f3c\u56fe\u5f62", "label": "\u76f8\u4f3c\u56fe\u5f62", "shape": "dot", "size": 10, "source_id": "chunk-5bc72352d3552516e17e5466f1f1fa7d", "title": "\u76f8\u4f3c\u56fe\u5f62\u7684\u672c\u8d28\u5728\u4e8e\u5bf9\u5e94\u89d2\u76f8\u7b49\u548c\u5bf9\u5e94\u8fb9\u6210\u6bd4\u4f8b\uff0c\u5e73\u884c\u7ebf\u5207\u5272\u5b9a\u7406\u662f\u5176\u57fa\u672c\u4e8b\u5b9e\u3002"}, {"color": "#75bc17", "created_at": 1752884046, "description": "\u51e0\u4f55\u57fa\u672c\u5b9a\u7406\u4e4b\u4e00\uff0c\u63cf\u8ff0\u5e73\u884c\u7ebf\u622a\u53d6\u6bd4\u4f8b\u7ebf\u6bb5\u7684\u5173\u7cfb\uff0c\u5728\u76f8\u4f3c\u6027\u8bc1\u660e\u4e2d\u88ab\u5f15\u7528\u3002", "entity_id": "\u5e73\u884c\u7ebf\u5207\u5272\u5b9a\u7406", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u5e73\u884c\u7ebf\u5207\u5272\u5b9a\u7406", "label": "\u5e73\u884c\u7ebf\u5207\u5272\u5b9a\u7406", "shape": "dot", "size": 10, "source_id": "chunk-d2d1fa534f69129f1d307d4162d5e6e4", "title": "\u51e0\u4f55\u57fa\u672c\u5b9a\u7406\u4e4b\u4e00\uff0c\u63cf\u8ff0\u5e73\u884c\u7ebf\u622a\u53d6\u6bd4\u4f8b\u7ebf\u6bb5\u7684\u5173\u7cfb\uff0c\u5728\u76f8\u4f3c\u6027\u8bc1\u660e\u4e2d\u88ab\u5f15\u7528\u3002"}, {"color": "#ad7197", "created_at": 1752884138, "description": "Congruent figures are special cases of similar figures with similarity ratio equal to 1.", "entity_id": "\u5168\u7b49\u56fe\u5f62", "entity_type": "UNKNOWN", "file_path": "ChuZhongShuXue.docx", "id": "\u5168\u7b49\u56fe\u5f62", "label": "\u5168\u7b49\u56fe\u5f62", "shape": "dot", "size": 10, "source_id": "chunk-a48b8be14cbd9c4d38521dbf6a089e16", "title": "Congruent figures are special cases of similar figures with similarity ratio equal to 1."}, {"color": "#0ac27a", "created_at": 1752884029, "description": "**Trigonometric Functions (\u4e09\u89d2\u51fd\u6570)** \n\nTrigonometric functions are fundamental mathematical tools that convert geometric problems into algebraic equations. In junior high (\u521d\u4e2d), these functions primarily describe the relationships between the sides and angles of right-angled triangles, serving as expressions of geometric ratios rather than strictly defined mathematical functions mapping real numbers to real numbers. This is due to angle measurement limitations at this level, which prevent them from being fully formalized as functions. \n\nIn senior high (\u9ad8\u4e2d), trigonometric functions are properly defined using radians, resolving the earlier ambiguities and aligning them with the rigorous function concept\u2014mapping real numbers to real numbers. 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Simpson Diversity Index is another biodiversity metric assessing species distribution within ecosystems."}, {"color": "#627b5d", "created_at": 1752884050, "description": "**Mathematics (\u6570\u5b66)** is a theoretical discipline that emphasizes symbolic representation and formal proofs, serving as a tool for observing, analyzing, and expressing relationships in the real world through mathematical models. 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His legacy includes contributions to definition theory and category theory, though his syllogistic approach faced criticism from figures like Francis Bacon. \n\nAristotle\u0027s works were compiled and translated into Chinese, underscoring his enduring global influence. His interdisciplinary scholarship shaped the trajectory of Western thought across metaphysics, natural science, and epistemology."}, {"color": "#a77251", "created_at": 1752884030, "description": "**Irrational Numbers (\u65e0\u7406\u6570)** \n\nIrrational numbers are a fundamental mathematical concept representing real numbers that cannot be expressed as a simple fraction or ratio of two integers. Examples include well-known constants such as \u03c0 (pi) and \u221a2 (the square root of 2). 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Examples include well-known constants such as \u03c0 (pi) and \u221a2 (the square root of 2). These numbers were first discovered by the Pythagoreans, particularly in the context of geometric constructions, such as the diagonal of a unit square (with side length 1), which revealed the existence of incommensurable lengths\u2014quantities that could not be precisely measured using whole-number ratios. \n\nBeyond their historical significance, irrational numbers play a crucial role in completing the real number system, extending beyond the limitations of rational fractions. Due to their non-terminating and non-repeating decimal expansions, they often require approximation in practical calculations. Mathematically, they demand rigorous definitions and are frequently discussed in relation to algebraic and geometric properties, such as roots of non-perfect squares or transcendental numbers like \u03c0. \n\nThe concept of irrational numbers unifies diverse observations\u2014from early Greek geometric discoveries to modern real analysis\u2014providing a coherent framework for understanding numbers that defy simple fractional representation."}, {"color": "#5243f3", "created_at": 1752884062, "description": "Perfect numbers are natural numbers equal to the sum of their proper divisors. They are linked to Mersenne primes through a formula established by Euler.", "entity_id": "\u5b8c\u6ee1\u6570", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u5b8c\u6ee1\u6570", "label": "\u5b8c\u6ee1\u6570", "shape": "dot", "size": 10, "source_id": "chunk-896c6ab0b0daee896907a3349e5c7811", "title": "Perfect numbers are natural numbers equal to the sum of their proper divisors. They are linked to Mersenne primes through a formula established by Euler."}, {"color": "#97ef0c", "created_at": 1752884039, "description": "**\u5706\u5468\u7387 (Pi, \u03c0)** is a fundamental mathematical constant representing the ratio of a circle\u0027s circumference to its diameter, as well as the ratio of a circle\u0027s area to the square of its radius. It is an important irrational number with a rich historical background spanning millennia. \n\nAncient civilizations, including Egypt, used approximations of \u5706\u5468\u7387 for geometric calculations involving circles, though it was not explicitly defined at the time. Over the centuries, mathematicians have refined its precision, with notable contributions from **Zu Chongzhi**, who calculated it with remarkable accuracy. \n\nToday, \u03c0 remains a cornerstone in mathematics, physics, and engineering, symbolizing the intrinsic relationship between circular geometry and numerical constants. Its value has been approximated to billions of digits, reflecting both its mathematical significance and enduring fascination.", "entity_id": "\u5706\u5468\u7387", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u5706\u5468\u7387", "label": "\u5706\u5468\u7387", "shape": "dot", "size": 10, "source_id": "chunk-58d4c3c0d2ac14aff5d53121bb780c43\u003cSEP\u003echunk-75f67b5b9db86524b3fadd3f9f5a2715\u003cSEP\u003echunk-c30b1193cd236ff32c7a33020186a433\u003cSEP\u003echunk-aca1faae06ff2fb9d92a04ee4d07e4f7", "title": "**\u5706\u5468\u7387 (Pi, \u03c0)** is a fundamental mathematical constant representing the ratio of a circle\u0027s circumference to its diameter, as well as the ratio of a circle\u0027s area to the square of its radius. It is an important irrational number with a rich historical background spanning millennia. \n\nAncient civilizations, including Egypt, used approximations of \u5706\u5468\u7387 for geometric calculations involving circles, though it was not explicitly defined at the time. Over the centuries, mathematicians have refined its precision, with notable contributions from **Zu Chongzhi**, who calculated it with remarkable accuracy. \n\nToday, \u03c0 remains a cornerstone in mathematics, physics, and engineering, symbolizing the intrinsic relationship between circular geometry and numerical constants. Its value has been approximated to billions of digits, reflecting both its mathematical significance and enduring fascination."