注：本文为 “复数 | 历史 / 演进” 相关文章合辑。

因 csdn 篇幅限制分篇连载，此为第二篇。

生料，不同的文章不同的点。

机翻，未校。

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History of Complex Numbers

复数的历史

The problem of complex numbers dates back to the 1st century, when Heron of Alexandria (about 75 AD) attempted to find the volume of a frustum of a pyramid, which required computing the square root of $81-144$ (though negative numbers were not conceived in the Hellenistic world).

复数问题可以追溯到公元 1 世纪，当时亚历山大港的希罗（约公元 75 年）试图计算平截头体的体积，这需要计算 $81-144$ 的平方根（尽管在希腊化时代还没有负数的概念）。

We also have the following quotation from Bhaskara Acharya (working in 486 AD), a Hindu mathematician: “The square of a positive number, also that of a negative number, is positive: and the square root of a positive number is two-fold, positive and negative; there is no square root of a negative number, for a negative number is not square.”

我们还可以看到印度数学家婆什迦罗・阿查里亚（活跃于公元 486 年）的一段话：“正数的平方是正数，负数的平方也是正数；正数的平方根有两个，一正一负；负数没有平方根，因为负数不是平方数。”

Later, around 850 AD, another Hindu mathematician, Mahavira Acharya, wrote: “As in the nature of things, a negative (quantity) is not a square (quantity), it has therefore no square root.”

后来，大约在公元 850 年，另一位印度数学家摩诃毗罗・阿查里亚写道：“从事物的本质来看，负数不是平方数，因此它没有平方根。”

In 1545, the Italian mathematician, physician, gambler, and philosopher Girolamo Cardano (1501-76) published his Ars Magna (The Great Art), in which he described algebraic methods for solving cubic and quartic equations.

1545 年，意大利数学家、医生、赌徒兼哲学家吉罗拉莫・卡尔达诺（1501-1576）出版了他的《大术》一书，书中阐述了求解三次和四次方程的代数方法。

This book was a great event in mathematics. In fact, it was the first major achievement in algebra in 3000 years, after the Babylonians showed how to solve quadratic equations.

这本书在数学界是一件大事。事实上，这是自巴比伦人展示如何求解二次方程以来，3000 年里代数学的首个重大成果。

Cardano also dealt with quadratics in his book. One of the problems that he called “manifestly impossible” is the following: Divide 10 into two parts whose product is 40; i.e., find the solution of $x+y=10$, $xy=40$, or, equivalently, the solution of the quadratic equation $40-x(10-x)=x^2-10x+40=0$ which has the roots $5+\sqrt{-15}$ and $5-\sqrt{-15}$.

卡尔达诺在书中也讨论了二次方程。他提出了一个他称之为 “显然不可能” 的问题：将 10 分成两部分，使它们的乘积为 40；即求解 $x+y=10$，$xy=40$，或者等价地，求解二次方程 $40-x(10-x)=x^2-10x+40=0$，该方程的根为 $5+\sqrt{-15}$ 和 $5-\sqrt{-15}$。

Cardano formally multiplied $5+\sqrt{-15}$ by $5-\sqrt{-15}$ and obtained 40; however, to calculations he said “putting aside the mental tortures involved.” He did not pursue the matter but concluded that the result was “as subtle as it is useless.”

卡尔达诺正式将 $5+\sqrt{-15}$ 与 $5-\sqrt{-15}$ 相乘并得到了 40；然而，对于这些计算，他说 “暂且抛开其中的精神折磨”。他没有继续深入研究这个问题，而是得出结论，认为这个结果 “既微妙又无用”。

This event was historic since it was the first time the square root of a negative number had been explicitly written down.

这一事件具有历史意义，因为这是首次明确写出负数的平方根。

For the cubic equation $x^3=ax+b$ the so-called Cardano formula is

对于三次方程 $x^3=ax+b$，所谓的卡尔达诺公式是

$$x=\sqrt[3]{\frac{b}{2}+\sqrt{{\left(\frac{b}{2}\right)}^2-{\left(\frac{a}{3}\right)}^3}}+\sqrt[3]{\frac{b}{2}-\sqrt{{\left(\frac{b}{2}\right)}^2-{\left(\frac{a}{3}\right)}^3}}.$$

When applied to the historic example $x^3=15x+4$, the formula yields

当将其应用于历史上的例子 $x^3=15x+4$ 时，该公式得出

$$x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}.$$

Although Cardano claimed that his general formula for the solution of the cubic equation was inapplicable in this case (because of the appearance of $\sqrt{-121}$), square roots of negative numbers could no longer be so lightly dismissed.

