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探秘数学常数:妙趣横生的数学常数

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数字是数学的基础,也是数学中极具魅力的部分。数学中的常数不计其数,本书通过精心挑选的约100个重要的数学常数,展现了丰富多彩的数学世界。读者通过阅读本书,不仅可以了解这些数学常数的有趣特点和重要应用,还可以了解这些数字背后的故事,感受到数学家天才般的智慧以及不懈探索的精神。
书中所介绍的知识与我们的生活有重要联系,读者可以看到数学是如何推动科学和技术的发展的;而且本书内容新颖,已涵盖本书印刷前数学研究的新进展和成果。
书中涉及国内外大量数学家、科学家、艺术家等重要人物,并对相关材料进行了认真梳理和考证,纠正了许多不正确的信息与观点。
本书作者从事数学教学与研究工作数十年,编写出版了20多部科普读物,发表文章500多万字,曾荣获国家科技进步奖二等奖等重要奖项,深受各方赞誉。
Content Description

数字中的各种常数令人敬畏,它们似乎是宇宙诞生之初“上帝”的精心选择。那一串串无限不循环的数字往往让人陷入无底洞般的哲学沉思:为什么这些数字不是别的,而偏偏就是这个子样呢?除了那些众所周知的基本常数之外,还有很多非主流的数学常数,它们奇异的特性同样具有浓重的神秘色彩。今天,就让我们打开《探秘数学常数:妙趣横生的数学常数》一起来看一看到底有哪些神秘的常数。
Author Description

陈梅,中学教师。2003年12月获重庆市重点中学中青年教师优质课大赛信息技术学科一等奖。曾担任“站在巨人肩上丛书”(共5册)编委和其中《科学偶然故事》的主编,该丛书获第一届北京市优秀科普作品奖以及第十八届(2002年度)北方十省市(区)优秀科技图书一等奖。担任“七彩学生文库·科学天梯丛书”(共10册)副总主编、编委和其中《科学趣味故事》的主编。参编的《说不尽的π》与《不可思议的e》(“好玩的数学丛书”10册中的两册)获2009年度国家科学技术进步奖二等奖。
Catalogue

第 1 章 精彩纷呈的自然数 / 1
1.10——是不是自然数 /1
1.23——石头垒成的数 /2
1.35——少年英雄为它呕心沥血 /14
1.46——四处游荡而又有时缺席 /24
1.57——吉凶一身,四海为家 /37
1.613——1 元美钞上的秘密 /47
1.7365——《难题》中的地球数 /58
1.81729——不同凡响的出租车数 /61
1.9142 857——下凡的圣数 /63
第2 章 并非绣花枕头的无理数 /72
2.1大海不能吞噬的秘密 /72
2.24 个著名的无理数 /76
2.3数系的发展:从自然数到超越数 /82
2.46 个著名的超越数 /84
第3 章 欧拉常数γ /91
3.1调和级数之谜 /92
3.2欧拉常数 γ /98
3.3用途广泛的 γ /106
3.4γ 与某些数学常数 /108
3.5“所有人的老师” /111
第4 章 孪生素数家族的常数 /116
4.1孪生素数的布朗常数 B2 和孪生素数常数 C2 /116
4.2孪生素数的布朗常数 B4 和表兄弟素数常数 C4 /124
4.3六素数与其他 /132
第5章其他有专用名称的数学常数 /134
5.1MRB 常数 /134
5.2迈塞尔-梅尔滕斯常数 M1 /136
5.3伯恩斯坦常数 β /137
5.4连分数常数 C1, C2, C3 /138
5.5恩布里-特雷费森常数 β* /139
5.6朗道-拉马努金常数 K /139
5.7卡塔兰常数 C /141
5.8勒让德常数 B'L /143
5.9威斯万纳斯常数 K /144
5.10阿佩里常数 a /145
5.11米尔斯常数 A /149
5.12拉马努金-索尔德内尔常数 μ /150
5.13巴克豪斯常数 B /151
5.14超黄金比 ψ /152
5.15厄尔多斯-波尔文常数 EB /153
5.16谢尔宾斯基常数 K /154
5.17辛钦常数 K0 /154
5.18斐波那契数倒数和常数 ψ /156
5.19费根鲍姆常数 δ 和 α /157
5.20卡普勒卡尔常数 Ka /159
第6 章 五彩缤纷的数列 /163
6.1完全数 /164
6.2“调和数” /170
6.3亲和数 /171
6.4幸运数 /178
6.5多角数 /178
6.6水仙花数、回归数和草数 /191
6.7幂数 /196
6.8黑格涅尔数 /202
6.9佩尔数 /204
6.10贝尔数与斯特林数 /207
6.11卡塔兰数 /210
6.12卡普勒卡尔数 /212
6.13无 8 数 /214
6.14皮索特数与塞勒姆数 /216
6.15快乐数与伤心数 /217
6.16史密斯数 /219
6.17弗里德曼数 /220
6.18马尔科夫数 /222
6.19卡迈克尔数 /223
6.20邹赛尔数 /224
6.21哈沙德数 /225
6.22自守数 /226
6.23基斯数 /228
6.24金蝉蜕壳数 /228
6.25冰雹数 /229
6.26累进可除数 /233
6.27普罗斯数、谢尔宾斯基数和里塞尔数 /236
6.28自我数 /239
6.29“魔术数” /240
6.30不可及数 /240
第7 章 琳琅满目的素数 /242
7.1素数的种类 /242
7.2梅森素数、双梅森素数和塔形梅森素数 /244
7.3费马素数 /251
7.4佩兰素数和佩兰伪素数 /252
7.5热尔曼素数、安全素数和危险素数 /254
7.6吸血鬼素数 /260
7.7回文素数和二面素数 /262
7.8循环素数、可逆素数、顺序素数和逆序素数 /267
7.9可截断素数 /270
7.10强素数 /272
7.11纽曼-山克斯-威廉斯素数 /274

