history of math

刻痕计数 Indentation counting

古埃及人发明象形数字 The ancient Egyptians invented hieroglyphic numerals.

两河流域出现楔形数字 The cuneiform script emerged in the Mesopotamian region.

古埃及人把一年定义为365天 The ancient Egyptians defined a year as 365 days.

尧命令曦叔制定历法，366天为一年，3年有一次闰月

中国黄河一带出现了幻方（就是现在魔方的前身）的最早记录《河图洛书》 The earliest record of the emergence of magic squares in the area of the Yellow River in China has been discovered.

古巴比伦人发明了60进制 The people of ancient Babylon invented the base-60 system. 同年，记录下了5组勾股数 In the same year, five sets of Pythagorean triplets were recorded.

埃及人计算出了棱台的体积 The Egyptians calculated the volume of a frustum. 同年出现了方程的计算 In the same year, the calculation of equations was carried out.

泰勒斯定理出现 Thales’ Theorem 主要内容： Main Content: 如果一个三角形的一个边是圆的直径，并且该三角形的第三个顶点在圆上（不与直径的端点重合）那么这个三角形一定是直角三角形，且直角位于该第三个顶点。 If one side of a triangle is the diameter of a circle, and the third vertex of the triangle lies on the circle (but does not coincide with the endpoints of the diameter), then this triangle must be a right triangle, and the right angle is located at the third vertex.

毕达哥拉斯定理（勾股定理） Pythagoras' Theorem 直角三角形中两直角边的平方和是斜边的平方 In a right-angled triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.

斯帕索斯，第一次发现了无理数 Irrational Number 芝诺悖论的提出 Zeno's paradox 阿吉里斯永远追不上乌龟 Achilles will never catch up with the tortoise.

雅典的瘟疫爆发，阿波罗神殿的神谕，提出了倍立方问题 The plague broke out in Athens. The oracle of the Apollo Temple raised the problem of doubling the cube.

比例理论 The Theory of Proportions 欧多克索斯，研究不可测量度 Eudoxus, the study of incommensurable magnitudes

亚里士多德证明了根号2是无理数 Aristotle proved that the square root of 2 is an irrational number.

欧几里得完成《几何原本》的写作 Euclid completed the writing of "Elements of Geometry".

球体积和表面积公式： Archimedes 阿基米德给出了球的体积和表面积公式 Formula for the volume and superficial area of a sphere 阿基米德又使用穷竭法来计算pai和抛物线下的面积 Archimedes also employed the method of exhaustion to calculate the value of pi and the area under the parabola. 埃拉托色尼，用相似计算出了地球的周长 Eratosthenes calculated the circumference of the Earth by means of similarity.

阿波罗尼斯奥完成《圆锥曲弦论》的写作 Apollonius of Perga completed the writing of "Conics".

三角函数 Trigonometric functions 希帕克斯制作出第一张三角函数表 Hipparchus produced the first table of trigonometric functions.

托勒密通过自己的定理，推出和差角公式重写三角函数表 Through his own theorems, Ptolemy derived the formula for the sum and difference of angles. Rewrite the table of trigonometric function

《算术》出世丢番图第一个使用代数符号，研究方程的解 Diophantus was the first to use algebraic symbols and study the solutions of equations.

中国人发明了算筹 The Chinese invented the counting rods.

音律学的出现：《管子》，用“三分损益法”确定了：宫商角zhi羽，分别对应Do Re Mi Sol La 宫商角 zhi 羽 Do Re Mi Sol La

《周髀算经》中国最古老的天文学著作，其中记载了勾股定理

《九章算术》出现： The "Nine Chapters on the Mathematical Art" appeared: 那时的中国人已经在用矩阵解线形方程了 At that time, Chinese people had already been using matrices to solve linear equations.

中国人发明了算盘 The Chinese invented the abacus.

刘徽，用了割圆法计算圆周率 Liu Hui used cyclotomy to calculate pi

印度出现了印度数字，也就是之后的阿拉伯数字 In India, the so-called "Indian numerals" emerged, which later became known as the Arabic numerals.