尽管卡尔达诺声称他求解三次方程的一般公式在这种情况下不适用（因为出现了 $\sqrt{-121}$），但负数的平方根再也不能被轻易忽视了。

Whereas for the quadratic equation (e.g., $x^2+1=0$) one could say that no solution exists, for the cubic $x^3=15x+4$ a real solution, namely $x=4$ does exist; in fact, the two other solutions, $-2\pm \sqrt{3}$, are also real.

对于二次方程（例如 $x^2+1=0$），人们可以说它没有解，而对于三次方程 $x^3=15x+4$，存在一个实数解 $x=4$；事实上，另外两个解 $-2\pm \sqrt{3}$ 也是实数。

It now remained to reconcile the formal and “meaningless” solution $x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$ of $x^3=15x+4$ found by using Cardano’s formula, with the solution $x=4$, found by inspection.

现在需要调和用卡尔达诺公式求出的 $x^3=15x+4$ 的形式上且 “无意义” 的解 $x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$ 与通过观察得到的解 $x=4$ 之间的矛盾。

The task was undertaken by the hydraulic engineer Rafael Bombelli (1526-73) about thirty years after the publication of Cardano’s work.

这项任务由水利工程师拉斐尔・邦贝利（1526-1573）在卡尔达诺的著作出版约 30 年后完成。

Bombelli had the “wild thought” that since the radicals $2+\sqrt{-121}$ and $2-\sqrt{-121}$ differ only in sign, the same might be true of their cube roots.

邦贝利有一个 “大胆的想法”，即由于根式 $2+\sqrt{-121}$ 和 $2-\sqrt{-121}$ 仅符号不同，它们的立方根可能也有同样的性质。

Thus, he let

于是，他令

$$\sqrt[3]{2+\sqrt{-121}}=a+\sqrt{-b}\text{ and }\sqrt[3]{2-\sqrt{-121}}=a-\sqrt{-b}$$

and proceeded to solve for $a$ and $b$ by manipulating these expressions according to the established rules for real variables.

并根据实数变量的既定规则对这些表达式进行运算，以求解 $a$ 和 $b$。

He deduced that $a=2$ and $b=1$ and thereby showed that, indeed,

他推导出 $a=2$ 且 $b=1$，从而证明，实际上

$$\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}=(2+\sqrt{-1})+(2-\sqrt{-1})=4.$$

Bombelli had thus given meaning to the “meaningless.” This event signaled the birth of complex numbers.

邦贝利就这样赋予了 “无意义” 的东西意义。这一事件标志着复数的诞生。

A breakthrough was achieved by thinking the unthinkable and daring to present it in public. Thus, the complex numbers forced themselves in connection with the solutions of cubic equations rather than the quadratic equations.

通过思考难以想象的事情并敢于公之于众，实现了突破。因此，复数是在求解三次方程而非二次方程的过程中应运而生的。

To formalize his discovery, Bombelli developed a calculus of operations with complex numbers. His rules, in our symbolism, are $(-i)(-i)=-1$ and

为了使他的发现形式化，邦贝利发展了一套复数运算规则。用我们现在的符号表示，他的规则是 $(-i)(-i)=-1$ 以及

$$(\pm 1)i=\pm i,(+i)(+i)=-1,(-i)(+i)=+1,$$

$$(\pm 1)(-i)=\mp i,(+i)(-i)=+1.$$

He also considered examples involving addition and multiplication of complex numbers, such as $8i+(-5i)=+3i$ and

他还考虑了涉及复数加法和乘法的例子，例如 $8i+(-5i)=+3i$ 以及

$$(\sqrt[3]{4+\sqrt{2i}})(\sqrt[3]{3+\sqrt{8i}})=\sqrt[3]{8+11\sqrt{2i}}.$$

Bombelli thus laid the foundation stone of the theory of complex numbers. However, his work was only the beginning of the saga of complex numbers.

邦贝利由此奠定了复数理论的基石。然而，他的工作仅仅是复数传奇的开端。

Although his book l’Algebra was widely read, complex numbers were shrouded in mystery, little understood, and often entirely ignored.

尽管他的《代数学》一书广为流传，但复数仍然笼罩在神秘之中，人们对其了解甚少，而且常常完全被忽视。

In fact, for complex numbers Simon Stevin (1548-1620) in 1585 remarked that “there is enough legitimate matter, even infinitely much, to exercise oneself without occupying oneself and wasting time on uncertainties.”