}, {"color": "#2b8db5", "created_at": 1752884030, "description": "A historical puzzle about traversing seven bridges without repetition, leading to the birth of graph theory and topology.\u003cSEP\u003e\u54e5\u5c3c\u65af\u5821\u4e03\u5ea7\u6865\u95ee\u9898\u662f\u62d3\u6251\u5b66\u7684\u53d1\u7aef\uff0c\u6d89\u53ca\u70b9\u4e0e\u7ebf\u7684\u7f51\u7edc\u5173\u7cfb\u3002\u003cSEP\u003e\u54e5\u5c3c\u65af\u5821\u4e03\u5ea7\u6865\u95ee\u9898\u662f\u6570\u5b66\u53f2\u4e0a\u7684\u8457\u540d\u95ee\u9898\uff0c\u63a2\u8ba8\u80fd\u5426\u4e0d\u91cd\u590d\u5730\u8d70\u904d\u4e03\u5ea7\u6865\uff0c\u6b27\u62c9\u901a\u8fc7\u62bd\u8c61\u5c06\u5176\u8f6c\u5316\u4e3a\u56fe\u8bba\u95ee\u9898\u3002", "entity_id": "\u54e5\u5c3c\u65af\u5821\u4e03\u5ea7\u6865\u95ee\u9898", "entity_type": "event", "file_path": "ChuZhongShuXue.docx", "id": "\u54e5\u5c3c\u65af\u5821\u4e03\u5ea7\u6865\u95ee\u9898", "label": "\u54e5\u5c3c\u65af\u5821\u4e03\u5ea7\u6865\u95ee\u9898", "shape": "dot", "size": 10, "source_id": "chunk-d79a25c9ff12f28251aed2da9132d65e\u003cSEP\u003echunk-5bc72352d3552516e17e5466f1f1fa7d\u003cSEP\u003echunk-a9f19b93d1f269d93ac2f8605d5d794b", "title": "A historical puzzle about traversing seven bridges without repetition, leading to the birth of graph theory and topology.\u003cSEP\u003e\u54e5\u5c3c\u65af\u5821\u4e03\u5ea7\u6865\u95ee\u9898\u662f\u62d3\u6251\u5b66\u7684\u53d1\u7aef\uff0c\u6d89\u53ca\u70b9\u4e0e\u7ebf\u7684\u7f51\u7edc\u5173\u7cfb\u3002\u003cSEP\u003e\u54e5\u5c3c\u65af\u5821\u4e03\u5ea7\u6865\u95ee\u9898\u662f\u6570\u5b66\u53f2\u4e0a\u7684\u8457\u540d\u95ee\u9898\uff0c\u63a2\u8ba8\u80fd\u5426\u4e0d\u91cd\u590d\u5730\u8d70\u904d\u4e03\u5ea7\u6865\uff0c\u6b27\u62c9\u901a\u8fc7\u62bd\u8c61\u5c06\u5176\u8f6c\u5316\u4e3a\u56fe\u8bba\u95ee\u9898\u3002"}, {"color": "#cf6f58", "created_at": 1752884043, "description": "Leonhard Euler (\u6b27\u62c9) was a renowned and prolific Swiss mathematician who made groundbreaking contributions across various fields of mathematics. In 1748, he published *Introductio in analysin infinitorum* (*Introduction to the Analysis of the Infinite*), where he established modern mathematical analysis as the science of functions, introduced the function notation *f(x)*, and standardized the modern polar coordinate system. Euler also played a key role in popularizing the use of the Greek letter *\u03c0* for the ratio of a circle\u0027s circumference to its diameter and named the natural logarithm base *e*. \n\nIn 1752, he articulated *Euler\u0027s formula*, which describes the relationship between the vertices, edges, and faces of polyhedra. Additionally, Euler proved significant connections between perfect numbers and Mersenne primes, identifying the 8th Mersenne prime. He also introduced the concept of transcendental numbers and conjectured that *\u03c0* is transcendental. \n\nEuler\u0027s work in graph theory was foundational, particularly his solution to the K\u00f6nigsberg bridge problem, which he abstracted into a graph theory problem, establishing fundamental principles for topology and network traversal (e.g., the conditions for traversing a graph in a single stroke). His contributions span multiple branches of mathematics, leaving a lasting legacy in analysis, number theory, topology, and beyond.", "entity_id": "\u6b27\u62c9", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u6b27\u62c9", "label": "\u6b27\u62c9", "shape": "dot", "size": 10, "source_id": "chunk-e310d39902c0e2129517ae5a93b19736\u003cSEP\u003echunk-d79a25c9ff12f28251aed2da9132d65e\u003cSEP\u003echunk-4dc0d317d51b12cedc9804d89a6f0d99\u003cSEP\u003echunk-a9f19b93d1f269d93ac2f8605d5d794b\u003cSEP\u003echunk-896c6ab0b0daee896907a3349e5c7811\u003cSEP\u003echunk-54a2947dce8bb8651e7af1f86e1e89a5\u003cSEP\u003echunk-89a00ca5535ebdb5d721e473eebbac4c\u003cSEP\u003echunk-3005fbbe56792da15c698f942a4b7305\u003cSEP\u003echunk-75f67b5b9db86524b3fadd3f9f5a2715\u003cSEP\u003echunk-b979930d4003be29c720b25f01498e92", "title": "Leonhard Euler (\u6b27\u62c9) was a renowned and prolific Swiss mathematician who made groundbreaking contributions across various fields of mathematics. In 1748, he published *Introductio in analysin infinitorum* (*Introduction to the Analysis of the Infinite*), where he established modern mathematical analysis as the science of functions, introduced the function notation *f(x)*, and standardized the modern polar coordinate system. Euler also played a key role in popularizing the use of the Greek letter *\u03c0* for the ratio of a circle\u0027s circumference to its diameter and named the natural logarithm base *e*. \n\nIn 1752, he articulated *Euler\u0027s formula*, which describes the relationship between the vertices, edges, and faces of polyhedra. Additionally, Euler proved significant connections between perfect numbers and Mersenne primes, identifying the 8th Mersenne prime. He also introduced the concept of transcendental numbers and conjectured that *\u03c0* is transcendental. \n\nEuler\u0027s work in graph theory was foundational, particularly his solution to the K\u00f6nigsberg bridge problem, which he abstracted into a graph theory problem, establishing fundamental principles for topology and network traversal (e.g., the conditions for traversing a graph in a single stroke). His contributions span multiple branches of mathematics, leaving a lasting legacy in analysis, number theory, topology, and beyond."}, {"color": "#ec429f", "created_at": 1752884049, "description": "K\u00f6nigsberg (now Kaliningrad) was the setting of the historical seven bridges problem, resolved by Euler.\u003cSEP\u003e\u54e5\u5c3c\u65af\u5821\u662f\u4e00\u4e2a\u5730\u7406\u4f4d\u7f6e\u7684\u540d\u79f0\uff0c\u4ee5\u5176\u4e03\u5ea7\u6865\u95ee\u9898\u800c\u95fb\u540d\uff0c\u8be5\u95ee\u9898\u88ab\u6b27\u62c9\u7528\u4e8e\u7814\u7a76\u7f51\u7edc\u7684\u4e00\u7b14\u753b\u95ee\u9898\u3002\u003cSEP\u003e\u54e5\u5c3c\u65af\u5821\u662f\u4e03\u5ea7\u6865\u95ee\u9898\u7684\u53d1\u6e90\u5730\uff0c\u5f53\u5730\u5c45\u6c11\u63d0\u51fa\u662f\u5426\u80fd\u5728\u4e00\u6b21\u6563\u6b65\u4e2d\u4e0d\u91cd\u590d\u5730\u8d70\u904d\u4e03\u5ea7\u6865\u7684\u95ee\u9898\u3002", "entity_id": "\u54e5\u5c3c\u65af\u5821", "entity_type": "geo", "file_path": "ChuZhongShuXue.docx", "id": "\u54e5\u5c3c\u65af\u5821", 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"description": "**\u5927\u6570\u636e (Big Data)** refers to extremely large and complex datasets that are characterized by high volume, variety, and velocity, posing challenges to traditional statistical and information science methods. 