《孙子算数》记载了鸡兔同笼的问题和韩信点兵 Use of Diophantine Equations and Chinese Remainder Theorem, CRT

祖冲之算出了pi的精确近似 Zu Chongzhi calculated the precise approximation of pi.

“代数之父”花拉子米，写出二次方程的求根公式 Al-Khwarizmi, the "Father of Algebra", derived the formula for solving quadratic equations. He systematically studied the solution methods of the quadratic equation of one variable And proposed the concepts of "al-jabr" (transposition) and "al-muqabala" (subtraction cancellation).

宋代的四大数学家： In Song Dynasty, there are lots of attributes in this period 这个时期出现了杨辉三角，《数书九章》，《测圆海镜》，《四元玉鉴》 1.秦九韶（1202—1261） • 代表作：《数书九章》 • 提出了“大衍求一术”，即中国剩余定理的系统化方法。 • 研究了高阶多项式方程的数值解法，相当于后来的霍纳法则（Horner’s method）。 • 发展了一次同余方程的解法，为后来的数学家提供了基础。 Qin Jiushao (1202–1261) • Masterpiece: Shu Shu Jiu Zhang (Mathematical Treatise in Nine Sections) • Developed the “Da Yan Rule”, which is essentially a systematic method for solving the Chinese Remainder Theorem. • Studied higher-order polynomial equations and propo

朱载育计算2的12次方，得到了十二平均律 Zhu Zaiyu calculated 2 to the power of 12 and obtained the Twelve-Tone Equal Temperament. 在十二平均律中，相邻两个音的频率比是相等的，即等比数列关系。是一个跟音乐中用到的数学概念，跟之前的五音一样 In the twelve-tone equal temperament, the frequency ratio between adjacent two notes is equal, which is an arithmetic progression relationship. It is a mathematical concept used in music, just like the previous five-tone system.

斐波那契著作《计算之书》把埃及，阿拉伯，希腊地区的数学思想传到欧洲 The book "The Book of Calculation" written by Fibonacci introduced the mathematical ideas from Egypt, Arabia and Greece to Europe.

《数学大全》横空出世 “现代会计之父”帕乔利，创造原始的符号代数 "Mathematics Masterpiece" Emerges Suddenly "Father of Modern Accounting" Pacioli created the original symbolic algebra.

卡尔达诺，发表三次方和四次方的求根公式 Cardano published the formulas for solving cubic and quartic equations.

邦别利，引入虚数来计算三次方根 Bonelli, introduced imaginary numbers to calculate cubic roots.

斯蒂文在欧洲普及十进制 Steven popularized the decimal system in Europe.

Napier发明了对数《奇妙的对数表的描述》 Napier invented logarithms “Wonderful logarithmic table“ 简化了乘除法的运算，他的对数被后人称为纳皮尔对数 By simplifying the operations of multiplication and division, his logarithms were later called Napierian logarithms.

V+F-E=2 V（Vertices）：顶点的数量 E（Edges）：棱的数量 F（Faces）：面的数量笛卡尔首先注意到这个特性，被后人称作“欧拉公式”

积分的思想卡瓦列里，用求和思想给出了幂函数曲线的积分公式，但是没有遵循严格的极限积分计算，知识单纯的加和 The idea of integration Cavalieri used the concept of summation to derive the integral formula for power function curves, but he did not follow the rigorous limit-based integration calculation. Instead, he merely performed simple summation.

微分思想：费马，在研究方程和曲线的关系时，知道了求极值的方法 Differential thought: When Fermat was studying the relationship between equations and curves, he discovered the method for finding extreme values.

1.费马提出著名的“费马猜想” Fermat put forward the famous "Fermat's Conjecture". 我已经找到了一种真正奇妙的证明，但此处空白太小，写不下 “I have found a truly wonderful proof, but the space here is too small to write it down” 2.解析几何的出现： The appearance of Analytic Geometry 同年，笛卡尔发表了《方法论》，但有趣的是这是一本哲学书，但是附录的时候提出了解析几何 In the same year, Descartes published "Discourse on the Method", but interestingly, this was a philosophical book, but in the appendix, he proposed analytic geometry.

