教育部考试中心以正式文件明确提出在高考数学考题中要体现数学文化，这是一个明显的信号：要求学生加强对数学文化知识的学习，自觉地、有针对性地重视对数学文化修养的提升。

本书创新性地从以数化人与人数学化两个角度来谈数学文化，对数学文化在数学历史、数学精神、数学应用、数学之美、数学语言、交汇拓展、数学游戏等各个主题上都作出了全面透彻的剖析和论述。除此之外，作者还结合历年高考数学试题进行具体的案例分析，以此让高考生能够精准消化吸收高考中的各种数学文化知识。

对高中生而言，这是一部让他们明确数学文化是什么怎么考怎么练，为备考数学文化提供清晰学习方向的红宝书；对数学教学工作者或研究者而言，这是一部系统揭示命制背景、展示解题过程、凸显数学文化价值的宝贵内参书；对于广大的数学文化爱好者而言，这又是一部让人崇尚真理，净化心灵，提升文化修养和文化品格的数学美书。

★为什么要学习数学文化？其实这个问题也等同于为什么要学习数学。数学本身就是一种文化，所以这本书旨在帮助学生形成良好的数学思维能力，掌握解决问题的方法，培养理性精神。

★现在的高考试题越来越注重考生解决问题的能力，倘若只会刷题而不能理解题中真意将会寸步难行。本书以文化为内核，将理论、方法、历史、文化融为一体，知识性、趣味性和实操性完美结合。

★数学家顾沛高度赞赏的数学文化典范之作，中国科学院院士陈佳洱、王大中联袂诚意推荐，适合高中生及教学工作者使用。

2016 年10 月8 日，教育部考试中心下发《关于2017 年普通高考考试大纲修订内容的通知》（教试中心函〔2016〕179 号），增加了数学文化的要求。这一文件的公布，引发了社会对数学文化的极大关注，成为教育界的热门话题。

事实上，当人们不满足于对数学本身的研究，开始从外部审视数学的价值时，对数学文化的研究就开始了。人们开始思考数学是什么与数学的价值是什么，并逐渐认识到数学已经成为人类文化的重要组成部分，在过去已经对人类文明做出了巨大的贡献，在未来也必将做出更大的贡献。

把数学作为文化来进行研究，国外要比我们早很多，比较有代表性的是美国的数学家怀尔德与数学史专家莫里斯·克莱因。怀尔德认为数学是一个由内在力量与外在力量共同作用，而处于不断发展和变化之中的文化系统。数学文化是由文化传统和数学本身所组成。这些观点在他的两部著作《数学概念的演化》和《作为一种文化系统的数学》中得以充分体现。莫里斯·克莱因在他的著作《西方文化中的数学》《数学：一种文化探索》《数学与知识的探求》中则系统论述了数学对西方文化、理性精神、现代人类思想发展所产生的影响。他侧重于数学与各种文化以及社会因素之间相互作用的分析。我国开展数学文化研究是近二三十年的事情，时间虽短，但是相关研究也取得了极大进展。徐利治探讨数学一般意义上存在的某些相同或不同的方法，开创了数学方法论的研究与教学；郑毓信教授在他的《数学文化学》中基于数学哲学观点构建数学文化学的理论体系，尝试从理论层面论述数学作为文化的内涵；齐民友在《数学与文化》一书中则论述了中西数学文化的差异。

我国教育主管部门对数学文化也很重视。早在2003 年，教育部颁布的《普通高中数学课程标准（实验）》中就突出强调了数学的文化价值数学是人类文化的重要组成部分，对数学文化给予了特别重视，要求数学文化贯穿整个高中数学课程并融入教学中。这次教育部考试中心又再次以发布正式文件的方式对数学文化重点提及，明确提出要在高考试题中考查数学文化，足以看出国家对数学文化的重视。

以往提及数学文化，大多停留在理论的层面，而且很多研究仅仅把史实、案例、故事等稍作整理，将数学史等同于数学文化。这种做法不过是给数学文化穿了件数学史外衣而已。类似的研究总给人以不接地气之嫌，没有引起人们的触动。但这次之所以引发对数学文化思考与研究的热潮，主要原因是考试主管部门从考试命题的角度提出了要求，第一次正式地明确要求把数学文化渗入数学试题，所以未来高考数学命题肯定会遵照执行。虽然我们提了很久的数学文化，但如何在数学试题中体现出来，对很多人来说还是比较陌生的。数学文化本来是个笼统的概念，传统意义上文化在文科中出现得较为普遍，在文科类的命题中是很容易做到的。而数学是逻辑的科学、思维的科学，如何把文化渗透其中，是一个摆在广大数学教育工作者面前的新课题。正是在这样的背景下，我根据多年积累的丰富资料和大量的研究工作，捉笔成书，对相关问题作出了系统的回答。