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He is celebrated for his work in number theory, including the formula for the sum of the first *n* natural numbers and his conjecture on the distribution of prime numbers, which laid the groundwork for the prime number theorem. Gauss also extended the fundamental theorem of arithmetic to complex numbers in his doctoral thesis, leading to the algebraic fundamental theorem. \n\nAs a professor at the University of G\u00f6ttingen, he influenced many later scholars, including David Hilbert. In geometry, Gauss introduced the concept of Gaussian curvature, a measure of a surface\u0027s intrinsic curvature, and developed a unified formula for calculating the sum of angles in a triangle. Additionally, he recognized the significance of studying positional geometry (*geometria situs*), inspiring his student Johann Benedict Listing to further develop the field of topology. \n\nGauss\u0027s legacy endures as one of history\u0027s greatest mathematicians, with his discoveries shaping modern mathematics, physics, and astronomy.", "entity_id": "\u9ad8\u65af", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u9ad8\u65af", "label": "\u9ad8\u65af", "shape": "dot", "size": 10, "source_id": "chunk-30c648659bc2a2999312fff54eb5d51a\u003cSEP\u003echunk-896c6ab0b0daee896907a3349e5c7811\u003cSEP\u003echunk-a9f19b93d1f269d93ac2f8605d5d794b\u003cSEP\u003echunk-33e4d77aeae0267e846e8c7ab9af2b41\u003cSEP\u003echunk-56d7d657ef131d893c933116865189b0\u003cSEP\u003echunk-f1e81718ff3ad9793c932f0681b6cfda", "title": "Carl Friedrich Gauss (\u9ad8\u65af), a renowned German mathematician, made profound contributions across multiple fields of mathematics and science. He is celebrated for his work in number theory, including the formula for the sum of the first *n* natural numbers and his conjecture on the distribution of prime numbers, which laid the groundwork for the prime number theorem. Gauss also extended the fundamental theorem of arithmetic to complex numbers in his doctoral thesis, leading to the algebraic fundamental theorem. \n\nAs a professor at the University of G\u00f6ttingen, he influenced many later scholars, including David Hilbert. In geometry, Gauss introduced the concept of Gaussian curvature, a measure of a surface\u0027s intrinsic curvature, and developed a unified formula for calculating the sum of angles in a triangle. Additionally, he recognized the significance of studying positional geometry (*geometria situs*), inspiring his student Johann Benedict Listing to further develop the field of topology. \n\nGauss\u0027s legacy endures as one of history\u0027s greatest mathematicians, with his discoveries shaping modern mathematics, physics, and astronomy."}, {"color": "#962b7c", "created_at": 1752884030, "description": "A triangular arrangement of binomial coefficients discovered by Yang Hui, used in combinatorics and probability.\u003cSEP\u003eYang Hui\u0027s triangle demonstrates binomial coefficients and combinatorial patterns.", "entity_id": "\u6768\u8f89\u4e09\u89d2\u5f62", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u6768\u8f89\u4e09\u89d2\u5f62", "label": "\u6768\u8f89\u4e09\u89d2\u5f62", "shape": "dot", "size": 10, "source_id": "chunk-edaeb942f2d7bc9b4f269a55a5bdd99f\u003cSEP\u003echunk-f1687ddd78928d213d5fd98272f5cfab", "title": "A triangular arrangement of binomial coefficients discovered by Yang Hui, used in combinatorics and probability.\u003cSEP\u003eYang Hui\u0027s triangle demonstrates binomial coefficients and combinatorial patterns."}, {"color": "#b938e8", "created_at": 1752884120, "description": "The triangle\u0027s patterns connect to rational approximations of irrational numbers.", "entity_id": "\u5706\u5468\u7387\u7684\u6709\u7406\u6570\u8868\u8fbe", "entity_type": "UNKNOWN", "file_path": "ChuZhongShuXue.docx", "id": "\u5706\u5468\u7387\u7684\u6709\u7406\u6570\u8868\u8fbe", "label": "\u5706\u5468\u7387\u7684\u6709\u7406\u6570\u8868\u8fbe", "shape": "dot", "size": 10, "source_id": "chunk-edaeb942f2d7bc9b4f269a55a5bdd99f", "title": "The triangle\u0027s patterns connect to rational approximations of irrational numbers."}, {"color": "#5ab4d9", "created_at": 1752884063, "description": "Chinese mathematician from the Song Dynasty who discovered the recursive method for binomial coefficients, known as Yang Hui\u0027s Triangle.\u003cSEP\u003e\u6768\u8f89is a mathematician from the Southern Song Dynasty known for the Yang Hui Triangle.", "entity_id": "\u6768\u8f89", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u6768\u8f89", "label": "\u6768\u8f89", "shape": "dot", "size": 10, "source_id": "chunk-4484c2c0903345428988e3c5a809c919\u003cSEP\u003echunk-f1687ddd78928d213d5fd98272f5cfab", "title": "Chinese mathematician from the Song Dynasty who discovered the recursive method for binomial coefficients, known as Yang Hui\u0027s Triangle.\u003cSEP\u003e\u6768\u8f89is a mathematician from the Southern Song Dynasty known for the Yang Hui Triangle."}, {"color": "#852334", "created_at": 1752884037, "description": "Ancient India is a historical civilization with significant contributions to mathematics, including the concept of zero.", "entity_id": "\u53e4\u5370\u5ea6", "entity_type": "geo", "file_path": "ChuZhongShuXue.docx", "id": "\u53e4\u5370\u5ea6", "label": "\u53e4\u5370\u5ea6", "shape": "dot", "size": 10, "source_id": "chunk-c38ffe600259aeb39e4227f260864859", "title": "Ancient India is a historical civilization with significant contributions to mathematics, including the concept of zero."}, {"color": "#f9c67e", "created_at": 1752884066, "description": "**Algebraic Numbers (\u4ee3\u6570\u6570)** \n\nAlgebraic numbers are real or complex numbers that serve as roots of non-zero polynomial equations with integer (or equivalently, rational) coefficients. They include all rational numbers, as well as certain irrational numbers such as \u221a2 and the golden ratio, which can be expressed as solutions to polynomial equations (e.g., \\( x^2 - 2 = 0 \\)). \n\nA key distinction in number theory is that algebraic numbers are contrasted with transcendental numbers (such as \u03c0 and e), which cannot be roots of such polynomials. This classification was notably emphasized by mathematicians like Euler. \n\nThe set of algebraic numbers is countably infinite, meaning its cardinality is the same as that of the natural numbers\u2014a significant result proven by Georg Cantor. While they encompass many irrational numbers, not all irrationals are algebraic; transcendental numbers form a separate, uncountable class. \n\nIn summary, algebraic numbers are a fundamental concept in mathematics, bridging algebra and number theory, with properties that distinguish them from both rational and transcendental numbers.", "entity_id": "\u4ee3\u6570\u6570", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u4ee3\u6570\u6570", "label": "\u4ee3\u6570\u6570", "shape": "dot", "size": 10, "source_id": "chunk-aca1faae06ff2fb9d92a04ee4d07e4f7\u003cSEP\u003echunk-4dc0d317d51b12cedc9804d89a6f0d99\u003cSEP\u003echunk-8b6f8917e6704256d88d931fd2d99d07\u003cSEP\u003echunk-a2266049e857fa018842eec4ce9a2e9d\u003cSEP\u003echunk-75f67b5b9db86524b3fadd3f9f5a2715", "title": "**Algebraic Numbers (\u4ee3\u6570\u6570)** \n\nAlgebraic numbers are real or complex numbers that serve as roots of non-zero polynomial equations with integer (or equivalently, rational) coefficients. They include all rational numbers, as well as certain irrational numbers such as \u221a2 and the golden ratio, which can be expressed as solutions to polynomial equations (e.g., \\( x^2 - 2 = 0 \\)). \n\nA key distinction in number theory is that algebraic numbers are contrasted with transcendental numbers (such as \u03c0 and e), which cannot be roots of such polynomials. This classification was notably emphasized by mathematicians like Euler. \n\nThe set of algebraic numbers is countably infinite, meaning its cardinality is the same as that of the natural numbers\u2014a significant result proven by Georg Cantor. While they encompass many irrational numbers, not all irrationals are algebraic; transcendental numbers form a separate, uncountable class. \n\nIn summary, algebraic numbers are a fundamental concept in mathematics, bridging algebra and number theory, with properties that distinguish them from both rational and transcendental numbers."