伽利略著作：《两名新科学的谈话》 Discourses and Mathematical Demonstrations Relating to Two New Sciences

帕斯卡，发明了第一台计算器，可以计算加减法 Pascal invented the first calculator which could perform addition and subtraction operations.

牛顿降生 Birth of Newton

帕斯卡和费马，在赌博中提出了数学期望的概念 Pascal and Fermat introduced the concept of mathematical expectation in the context of gambling. AS数学中在S的时候会学到 In AS Mathematics, you will learn about S at that time.

沃利斯的著作《无穷算数》提出了沃利斯公式 Wallis' book "Infinite Series" presented Wallis' formula. 这个公式逐渐收敛至 pi/2 This formula gradually converges to pi/2.

牛顿提出了广义二项式定理，研究流数术，后来演变成微积分 Newton proposed the generalized binomial theorem and studied the calculus of variations. Later, it evolved into calculus. 在牛顿的流数术中： In his theory: •流形（Fluent）：表示随时间变化的变量，相当于现代数学中的函数 Manifold (Fluent): Denotes variables that change over time, equivalent to functions in modern mathematics. •流率（Fluxion）：表示流形的变化率，相当于现代数学中的导数 Fluxion: It represents the rate of change of a manifold, which is equivalent to the derivative in modern mathematics. •二次流率（Second Fluxion）：相当于二阶导数 Second

牛顿使用了极坐标 Newton used polar coordinates 同年，莱布尼茨发明了乘法计算器，可以计算加减乘除 In the same year, Leibniz invented a multiplication calculator which could perform addition, subtraction, multiplication and division.

莱布尼茨提出了二进制 Leibniz proposed the binary system.

莱布尼茨发表了第一篇关于微分学的论文：《Nova Methodus pro Maximis et Minimis》（关于求极大值与极小值以及切线的新方法） Leibniz published the first paper on differential calculus.

继1684年之后，莱布尼茨又发表了第一篇关于积分的论文：《论深奥的几何与不可分量的分析及无穷》 After 1684, Leibniz published his first paper on integration: "On the Sublime Geometry and the Analysis of Incommensurable Magnitudes and Infinitesimals"

牛顿发表《自然哲学之数学原理》，用数学方法揭示了万有引力定律 Newton published "Mathematical Principles of Natural Philosophy", using mathematical methods to reveal the law of universal gravitation.

洛必达法则：约翰·伯努利发现的，但是由于生计所迫，卖给了洛必达 L'Hôpital's Rule: It was discovered by Johann Bernoulli, but due to financial necessity, he sold it to L'Hôpital.

大数定律由雅各布·伯努利提出 The Law of Large Numbers Proposed by Jacob Bernoulli 样本均值收敛于期望值 The sample mean converges to the expected value.

泰勒级数《Methodus Incrementorum Directa et Inversa》用多项式逼近任意光滑函数 Taylor series "Methodus Incrementorum Directa et Inversa" Approximating any smooth function by polynomials

哥尼斯堡七桥问题欧拉在研究它的时候开创了拓扑学 The Seven Bridges of Königsberg Problem When Euler studied it, he initiated the field of topology.

哥德巴赫猜想；哥德巴赫在写给欧拉的心中提出 Goldbach's Conjecture; Goldbach put forward in his letter to Euler. 任一大于2的偶数均可表示为两个素数之和 Any even number greater than 2 can be expressed as the sum of two prime numbers.

波动方程达朗贝尔，通过研究弦的振动，给出了波动方程 Wave equation D'Alembert, through his study of the vibration of a string, derived the wave equation 同年发生了微积分论战，伯克莱批判了微积分，牛顿和莱布尼茨大战三百回合 In the same year, there was a debate on calculus. Berkeley criticized calculus.Newton and Leibniz engaged in a three-hundred-round battle. 这也同样促使后来的人对微积分的定义进一步完善 This also prompted later people to further refine the definition of calculus.