Introduction

著名数学家哥德尔 18 岁进入维也纳大学时学习的是理论物理专业,他常到位于维也纳第九区斯特鲁德尔霍夫大街 4 号的理论物理研究所 4 楼的大教室聆听奥地利物理学家蒂林的课。有一次,他参加了德国哲学家、物理学家施里克介绍罗素数论著作的研讨会,由此对数理逻辑产生了浓厚的兴趣。数的吸引力如此之大,竟致哥德尔改换门庭。入校两年后,哥德尔放弃了物理学专业,改学数学。 5 年以后的 1931 年, 25 岁的哥德尔发表了震惊数学界的不完备定理。而在今天,用他的姓氏命名的哥德尔奖从 1993 年起每年都在颁发。

提及数学,我们自然无法回避数学常数。最著名的数学常数当属圆周率 π、黄金分割数φ和自然对数的底 e。以色列数学史家马奥尔在《无穷之旅——关于无穷大的文化史》一书中称它们为“ 3 个著名的无理数”。在美国加州谷歌公司总部的 4 座办公大楼中,有 3 座是以这 3 个数学符号命名的。当然,数学中重要的常数远不止于此。
数学常数是古老而又年轻的数论的重要组成部分。虽然截至目前已经取得了许多重大的数论研究成果,但是还有不少谜团让人看朱成碧,不辨五色。正如日本著名数学家弥永昌吉所说,数论的 “大部分仍然笼罩在神奇的面纱之下”。因此,需要我们继续不断探索。“苔花如米小,也学牡丹开。”本丛书作者虽然明知力所不逮,但凭借着对数学常数的浓厚兴趣,自 1982 年开始,断续经过33年的努力, 终于编写成了这套系统介绍数学常数的普及读物。这套书包括 《说不尽的圆周率》《不可思议的自然对数》《奥妙无穷的黄金分割》《妙趣横生的数学常数》。

虽然目前国内外出版了一些介绍上述三大数学常数的图书,但总体来说不够系统全面,许多有趣的知识没有罗列其中。本套图书希望能在以下方面做出有益探索。
首先,本套图书以 π、φ、 e 等主要的数学常数为主线,但又不限于此,内容涵盖重要数学概念和数学思想的形成、著名公式和定理的证明以及它们在各学科和生活中的应用。单是书中涉及的数学家以及相关的科学家、哲学家等就有 2000 多位,读者可以从书中了解这些著名人物在数学研究中艰难的探索历程(而心怀感恩之情)、所取得的重要成果以及逸闻趣事。
其次,本套图书在化繁为简、深入浅出方面进行了积极的尝试,力求消除“数学是可怕的学科”这样的误解,把“可怕”变为“可爱”,让读者充分感受到数学之真、之美、之乐、之用,感受到数学的魅力。中国数学家张景中在其所著的《数学与哲学》一书开篇就指出:“联系数学的发展历史学习数学哲学,有趣又有效。”在编写过程中,作者也尽力将数学史和科学史内容融入书中,从而梳理出重要数学思想或者数学定理的发展脉络,以及著名数学难题的求解历程。读者感受到的将不再是仅以定理、证明、计算面孔出现的枯燥乏味的脑力训练,而将看到一个有血有肉、充满活力与无限乐趣的新形象。
再次,知识驱动着人类文明的发展,数学作为我们理解和探索科学以及世界的重要工具发挥了重要作用。知识的获取过程是艰辛的,但也充满着乐趣,数学更是如此。本套图书在介绍数学知识之外,更加注重表现数学家的优秀品质以及探索精神,希望读者在享用前人所留下的宝贵精神财富的同时能有一些感触或感悟。
除了作者团队的协作与努力之外,许多专家、学者、朋友及相关人士也对本套图书的编写与出版给与了帮助。在此感谢张景中、李敏、郭书春、宁挺、梁宗巨、张奠宙、査有良、吴振奎、丘和、曾润生、王青建、邹大海、彭定才、潘宁、邓文华、陈文伟、贾小勇、吴至友、罗明、安克·毛雷尔等。由于篇幅所限,还有不少人士的姓名在此未能提及,一并表示感谢。
限于作者的陋见,书中难免存在疏漏与不当之处,请读者批评指正。
爱因斯坦说:“不要担心你在数学上遇到的困难,我敢保证我遇到的困难比你大得多。”让我们以此共勉,并慢慢爱上有趣又好玩的数学吧。

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Jingdong book