}, {"color": "#9a6e04", "created_at": 1752884066, "description": "\u67f3\u7ef4\u5c14(Liouville) proved the existence of transcendental numbers using constructive methods in 1844.", "entity_id": "\u67f3\u7ef4\u5c14", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u67f3\u7ef4\u5c14", "label": "\u67f3\u7ef4\u5c14", "shape": "dot", "size": 10, "source_id": "chunk-4dc0d317d51b12cedc9804d89a6f0d99", "title": "\u67f3\u7ef4\u5c14(Liouville) proved the existence of transcendental numbers using constructive methods in 1844."}, {"color": "#5eafc3", "created_at": 1752884067, "description": "Formalized in 1872 by Cantor and Dedekind, resolving continuity and\u8fd0\u7b97issues.", "entity_id": "\u5b9e\u6570\u7406\u8bba", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u5b9e\u6570\u7406\u8bba", "label": "\u5b9e\u6570\u7406\u8bba", "shape": "dot", "size": 10, "source_id": "chunk-4dc0d317d51b12cedc9804d89a6f0d99", "title": "Formalized in 1872 by Cantor and Dedekind, resolving continuity and\u8fd0\u7b97issues."}, {"color": "#5297de", "created_at": 1752884067, "description": "Ancient civilization that used integers and fractions empirically before formal mathematical rigor.", "entity_id": "\u57c3\u53ca\u4eba", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u57c3\u53ca\u4eba", "label": "\u57c3\u53ca\u4eba", "shape": "dot", "size": 10, "source_id": "chunk-4dc0d317d51b12cedc9804d89a6f0d99", "title": "Ancient civilization that used integers and fractions empirically before formal mathematical rigor."}, {"color": "#7fbf37", "created_at": 1752884067, "description": "Finite decimals are used to approximate irrational numbers in real-life calculations.", "entity_id": "\u6709\u9650\u5c0f\u6570", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u6709\u9650\u5c0f\u6570", "label": "\u6709\u9650\u5c0f\u6570", "shape": "dot", "size": 10, "source_id": "chunk-89a00ca5535ebdb5d721e473eebbac4c", "title": "Finite decimals are used to approximate irrational numbers in real-life calculations."}, {"color": "#64e580", "created_at": 1752884067, "description": "Georg Cantor was a German mathematician who contributed to the theory of irrational numbers and real numbers.", "entity_id": "\u5fb7\u56fd\u6570\u5b66\u5bb6\u5eb7\u6258", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u5fb7\u56fd\u6570\u5b66\u5bb6\u5eb7\u6258", "label": "\u5fb7\u56fd\u6570\u5b66\u5bb6\u5eb7\u6258", "shape": "dot", "size": 10, "source_id": "chunk-89a00ca5535ebdb5d721e473eebbac4c", "title": "Georg Cantor was a German mathematician who contributed to the theory of irrational numbers and real numbers."}, {"color": "#374a4d", "created_at": 1752884037, "description": "**Ancient Egypt (\u53e4\u57c3\u53ca)** was a historical civilization located in arid northern Africa, profoundly influenced by the Nile River\u0027s annual flooding, which shaped its agriculture and spurred the development of practical geometry. This ancient civilization made significant contributions to mathematics, including the use of fractions, the approximation of irrational numbers in practical calculations, and the independent development of geometric principles. Notably, Ancient Egypt applied Pythagorean triples in the construction of its iconic pyramids. \n\nThe civilization also excelled in astronomical knowledge, creating the Sothic (Sirius-based) calendar and mastering a precise solar calendar, with calculations of the annual cycle aligning with those of ancient China. Its advancements in timekeeping were closely tied to the Nile\u0027s flooding cycles, which were central to its agricultural and societal organization. \n\nOverall, Ancient Egypt was a pioneering civilization that combined practical mathematical and geometric innovations with astronomical and calendrical expertise, all deeply rooted in its environmental and cultural context.", "entity_id": "\u53e4\u57c3\u53ca", "entity_type": "geo", "file_path": "ChuZhongShuXue.docx", "id": "\u53e4\u57c3\u53ca", "label": "\u53e4\u57c3\u53ca", "shape": "dot", "size": 10, "source_id": "chunk-fcc8429be9c152ab7fd3a6e6daac3659\u003cSEP\u003echunk-89a00ca5535ebdb5d721e473eebbac4c\u003cSEP\u003echunk-c30b1193cd236ff32c7a33020186a433\u003cSEP\u003echunk-72fa7859d7da3a73a7558cc01fd39ae3\u003cSEP\u003echunk-fa7a7a68a2388eb9ade79559ac5334d0\u003cSEP\u003echunk-c38ffe600259aeb39e4227f260864859\u003cSEP\u003echunk-b86ecc17467c79a88d8dfa681dcdb3ec", "title": "**Ancient Egypt (\u53e4\u57c3\u53ca)** was a historical civilization located in arid northern Africa, profoundly influenced by the Nile River\u0027s annual flooding, which shaped its agriculture and spurred the development of practical geometry. This ancient civilization made significant contributions to mathematics, including the use of fractions, the approximation of irrational numbers in practical calculations, and the independent development of geometric principles. Notably, Ancient Egypt applied Pythagorean triples in the construction of its iconic pyramids. \n\nThe civilization also excelled in astronomical knowledge, creating the Sothic (Sirius-based) calendar and mastering a precise solar calendar, with calculations of the annual cycle aligning with those of ancient China. Its advancements in timekeeping were closely tied to the Nile\u0027s flooding cycles, which were central to its agricultural and societal organization. \n\nOverall, Ancient Egypt was a pioneering civilization that combined practical mathematical and geometric innovations with astronomical and calendrical expertise, all deeply rooted in its environmental and cultural context."}, {"color": "#0ba3f2", "created_at": 1752884038, "description": "Ancient China is referenced as an early civilization that approximated irrational numbers in practical calculations.\u003cSEP\u003e\u53d1\u5c55\u51fa\u72ec\u7279\u7684\u5206\u6570\u7cfb\u7edf\u7528\u4e8e\u5929\u6587\u5386\u6cd5\u8ba1\u7b97", "entity_id": "\u4e2d\u56fd\u53e4\u4ee3", "entity_type": "geo", "file_path": "ChuZhongShuXue.docx", "id": "\u4e2d\u56fd\u53e4\u4ee3", "label": "\u4e2d\u56fd\u53e4\u4ee3", "shape": "dot", "size": 10, "source_id": "chunk-89a00ca5535ebdb5d721e473eebbac4c\u003cSEP\u003echunk-121f6aa86f389551192a9dd60f07ea42", "title": 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As a university press, it specializes in publishing a wide range of scholarly and educational materials, including translations of influential works such as *Hilbert\u0027s Foundations of Geometry*, *Descartes\u0027 Geometry*, and *The Thirteen Classics Annotated\u00b7Commentary on the Book of Documents*. Affiliated with Peking University, the press is recognized for its contributions to academic publishing in China, producing high-quality works that support both domestic and international scholarly communities. 