《无穷小分析理论》欧拉在这本书提出欧拉恒等式 "Infinitesimal Analysis Theory" by Euler. In this book, Euler presented Euler's identity. 发表了现在标准的通过反函数来处理对数的方法弄清楚了指数和对数的关系，也推导出了e的数值 Published the current standard method of handling logarithms through inverse functions. Having clarified the relationship between exponents and logarithms, we also derived the value of e. 欧拉公式像一座桥梁，连接了微积分中离散与连续、实数与复数的世界 Euler's formula is like a bridge that connects the discrete and continuous realms, as well as the real and complex worlds in calculus.

拉格朗日，提出解决变分问题的一般方法 Lagrange proposed a general method for solving variational problems.

贝叶斯定理的发表，但是实在贝叶斯去世后2年 The Bayesian theorem was published, but it was actually two years after Bayes' death. In A2 mathmatics and Further mathmatics, students will learn that

欧拉去世 Euler passed away

高斯降生 Birth of Gaussian

拉普拉斯，在研究天体物理的时候写出了位势方程 Laplace, while conducting research on astrophysics, formulated the potential equation. 位势方程犹如一座桥梁，连接了纯数学的抽象理论与应用数学的具体问题 Let the abstract theories in pure mathmatics lively in our real life

高斯提出最小二乘法，分析出正态分布曲线 Gauss proposed the method of least squares and analyzed the normal distribution curve. 同年，同一个人，尺规作出正17边形的图像 In the same year, he draw the Regular heptadecagon with ruler and compasses for the first time in human history 把几何问题转化为代数问题 In a result, transfer a problem of geometry to a question of algebra 拉格朗日插值 Lagrange's interpolation

勒让德提出了素数的渐近分布的猜想（素数定理），虽然不精确，但是为后来的人奠定了思想基础 Legendre put forward the conjecture about the asymptotic distribution of prime numbers. （The Prime Number） TheoremAlthough it was not precise, it laid the ideological foundation for later researchers.

代数基本定理 The Fundamental Theorem of Algebra 在高斯的大学论文中给出了严格的定义 A rigorous definition was provided in Gauss's doctoral dissertation.

高斯整理了自己和前人在数学方面的成就 Gauss compiled the achievements of himself and predecessors in mathematics.

拉普拉斯证明了中心极限定理 Laplace proved the Central Limit Theorem.

柯西给出了极限的数学定义，终于给微积分了一个温暖的家 Cauchy provided the mathematical definition of limit, and finally gave calculus a warm abode.

傅立叶在研究热传导的时候，提出了傅里叶级数和热方程 When Fourier was studying heat conduction, he proposed the Fourier series and the heat equation.

阿贝尔，证明了五次及以上的多项式方程没有求根公式 Abel proved that there is no general formula for solving polynomial equations of degree five or higher.

高斯证明了高斯绝妙定理，是古典微分几何的里程碑 Gauss proved Gauss's wonderful theorem, and it was a milestone in classical differential geometry. 为黎曼几何、广义相对论奠定基础 Laying the foundation for Riemannian geometry and general relativity

格林在研究电磁学的时候发现了格林公式，简化了物理场计算 While studying electromagnetism, Green discovered Green's formula, which simplified the calculation of physical fields.

罗巴切夫斯基，改写第五公设，非欧几何的出现 Robaczewski, modifying the fifth postulate, the emergence of non-Euclidean geometry

刘维尔发现第一个超越数 Lambert discovered the first transcendental number. 代表了数学中“不可构造”的深层复杂性 It represents the profound complexity of the "unconstructible" aspect in mathematics.

纳维·斯托克斯提出了不可压缩流体的运动方程 Navier-Stokes proposed the equations of motion for incompressible fluids.

布尔代数 Boolean algebra

黎曼提出流行概念，是广义相对论的数学基础 Riemann's proposal of the popular concept served as the mathematical foundation of the general theory of relativity.