《高考中的数学文化》一书共分八章。第一章为数学文化总论，主要阐述了数学文化的含义、内容和基本特性等问题；后面七章分别为数学历史、数学精神、数学应用、数学之美、数学语言、交汇拓展、数学游戏，遴选近十五年高考数学中涉及数学文化的试题进行分类赏析，系统揭示命制背景，展示解题过程，凸显数学文化价值。全书尽可能把数学文化出题的方向一网打尽，让读者明确数学文化是什么怎么考怎么练，为备考数学文化提供清晰的复习方向。同时每个主题后面提供与主题相关的数学文化练习题，方便读者练习巩

固使用，从而更有针对性地复习。本书的编写具有实用性、针对性和时效性三个鲜明特点，密切结合当下学生、教师对数学文化试题认识相对陌生的客观实际，对高考中曾经出现的数学文化试题进行分类赏析，系统阐述最新考纲考查数学文化的内涵，同时为学生备考提供科学性的建议。

我国对数学文化的系统研究时间不长，尤其是数学文化与试题的结合研究更是不成熟。此书尽管对数学文化的理论探讨与应用研究取得了较大突破，然而也不免出现不完善或者纰漏之处。但我相信，此书的问世将会受到广大同行的重视和学生的欢迎。在此，我也恳请读者批评雅正。

齐龙新，北京市数学特级教师，山东省教学能手，山东省新课程研修团队专家，山东省基础教育教师培训专家，山东省优质课一等奖、教育部一师一优课一等奖获得者，现为北京市育英学校经济学实验班班主任。作者致力于新数学自学辅导教学法的实践与研究，所编自学教材曾获得山东省优秀课程资源一等奖，在日常教学实践中也取得了非常显著的教学效果。其微信个人公众号龙新数学(zxfd66)曾经获得2016年数学文化杂志社主办的携手北大数学文化节最红公众号评选全国第二名。

把数学教得通俗易懂，让学生学得幸福有趣是作者一以贯之的精致追求，其教学风格独特，教法细腻、技巧纯熟，教学成绩突出，与学生之间亦师亦友，深得学生的喜爱。作者教学之余写出了数十篇教育教学论文，撰写了《初高中数学衔接教材》等多部教研论著。

第一章 数学文化总论 ··················································································· 1
第一节 什么是数学文化 ········································································ 1
第二节 数学文化的四个层面 ································································· 3
一、知识系统 ······················································································· 4
二、工具系统 ······················································································· 4
三、价值系统 ······················································································· 5
四、史实系统 ······················································································· 7
第三节 数学文化的人本特性 ································································· 7
一、精神特性 ······················································································· 8
二、物质特性 ······················································································· 8
三、行为特性 ······················································································· 9
四、审美特性 ······················································································· 9
第二章 数学历史 ························································································ 11
第一节 数学名著 ················································································ 11
一、《九章算术》 ················································································ 12
二、《数书九章》 ················································································ 19
三、《算数书》 ··················································································· 24
第二节 数学故事 ················································································ 27
一、毕达哥拉斯的故事 ········································································· 27
二、高斯的故事 ·················································································· 36
三、哥德巴赫的故事 ············································································ 41
第三节 数学名题 ················································································ 44
一、米勒问题 ····················································································· 45
二、斐波那契数列 ··············································································· 57
三、阿基米德穷竭法 ············································································ 66
四、柯西不等式 ·················································································· 73
五、角谷猜想 ····················································································· 80
六、阿波罗尼奥斯圆 ············································································ 85
七、回文数 ························································································ 93
八、勾股数 ························································································ 98
九、平均数 ······················································································· 101
十、皮克定理 ···················································································· 108
十一、数字黑洞 ················································································· 114
十二、伯努利不等式 ··········································································· 119
十三、正整数方幂和 ··········································································· 124
十四、切比雪夫多项式 ········································································ 130
十五、马尔科夫定理 ··········································································· 135
十六、四色问题 ················································································· 143
十七、莱布尼茨三角形 ········································································ 146
十八、黄金分割比 ·············································································· 151
第三章 数学精神 ······················································································· 155
第一节 理性求真 ··············································································· 155
第二节 创新意识 ··············································································· 167
第四章 数学应用 ·······················