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It is renowned for publishing high-profile works such as the Chinese translation of *Science and Civilisation in China* (*\u300a\u4e2d\u56fd\u79d1\u5b66\u6280\u672f\u53f2\u300b*) and *History of Mathematical Symbols*, as well as the Chinese version of David Hilbert\u0027s *Foundations of Geometry*. As a prominent publisher in China, \u79d1\u5b66\u51fa\u7248\u793e focuses on disseminating authoritative scientific knowledge and academic resources, contributing significantly to the advancement of science and technology in the region. 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In modern usage, \"\u4e2d\u56fd\" refers to the nation of China, with its name preserving this historical and cultural significance. \n\nThis summary integrates China\u2019s mathematical heritage, astronomical associations, and the etymology of its name into a coherent overview, resolving any apparent contradictions by contextualizing the diverse descriptions within its long and continuous historical narrative.", "entity_id": "\u4e2d\u56fd", "entity_type": "geo", "file_path": "ChuZhongShuXue.docx", "id": "\u4e2d\u56fd", "label": "\u4e2d\u56fd", "shape": "dot", "size": 10, "source_id": "chunk-c64e6f44879da2fe257177d9e60e88f1\u003cSEP\u003echunk-3005fbbe56792da15c698f942a4b7305\u003cSEP\u003echunk-f65f3e755944ffb1fd1280feebd5aa52\u003cSEP\u003echunk-fa7a7a68a2388eb9ade79559ac5334d0\u003cSEP\u003echunk-c38ffe600259aeb39e4227f260864859\u003cSEP\u003echunk-75f67b5b9db86524b3fadd3f9f5a2715", "title": "China (\u4e2d\u56fd) is a country with a profound historical and cultural legacy, particularly in the fields of mathematics, astronomy, and early scientific advancements. As an ancient civilization, it independently discovered trigonometric relationships for practical measurements and made significant contributions to mathematics, including the early use of fractions and decimals. China is the geographic origin of renowned mathematician Zu Chongzhi\u2019s work and classical texts such as the *Zhou Bi Suan Jing*, which reflects its deep engagement with astronomical and mathematical knowledge. \n\nThe name \"\u4e2d\u56fd\" (Zh\u014dnggu\u00f3), meaning \"Middle Kingdom\" or \"Central State,\" has historical roots tied to the concept of being the \"center of the world\" (\u5929\u4e0b\u4e4b\u4e2d) in ancient Chinese cosmology. Its earliest recorded use appears on the He Zun vessel from the Western Zhou Dynasty (c. 11th\u20138th century BCE), linking the name to both astronomical observations and early state identity. In modern usage, \"\u4e2d\u56fd\" refers to the nation of China, with its name preserving this historical and cultural significance. \n\nThis summary integrates China\u2019s mathematical heritage, astronomical associations, and the etymology of its name into a coherent overview, resolving any apparent contradictions by contextualizing the diverse descriptions within its long and continuous historical narrative."}, {"color": "#79cbe8", "created_at": 1752884038, "description": "Unit fraction representation expresses fractions as sums of distinct unit fractions.", "entity_id": "\u5355\u5206\u6570\u8868\u8fbe", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u5355\u5206\u6570\u8868\u8fbe", "label": "\u5355\u5206\u6570\u8868\u8fbe", "shape": "dot", "size": 10, "source_id": "chunk-c38ffe600259aeb39e4227f260864859", "title": "Unit fraction representation expresses fractions as sums of distinct unit fractions."}, {"color": "#2ea073", "created_at": 1752884038, "description": "\u5f3a\u8c03\u5206\u6570\u8ba1\u91cf\u5355\u4f4d\u8868\u8fbe\u7684\u6559\u5b66\u65b9\u6cd5", "entity_id": "\u65e5\u672c\u6570\u5b66\u6559\u80b2", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u65e5\u672c\u6570\u5b66\u6559\u80b2", "label": "\u65e5\u672c\u6570\u5b66\u6559\u80b2", "shape": "dot", "size": 10, "source_id": "chunk-121f6aa86f389551192a9dd60f07ea42", "title": "\u5f3a\u8c03\u5206\u6570\u8ba1\u91cf\u5355\u4f4d\u8868\u8fbe\u7684\u6559\u5b66\u65b9\u6cd5"}, {"color": "#5a1c01", "created_at": 1752884040, "description": "\u9664\u6cd5\u662f\u6570\u5b66\u57fa\u672c\u8fd0\u7b97\u4e4b\u4e00\uff0c\u53ef\u4ee5\u7528\u5206\u6570\u5f62\u5f0f\u8868\u793a\u5176\u8fd0\u7b97\u8fc7\u7a0b\u548c\u7ed3\u679c\u3002", "entity_id": "\u9664\u6cd5", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u9664\u6cd5", "label": "\u9664\u6cd5", "shape": "dot", "size": 10, "source_id": "chunk-2820dfe8496edbd1e66a24c26cdd1859", "title": "\u9664\u6cd5\u662f\u6570\u5b66\u57fa\u672c\u8fd0\u7b97\u4e4b\u4e00\uff0c\u53ef\u4ee5\u7528\u5206\u6570\u5f62\u5f0f\u8868\u793a\u5176\u8fd0\u7b97\u8fc7\u7a0b\u548c\u7ed3\u679c\u3002"}, {"color": "#2177a2", "created_at": 1752884040, "description": "\u7b49\u5f0f\u4fdd\u6301\u76f8\u7b49\u7684\u57fa\u672c\u6027\u8d28\uff0c\u7528\u4e8e\u8bc1\u660e\u5206\u6570\u4e0e\u9664\u6cd5\u7684\u7b49\u4ef7\u6027\u3002", "entity_id": "\u7b49\u5f0f\u7684\u57fa\u672c\u6027\u8d28", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u7b49\u5f0f\u7684\u57fa\u672c\u6027\u8d28", "label": "\u7b49\u5f0f\u7684\u57fa\u672c\u6027\u8d28", "shape": "dot", "size": 10, "source_id": "chunk-2820dfe8496edbd1e66a24c26cdd1859", "title": "\u7b49\u5f0f\u4fdd\u6301\u76f8\u7b49\u7684\u57fa\u672c\u6027\u8d28\uff0c\u7528\u4e8e\u8bc1\u660e\u5206\u6570\u4e0e\u9664\u6cd5\u7684\u7b49\u4ef7\u6027\u3002"}, {"color": "#1aa038", "created_at": 1752884040, "description": "\u6570\u5b66\u57fa\u672c\u516c\u7406\uff0c\u7528\u4e8e\u63a8\u5bfc\u5206\u6570\u4e0e\u9664\u6cd5\u7684\u7b49\u4ef7\u5173\u7cfb\u3002", "entity_id": "\u7b49\u91cf\u7684\u7b49\u91cf\u76f8\u7b49", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u7b49\u91cf\u7684\u7b49\u91cf\u76f8\u7b49", "label": "\u7b49\u91cf\u7684\u7b49\u91cf\u76f8\u7b49", "shape": "dot", "size": 10, "source_id": "chunk-2820dfe8496edbd1e66a24c26cdd1859", "title": "\u6570\u5b66\u57fa\u672c\u516c\u7406\uff0c\u7528\u4e8e\u63a8\u5bfc\u5206\u6570\u4e0e\u9664\u6cd5\u7684\u7b49\u4ef7\u5173\u7cfb\u3002"}, {"color": "#8c17ae", "created_at": 1752884037, "description": "Ptolemy (\u6258\u52d2\u5bc6) was an ancient Greek scholar renowned for his contributions to astronomy, mathematics, and geography. As a prominent figure associated with the Library of Alexandria, he authored the influential work *Almagest*, in which he divided the circle into 360 degrees and established the 60-based system of minutes and seconds, a foundational concept that continues to influence modern timekeeping and angular measurement. While he employed the sexagesimal (60-base) system for fractional parts of angles, he did not extend this system to integer values. \n\nIn addition to his astronomical achievements, Ptolemy made significant advancements in geography by designing a system of curved latitude and longitude lines, which improved mapmaking techniques. His multidisciplinary work bridged the fields of astronomy, mathematics, and cartography, leaving a lasting legacy in both ancient and modern scientific