矩阵，凯引入了矩阵，并给出了运算法则 Matrix, Cayley introduced matrix and presented the operation rules.

黎曼猜想： Riemann Hypothesis: 非平凡零点的实部是1/2 The real part of the non-trivial zeros is 1/2. 1. 如果黎曼猜想成立，它将极大地改进素数分布的误差估计，并为数论的许多未解问题提供强有力的工具 If the Riemann Hypothesis holds, it will significantly improve the error estimation of prime number distribution and provide powerful tools for many unsolved problems in number theory 2. 如果它不成立，将会导致许多数论理论的重大修改，并可能影响现代密码学和计算方法 If it does not hold, it will lead to major modifications in many number theory theories and may affect modern cryptography and computational methods

维尔斯特拉斯创造了一个处处连续但是处处不可导的函数 Vil'sta constructed a function that is continuous everywhere but nowhere differentiable.

康托证明连续统的不可数性，提出连续统假设 Cantor proved the uncountability of the continuum and proposed the Continuum Hypothesis.

庞加莱，推广了贝蒂数，奠定了代数拓扑学的基础 Poincaré promoted the concept of Becham numbers and laid the foundation of algebraic topology.

希尔伯特提出了23个还没有解决的数学问题 Hilbert put forward 23 mathematical problems that remain unsolved to this day. 同年，皮尔逊推广卡方检验，数理统计就此诞生 In the same year, Pearson popularized the chi-square test and mathematical statistics thus came into being.

勒贝格建立测度理论，定义勒贝格积分 Lebesgue established the theory of measure and defined the Lebesgue integral. 同年，庞加莱提出庞加莱猜想 In the same year, Poincaré put forward the Poincaré Conjecture

马尔可夫提出马尔可夫过程，随机过程就此产生 Markov proposed the Markov process, and thus the random process came into being.

豪斯道夫发表《集合论基础》，创立点集拓扑学 Hausdorff published "Foundations of Set Theory", and founded point-set topology.

哥德尔提出哥德尔不完备理论： Gödel put forward Gödel's Incompleteness Theory: “一致”和“完备”不能同时满足 "Consistency" and "completeness" cannot be satisfied simultaneously. 同年，科莫哥洛夫建立公理化概率论，标志着现代概率论的创立 In the same year, Kolmogorov established axiomatic probability theory, marking the establishment of modern probability theory.

勒维在赌博中提出“鞅”这一概念 Levy introduced the concept of "martingale" in the context of gambling.

图灵描述了一种代替人类计算的机器 Turing described a machine that substitutes for human computing.

冯诺伊曼建立了二人零和博弈理论 Von Neumann established the theory of two-person zero-sum game.

以冯诺伊曼为首的科学家们开发了世界上第一台计算机 Scientists led by von Neumann developed the world's first computer.

维纳发表《控制论》，宣告了现代控制论的诞生 Wiener published "Cybernetics", announcing the birth of modern cybernetics.

纳什在《非合作博弈论》中，提出了纳什均衡的博弈理论 In "Non-cooperative Game Theory", Nash proposed the game theory of Nash equilibrium.

布莱克与舒尔斯，给出了期权定价模型 Black and Scholes presented the option pricing model.

怀斯曼发表费马猜想的证明，费马大定理成立 Wiesman published the proof of Fermat's conjecture and Fermat's Last Theorem was thus established.

贝雷尔曼证明庞加莱猜想 Belermann proved the Poincaré conjecture.