traditions. \n\n(Note: Contradictions regarding the use of the 60-base system were resolved by clarifying its application to fractional rather than integer values.)", "entity_id": "\u6258\u52d2\u5bc6", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u6258\u52d2\u5bc6", "label": "\u6258\u52d2\u5bc6", "shape": "dot", "size": 10, "source_id": "chunk-fcc8429be9c152ab7fd3a6e6daac3659\u003cSEP\u003echunk-577ed72eade6a6cbd00f14e981bf4078\u003cSEP\u003echunk-6782332859e0bdef19c72a7367743352\u003cSEP\u003echunk-c38ffe600259aeb39e4227f260864859\u003cSEP\u003echunk-121f6aa86f389551192a9dd60f07ea42", "title": "Ptolemy (\u6258\u52d2\u5bc6) was an ancient Greek scholar renowned for his contributions to astronomy, mathematics, and geography. As a prominent figure associated with the Library of Alexandria, he authored the influential work *Almagest*, in which he divided the circle into 360 degrees and established the 60-based system of minutes and seconds, a foundational concept that continues to influence modern timekeeping and angular measurement. While he employed the sexagesimal (60-base) system for fractional parts of angles, he did not extend this system to integer values. \n\nIn addition to his astronomical achievements, Ptolemy made significant advancements in geography by designing a system of curved latitude and longitude lines, which improved mapmaking techniques. His multidisciplinary work bridged the fields of astronomy, mathematics, and cartography, leaving a lasting legacy in both ancient and modern scientific traditions. \n\n(Note: Contradictions regarding the use of the 60-base system were resolved by clarifying its application to fractional rather than integer values.)"}, {"color": "#26339a", "created_at": 1752884042, "description": "Liu Hui (\u7ea6225-295) was a great mathematician of the Wei-Jin period, from Linzi, Shandong.\u003cSEP\u003eLiu Hui is a historical figure and mathematician, known for his commentary on \u0027The Nine Chapters on the Mathematical Art\u0027.\u003cSEP\u003e\u5218\u5fbd\u662f\u6c49\u4ee3\u6570\u5b66\u5bb6\uff0c\u66fe\u6ce8\u91ca\u300a\u4e5d\u7ae0\u7b97\u672f\u300b\uff0c\u5e76\u5728\u516c\u5143263\u5e74\u5de6\u53f3\u4e3a\u5176\u8bbe\u7f6e\u0027\u65b9\u7a0b\u0027\u7bc7\uff0c\u5bf9\u53e4\u4ee3\u6570\u5b66\u7684\u53d1\u5c55\u6709\u91cd\u8981\u8d21\u732e\u3002", "entity_id": "\u5218\u5fbd", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u5218\u5fbd", "label": "\u5218\u5fbd", "shape": "dot", "size": 10, "source_id": "chunk-bc737ed018919d513c346a3c74b449aa\u003cSEP\u003echunk-7f35e11154694cb00fbc4308a33a36a3\u003cSEP\u003echunk-8536b48102e07677da2af463a726f4c6", "title": "Liu Hui (\u7ea6225-295) was a great mathematician of the Wei-Jin period, from Linzi, Shandong.\u003cSEP\u003eLiu Hui is a historical figure and mathematician, known for his commentary on \u0027The Nine Chapters on the Mathematical Art\u0027.\u003cSEP\u003e\u5218\u5fbd\u662f\u6c49\u4ee3\u6570\u5b66\u5bb6\uff0c\u66fe\u6ce8\u91ca\u300a\u4e5d\u7ae0\u7b97\u672f\u300b\uff0c\u5e76\u5728\u516c\u5143263\u5e74\u5de6\u53f3\u4e3a\u5176\u8bbe\u7f6e\u0027\u65b9\u7a0b\u0027\u7bc7\uff0c\u5bf9\u53e4\u4ee3\u6570\u5b66\u7684\u53d1\u5c55\u6709\u91cd\u8981\u8d21\u732e\u3002"}, {"color": "#34c3e9", "created_at": 1752884067, "description": "Discriminant (b\u00b2-4ac) determining nature of roots in quadratic equations.", "entity_id": "\u5224\u522b\u5f0f", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u5224\u522b\u5f0f", "label": "\u5224\u522b\u5f0f", "shape": "dot", "size": 10, "source_id": "chunk-a2266049e857fa018842eec4ce9a2e9d", "title": "Discriminant (b\u00b2-4ac) determining nature of roots in quadratic equations."}, {"color": "#711edb", "created_at": 1752884064, "description": "The Nile River was central to ancient Egyptian civilization, with its annual flooding influencing agricultural and calendar systems.\u003cSEP\u003e\u5c3c\u7f57\u6cb3(Nile River) flooded annually, transforming Egypt\u0027s landscape and necessitating land measurement techniques, which led to geometry\u0027s invention.", "entity_id": "\u5c3c\u7f57\u6cb3", "entity_type": "geo", "file_path": "ChuZhongShuXue.docx", "id": "\u5c3c\u7f57\u6cb3", "label": "\u5c3c\u7f57\u6cb3", "shape": "dot", "size": 10, "source_id": "chunk-c30b1193cd236ff32c7a33020186a433\u003cSEP\u003echunk-72fa7859d7da3a73a7558cc01fd39ae3", "title": "The Nile River was central to ancient Egyptian civilization, with its annual flooding influencing agricultural and calendar systems.\u003cSEP\u003e\u5c3c\u7f57\u6cb3(Nile River) flooded annually, transforming Egypt\u0027s landscape and necessitating land measurement techniques, which led to geometry\u0027s invention."}, {"color": "#8e7775", "created_at": 1752884047, "description": "**Herodotus (\u5e0c\u7f57\u591a\u5fb7)** was an ancient Greek historian best known for his seminal work *Histories* (*\u5386\u53f2*), which documented a wide range of historical, cultural, and geographical observations from the ancient world. He is particularly noted for his descriptions of the **Seven Wonders**, as well as detailed accounts of regions such as **Babylon** and **Egypt**. In his writings on Egypt, Herodotus explored the significance of the Nile floods and their influence on the development of geometry. While his works provide invaluable insights into antiquity, some of his accounts, such as those on Babylon, have been questioned for their accuracy. Nevertheless, Herodotus remains a foundational figure in historiography, often referred to as the \"Father of History\" for his pioneering approach to recording historical events.", "entity_id": "\u5e0c\u7f57\u591a\u5fb7", "entity_type": "person", "file_path": "ChuZhongShuXue.docx", "id": "\u5e0c\u7f57\u591a\u5fb7", "label": "\u5e0c\u7f57\u591a\u5fb7", "shape": "dot", "size": 10, "source_id": "chunk-5bfa7384dc42f0162f843b05c24e2ea8\u003cSEP\u003echunk-fa7a7a68a2388eb9ade79559ac5334d0\u003cSEP\u003echunk-8eefc105e8cce456308ce1260d081389\u003cSEP\u003echunk-c30b1193cd236ff32c7a33020186a433", "title": "**Herodotus (\u5e0c\u7f57\u591a\u5fb7)** was an ancient Greek historian best known for his seminal work *Histories* (*\u5386\u53f2*), which documented a wide range of historical, cultural, and geographical observations from the ancient world. He is particularly noted for his descriptions of the **Seven Wonders**, as well as detailed accounts of regions such as **Babylon** and **Egypt**. In his writings on Egypt, Herodotus explored the significance of the Nile floods and their influence on the development of geometry. While his works provide invaluable insights into antiquity, some of his accounts, such as those on Babylon, have been questioned for their accuracy. Nevertheless, Herodotus remains a foundational figure in historiography, often referred to as the \"Father of History\" for his pioneering approach to recording historical events."}, {"color": "#30e2e0", "created_at": 1752884070, "description": "Sirius (\u5929\u72fc\u661f) is the brightest star in the night sky, used by ancient Egyptians to create their Sothic calendar.", "entity_id": "\u5929\u72fc\u661f", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u5929\u72fc\u661f", "label": "\u5929\u72fc\u661f", "shape": "dot", "size": 10, "source_id": "chunk-72fa7859d7da3a73a7558cc01fd39ae3", "title": "Sirius (\u5929\u72fc\u661f) is the brightest star in the night sky, used by ancient Egyptians to create their Sothic calendar."