1624 年，布里格斯出版了他的《对数算术》（Arithmetica Logarithmica），该书采用开本形式出版，其中包含了 30000 个自然数的对数，精确到小数点后 14 位。 In 1624, Briggs published his Arithmetica Logarithmica, in folio, a work containing the logarithms of thirty thousand natural numbers to fourteen decimal places

1649 年，阿尔方斯·安东尼奥·德·萨拉萨（Alphonse Antonio de Sarasa），曾是格雷戈尔·德·圣文森特（Gregoire de Saint-Vincent）的学生，将对数与双曲线的求积问题联系了起来发现双曲线下的面积就是log（x）他们叫他：Hyperbolic Logarithms 但是并没有给出计算公式 In 1649, Alphonse Antonio de Sarasa, a former student of Grégoire de Saint-Vincent,[8] related logarithms to the quadrature of the hyperbola It is discovered that the area under the hyperbola is log(x)，where x is the x-coordinate of the point on the hyperbola They called it hyperbolic Logarithms but they did not give a formula to calculate

尼古拉斯·墨卡托也独立发现了它，并在 1668 年的论文《对数技术》中收录了该级数对于小值的值并给他取名自然对数 Nicholas Mercator also independently discovered Mercator series , and included values of the series for small values in his treatise Logarithmotechnia Give it a name of Natural Logarithms 牛顿和约翰内斯·胡德再1655年也发现了这个级数但是并没有发表 (Newton also did that with Johannes Hudde in 1655 but they didn't published it. )

亚里士多德知道力可以表示成向量，但他并没有应用 Aristotle was aware that force could be expressed as a vector, but he did not apply this knowledge.

欧几里得在几何原本中也使用了线段来表示长度和方向 Euclid also employed line segments to represent lengths and directions in his Elements of Geometry.

阿尔冈以AB（上加箭头）表示有向线段 Algonquin uses AB (with arrow on the top) to represent a directed line segment.

莫比乌斯，以AB（上加箭头）表示起点为A，终点为B的向量 Möbius, using AB (with an arrow above) to denote a vector where A is the starting point and B is the ending point.

威廉·罗恩·哈密尔顿提出四元数，试图用四元数统一表示空间中的方向和旋转。他提出的四维代数结构中包括了“向量”的概念 William Rowan Hamilton proposed quaternions, attempting to represent directions and rotations in space uniformly using quaternions. The four-dimensional algebraic structure he proposed included the concept of "vectors".

吉布斯在耶鲁大学教授时撰写了向量分析的讲义 When Gibbs was teaching at Yale University, he wrote lecture notes on vector analysis. 1901年，他的学生把讲义《Vector Analysis》整理后出版 In 1901, his students compiled the lecture notes and published them（《Vector Analysis》）

重写了麦克斯韦方程，把原来20个标量方程简化为4个矢量方程 Oliver Heaviside rewrote the Maxwell's equations, reducing the original 20 scalar equations to 4 vector equations.

希尔伯特引入希尔伯特空间，开创无穷维向量空间理论 Hilbert introduced Hilbert space and initiated the theory of infinite-dimensional vector spaces.

斯特凡·巴拿赫引入了巴拿赫空间，即带有范数的完备向量空间 Stefan Banach introduce the concept of Banach space 在同一年冯诺伊曼发表《Mathematical Foundations of Quantum Mechanics》将粒子的“状态”定义为希尔伯特空间中的向量，观测量对应为自伴算子 In the same year, von Neumann published "Mathematical Foundations of Quantum Mechanics", in which he defined the "state" of a particle as a vector in a Hilbert space, and the observables corresponded to self-adjoint operators.

艾米·诺特推动了代数结构的研究。虽然她不直接叫“向量空间”，但她的抽象代数思想影响了向量空间的公理化和一般化 Amy Nott promoted the research on algebraic structures. Although she did not directly call it "vector space", her abstract algebraic ideas influenced the axiomatization and generalization of vector space.

加勒特·伯克霍夫和桑姆纳·麦克雷发表了《A Survey of Modern Algebra》强调了“向量空间”的公理化结构和代数运算，推动了数学教育和体系化发展。 Garrett Birkhoff and Saunders Mac Lane published "A Survey of Modern Algebra", which emphasized the axiomatic structure and algebraic operations of "vector spaces", and promoted the development of mathematics education and systematization.

微积分时代 The era of calculus

大革命时代 The era of the Great Revolution

奥利弗·海维赛德发表了大量的文章，引入现代向量形式 Oliver Heaviside published a large number of articles and introduced modern vector forms.

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