}, {"color": "#bf6896", "created_at": 1752884071, "description": "The heliacal rising of Sirius, marking the Egyptian New Year and Nile flood season.", "entity_id": "\u5055\u65e5\u5347", "entity_type": "event", "file_path": "ChuZhongShuXue.docx", "id": "\u5055\u65e5\u5347", "label": "\u5055\u65e5\u5347", "shape": "dot", "size": 10, "source_id": "chunk-72fa7859d7da3a73a7558cc01fd39ae3", "title": "The heliacal rising of Sirius, marking the Egyptian New Year and Nile flood season."}, {"color": "#af04e6", "created_at": 1752884070, "description": "**The 24 Solar Terms (\u4e8c\u5341\u56db\u8282\u6c14)** are a traditional Chinese lunisolar calendar system developed in ancient China based on solar observations. This system divides the year into 24 distinct periods, each reflecting seasonal changes and natural phenomena. Primarily used for agricultural planning, the 24 Solar Terms guide farming activities, such as planting and harvesting, by aligning them with optimal climatic conditions. Additionally, they serve as a framework for organizing daily life and cultural practices in accordance with the natural cycle. The terms are deeply rooted in Chinese history and remain significant in both agricultural and cultural contexts today. \n\nThe descriptions emphasize their dual role: as a scientific timekeeping system tied to the solar calendar and as a practical tool for synchronizing human activities with seasonal rhythms. While some descriptions highlight their agricultural focus, others note their broader application in daily life, reinforcing their enduring cultural and functional importance. Contradictions are minimal, with all sources agreeing on their origin, purpose, and solar-based structure.", "entity_id": "\u4e8c\u5341\u56db\u8282\u6c14", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u4e8c\u5341\u56db\u8282\u6c14", "label": "\u4e8c\u5341\u56db\u8282\u6c14", "shape": "dot", "size": 10, "source_id": "chunk-c64e6f44879da2fe257177d9e60e88f1\u003cSEP\u003echunk-fcc8429be9c152ab7fd3a6e6daac3659\u003cSEP\u003echunk-72fa7859d7da3a73a7558cc01fd39ae3\u003cSEP\u003echunk-67ccc979a5e11b9d77c4b239a09538c8", "title": "**The 24 Solar Terms (\u4e8c\u5341\u56db\u8282\u6c14)** are a traditional Chinese lunisolar calendar system developed in ancient China based on solar observations. This system divides the year into 24 distinct periods, each reflecting seasonal changes and natural phenomena. Primarily used for agricultural planning, the 24 Solar Terms guide farming activities, such as planting and harvesting, by aligning them with optimal climatic conditions. Additionally, they serve as a framework for organizing daily life and cultural practices in accordance with the natural cycle. The terms are deeply rooted in Chinese history and remain significant in both agricultural and cultural contexts today. \n\nThe descriptions emphasize their dual role: as a scientific timekeeping system tied to the solar calendar and as a practical tool for synchronizing human activities with seasonal rhythms. While some descriptions highlight their agricultural focus, others note their broader application in daily life, reinforcing their enduring cultural and functional importance. Contradictions are minimal, with all sources agreeing on their origin, purpose, and solar-based structure."}, {"color": "#c79f5d", "created_at": 1752884073, "description": "An Egyptian mathematical papyrus containing Pythagorean problems.\u003cSEP\u003e\u53e4\u57c3\u53ca\u6570\u5b66\u6587\u732e", "entity_id": "\u5361\u547c\u6069\u7eb8\u8349\u4e66", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u5361\u547c\u6069\u7eb8\u8349\u4e66", "label": "\u5361\u547c\u6069\u7eb8\u8349\u4e66", "shape": "dot", "size": 10, "source_id": "chunk-b86ecc17467c79a88d8dfa681dcdb3ec\u003cSEP\u003echunk-b459f08681c0cd66ebd633869b2b1be7", "title": "An Egyptian mathematical papyrus containing Pythagorean problems.\u003cSEP\u003e\u53e4\u57c3\u53ca\u6570\u5b66\u6587\u732e"}, {"color": "#b4ec75", "created_at": 1752884038, "description": "A famous Chinese mathematician and astronomer who calculated the value of pi with high precision.\u003cSEP\u003eZu Chongzhi was a Chinese mathematician and 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"\u6cb3\u5357\u7701\u767b\u5c01\u53bf\u544a\u6210\u9547is believed to be the ancient \u0027center of the world\u0027 as determined by Zhou Dynasty astronomers using gnomon measurements."}, {"color": "#237b95", "created_at": 1752884075, "description": "A location in China at approximately 34\u00b0N latitude and 113\u00b0E longitude, used as a reference point in Zhou Dynasty geographical calculations.\u003cSEP\u003eDengfeng Gaocheng is a location in China with a latitude of 34\u00b023\u2032 north, historically significant for astronomical observations.\u003cSEP\u003e\u767b\u5c01\u544a\u6210is a location in China, referenced for its historical significance in geographical measurements and the determination of the \u0027center of the world.\u0027", "entity_id": "\u767b\u5c01\u544a\u6210", "entity_type": "geo", "file_path": "ChuZhongShuXue.docx", "id": "\u767b\u5c01\u544a\u6210", "label": "\u767b\u5c01\u544a\u6210", "shape": "dot", "size": 10, "source_id": 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figure in ancient Chinese culture, revered as one of the foundational ancestors of Chinese civilization. He is credited with significant contributions to astronomy and calendrical systems, particularly the establishment of celestial measurements and the **Zhou Tian calendar system** (\u5468\u5929\u5386\u5ea6). Additionally, he is often associated with the origins of the **Gai Tian theory** (\u76d6\u5929\u8bf4), an early Chinese cosmological model. Widely regarded as a cultural and intellectual pioneer, \u4f0f\u7fb2 is celebrated as a key figure in the development of early Chinese knowledge, particularly in the realms of astronomy, calendrics, and the broader foundations of Chinese civilization. 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Widely regarded as a cultural and intellectual pioneer, \u4f0f\u7fb2 is celebrated as a key figure in the development of early Chinese knowledge, particularly in the realms of astronomy, calendrics, and the broader foundations of Chinese civilization. His legacy endures as a symbol of wisdom and innovation in Chinese mythology and history."}, {"color": "#192cb8", "created_at": 1752884072, "description": "**\u5468\u5929\u5386\u5ea6 (Zh\u014du Ti\u0101n L\u00ec D\u00f9)** is an ancient Chinese astronomical concept and celestial measurement system discussed in the *\u5468\u9ac0\u7b97\u7ecf (Zhoubi Suanjing)*. It refers to the period during which the stars complete a full rotation around the North Star (Polaris), equivalent to the number of days in a solar year (approximately 365.25 days). This concept was foundational in early Chinese astronomy, illustrating the cyclical nature of celestial movements. \n\nTraditionally, **\u5468\u5929\u5386\u5ea6** is attributed to **\u5e96\u727a\u6c0f (Fuxi)**, a mythical figure in Chinese culture, and was preserved and developed through the Shang and Zhou dynasties. While some descriptions emphasize its role as a measurement system, others highlight its conceptual significance in understanding stellar cycles. Despite slight variations in phrasing, the core idea remains consistent: it represents the celestial rotation period tied to the solar year, reflecting the integration of astronomy and calendrics in ancient China. \n\nThe system underscores the early Chinese emphasis on celestial observation and its application in timekeeping and cosmological frameworks. No major contradictions exist among the descriptions; rather, they complement each other by providing historical, conceptual, and practical perspectives on **\u5468\u5929\u5386\u5ea6**.", "entity_id": "\u5468\u5929\u5386\u5ea6", "entity_type": "category", "file_path": "ChuZhongShuXue.docx", "id": "\u5468\u5929\u5386\u5ea6", "label": "\u5468\u5929\u5386\u5ea6", "shape": "dot", "size": 10, "source_id": "chunk-38d881a4f0d10996dabecdb3f4bf659a\u003cSEP\u003echunk-fcc8429be9c152ab7fd3a6e6daac3659\u003cSEP\u003echunk-b86ecc17467c79a88d8dfa681dcdb3ec\u003cSEP\u003echunk-67ccc979a5e11b9d77c4b239a09538c8", "title": "**\u5468\u5929\u5386\u5ea6 (Zh\u014du Ti\u0101n L\u00ec D\u00f9)** is an ancient Chinese astronomical concept and celestial measurement system discussed in the *\u5468\u9ac0\u7b97\u7ecf (Zhoubi Suanjing)*. It refers to the period during which the stars complete a full rotation around the North Star (Polaris), equivalent to the number of days in a solar year (approximately 365.25 days). This concept was foundational in early Chinese astronomy, illustrating the cyclical nature of celestial movements. \n\nTraditionally, **\u5468\u5929\u5386\u5ea6** is attributed to **\u5e96\u727a\u6c0f (Fuxi)**, a mythical figure in Chinese culture, and was preserved and developed through the Shang and Zhou dynasties. While some descriptions emphasize its role as a measurement system, others highlight its conceptual significance in understanding stellar cycles. Despite slight variations in phrasing, the core idea remains consistent: it represents the celestial rotation period tied to the solar year, reflecting the integration of astronomy and calendrics in ancient China. \n\nThe system underscores the early Chinese emphasis on celestial observation and its application in timekeeping and cosmological frameworks. No major contradictions exist among the descriptions; rather, they complement each other by providing historical, conceptual, and practical perspectives on **\u5468\u5929\u5386\u5ea6**."}, {"color": "#d0efda", "created_at": 1752884071, "description": "**\u590f\u81f3 (Summer Solstice)** is an important astronomical event in ancient Chinese cosmology, marking the longest day of the year and serving as a key reference point in traditional Chinese calendrical systems. It is characterized as the solar term with the shortest shadow length (\u65e5\u5f71\u6700\u77ed\u7684\u8282\u6c14), reflecting its significance in astronomical observations and measurements. Ancient Chinese astronomers utilized the summer solstice for precise calculations, including tracking solar phenomena. Notably, it is associated with unique shadow effects at the Tropic of Cancer (\u5317\u56de\u5f52\u7ebf), where the sun appears directly overhead during this time. While the summer solstice played a role in Eratosthenes\u0027 measurement of Earth\u0027s circumference, the descriptions primarily emphasize its importance in Chinese astronomical traditions, where it served as a critical marker for timekeeping, agriculture, and cosmological studies. 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\n\n\u8be5\u4e66\u4e0d\u4ec5\u662f\u51e0\u4f55\u5b66\u9886\u57df\u7684\u5960\u57fa\u6027\u6587\u672c\uff0c\u8fd8\u5bf9\u73b0\u4ee3\u6570\u5b66\u7684\u516c\u7406\u5316\u65b9\u6cd5\u4ea7\u751f\u4e86\u6df1\u8fdc\u5f71\u54cd\u3002\u5e0c\u5c14\u4f2f\u7279\u901a\u8fc7\u300a\u51e0\u4f55\u57fa\u7840\u300b\u5c1d\u8bd5\u5efa\u7acb\u4e86\u4e00\u4e2a\u4e25\u8c28\u7684\u51e0\u4f55\u516c\u7406\u4f53\u7cfb\uff0c\u63a8\u52a8\u4e86\u6570\u5b66\u57fa\u7840\u7814\u7a76\u7684\u8fdb\u6b65\u3002\u5c3d\u7ba1\u8868\u8ff0\u65b9\u5f0f\u7565\u6709\u5dee\u5f02\uff0c\u4f46\u6240\u6709\u63cf\u8ff0\u5747\u786e\u8ba4\u8be5\u4e66\u7531\u5e0c\u5c14\u4f2f\u7279\u6240\u8457\uff0c\u5e76\u5f3a\u8c03\u5176\u5728\u516c\u7406\u5316\u51e0\u4f55\u548c\u6570\u5b66\u65b9\u6cd5\u8bba\u4e2d\u7684\u6838\u5fc3\u5730\u4f4d\u3002 \n\n**Key Points:** \n- \u4f5c\u8005\uff1a\u5927\u536b\u00b7\u5e0c\u5c14\u4f2f\u7279 \n- \u8457\u4f5c\uff1a\u300a\u51e0\u4f55\u57fa\u7840\u300b\uff08*Foundations of Geometry*\uff09 \n- 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\n\n\u300a\u51e0\u4f55\u57fa\u7840\u300b\uff08*Foundations of Geometry*\uff09\u662f\u5fb7\u56fd\u6570\u5b66\u5bb6\u5927\u536b\u00b7\u5e0c\u5c14\u4f2f\u7279\uff08David Hilbert\uff09\u6240\u8457\u7684\u4e00\u90e8\u5177\u6709\u6df1\u8fdc\u5f71\u54cd\u7684\u51e0\u4f55\u5b66\u8457\u4f5c\u3002\u5e0c\u5c14\u4f2f\u7279\u5728\u8be5\u4e66\u4e2d\u7cfb\u7edf\u6027\u5730\u6539\u9769\u4e86\u4f20\u7edf\u7684\u6b27\u51e0\u91cc\u5f97\u51e0\u4f55\u516c\u7406\u4f53\u7cfb\uff0c\u901a\u8fc7\u5f15\u5165\u540d\u4e49\u5b9a\u4e49\u548c\u4e94\u7ec4\u516c\u7406\uff0c\u5f25\u8865\u4e86\u539f\u6709\u4f53\u7cfb\u7684\u7f3a\u9677\uff0c\u4ece\u800c\u5f7b\u5e95\u9769\u65b0\u4e86\u51e0\u4f55\u5b66\u7684\u7814\u7a76\u65b9\u6cd5\u3002 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"description": "Hilbert formalized geometric axioms, such as the relationship between points and lines in space.", "file_path": "ChuZhongShuXue.docx", "from": "\u5e0c\u5c14\u4f2f\u7279", "keywords": "axiomatic system,mathematical formalization", "source_id": "chunk-afc841e1a6c397cedfaba45e3a346fb6", "title": "Hilbert formalized geometric axioms, such as the relationship between points and lines in space.", "to": "\u516c\u7406", "width": 7.0}, {"created_at": 1752884124, "description": "\u5e0c\u5c14\u4f2f\u7279\u63d0\u51fa\u540d\u4e49\u5b9a\u4e49\u65b9\u6cd5\u6539\u8fdb\u51e0\u4f55\u5b66\u57fa\u7840\u5bf9\u8c61\u7684\u5b9a\u4e49\u3002", "file_path": "ChuZhongShuXue.docx", "from": "\u5e0c\u5c14\u4f2f\u7279", "keywords": "\u6570\u5b66\u57fa\u7840,\u7406\u8bba\u521b\u65b0", "source_id": "chunk-665ebf4a9a719e0056c9e54525078072", "title": "\u5e0c\u5c14\u4f2f\u7279\u63d0\u51fa\u540d\u4e49\u5b9a\u4e49\u65b9\u6cd5\u6539\u8fdb\u51e0\u4f55\u5b66\u57fa\u7840\u5bf9\u8c61\u7684\u5b9a\u4e49\u3002", "to": "\u540d\u4e49\u5b9a\u4e49", "width": 7.0}, {"created_at": 1752884125, "description": "Hilbert developed rigorous congruence axioms to address deficiencies in Euclid\u0027s treatment of geometric equality.\u003cSEP\u003eHilbert developed the congruence axioms to replace Euclid\u0027s ambiguous fourth axiom with precise logical definitions.", "file_path": "ChuZhongShuXue.docx", "from": "\u5e0c\u5c14\u4f2f\u7279", "keywords": "axiomatic precision,axiomatic reform,logical rigor,mathematical rigor", "source_id": "chunk-e91dee450ab50cb1a44f08fe171c809b\u003cSEP\u003echunk-33e4d77aeae0267e846e8c7ab9af2b41", "title": "Hilbert developed rigorous congruence axioms to address deficiencies in Euclid\u0027s treatment of geometric equality.\u003cSEP\u003eHilbert developed the congruence axioms to replace Euclid\u0027s ambiguous fourth axiom with precise logical definitions.", "to": "\u5408\u540c\u516c\u7406", "width": 18.0}, {"created_at": 1752884159, "description": "Hilbert\u0027s 7th problem 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