事实上，1585 年，西蒙・斯蒂文（1548-1620）针对复数评论道：“有足够多合理的内容，甚至是无穷无尽的，可供人们钻研，而无需在不确定的事物上花费时间。”

John Wallis (1616-1703), an astronomer, physicist, horologist, and writer of early science fiction, had pondered and puzzled over the meaning of imaginary numbers in geometry.

约翰・沃利斯（1616-1703），天文学家、物理学家、钟表学家以及早期科幻小说作家，曾对虚数在几何学中的意义进行思考并感到困惑。

He wrote, “These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen) are reputed to imply that the Case proposed is Impossible.”

他写道：“这些通常被称为虚数的量，源自负数平方的假设平方根（如果出现的话），人们认为这意味着所提出的问题是不可能的。”

Gottfried Wilhelm von Leibniz (1646-1716) made the following statement in 1702: “The imaginary numbers are a fine and wonderful refuge of the Divine Sprit, almost an amphibian between being and nonbeing.”

戈特弗里德・威廉・莱布尼茨（1646-1716）在 1702 年发表了如下言论：“虚数是圣灵的美妙避难所，几乎是介于存在与不存在之间的两栖物。”

Christiaan Huygens (1629-95), a prominent Dutch mathematician, was just as puzzled as Leibniz. In reply to a query he wrote to Leibniz: “One would never have believed that $\sqrt{1+\sqrt{-3}}+\sqrt{1-\sqrt{-3}}=\sqrt{6}$ and there is something hidden in this which is incomprehensible to us.”

克里斯蒂安・惠更斯（1629-1695），著名的荷兰数学家，和莱布尼茨一样困惑。在回复莱布尼茨的询问时，他写道：“人们绝不会相信 $\sqrt{1+\sqrt{-3}}+\sqrt{1-\sqrt{-3}}=\sqrt{6}$，其中一定隐藏着我们无法理解的东西。”

Leonhard Euler (1707-83) was candidly astonished by the remarkable fact that expressions such as $\sqrt{-1}$, $\sqrt{-2}$ etc., are neither nothing, nor greater than nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible.

莱昂哈德・欧拉（1707-1783）坦率地对诸如 $\sqrt{-1}$、$\sqrt{-2}$ 等表达式既非零，又非大于零，也非小于零这一显著事实感到惊讶，这必然使它们成为虚数或不可能的数。

In fact, he was confused by the absurdity $\sqrt{-4}\sqrt{-9}=(2i)(3i)=6i^2=-6$ while $\sqrt{(-4)(-9)}=\sqrt{36}=6\ne -6$.

事实上，他对 $\sqrt{-4}\sqrt{-9}=(2i)(3i)=6i^2=-6$ 而 $\sqrt{(-4)(-9)}=\sqrt{36}=6\ne -6$ 这种荒谬情况感到困惑。

Similar doubts concerning the meaning and legitimacy of complex numbers persisted for two and a half centuries.

关于复数的意义和合理性的类似疑虑持续了两个半世纪。

Nevertheless, during the same period complex numbers were extensively used and a considerable amount of theoretical work was done by such distinguished mathematicians as Ren´e Descartes (1596-1650) (who coined the term imaginary number, before him these numbers were called sophisticated or subtle), and Euler (who was the first to designate $\sqrt{-1}$ as $i$); Abraham de Moivre (1667-1754) in 1730 noted that the complicated identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be simply reexpressed by the well-known formula $(\cos \theta +i\sin \theta {)}^n=\cos n\theta +i\sin n\theta$ and many others.

尽管如此，在同一时期，复数被广泛应用，许多杰出的数学家开展了大量理论研究工作，比如勒内・笛卡尔（1596-1650，他创造了 “虚数” 这个术语，在他之前这些数被称为 “晦涩的” 或 “微妙的”），以及欧拉（他第一个用 $i$ 表示 $\sqrt{-1}$）；亚伯拉罕・棣莫弗（1667-1754）在 1730 年指出，将一个角的整数倍的三角函数与该角三角函数的幂联系起来的复杂恒等式，可以用著名的公式 $(\cos \theta +i\sin \theta {)}^n=\cos n\theta +i\sin n\theta$ 简单地重新表达，还有许多其他数学家也做出了贡献。

Complex numbers also found applications in map projection by Johann Heinrich Lambert (1728-77) and by Jean le Rond d’Alembert (1717-83) in hydrodynamics.

约翰・海因里希・兰伯特（1728-1777）将复数应用于地图投影，让・勒朗・达朗贝尔（1717-1783）则将其应用于流体动力学。

The desire for a logically satisfactory explanation of complex numbers became manifest in the latter part of the 18th century, on philosophical, if not on utilitarian grounds.

在 18 世纪后期，即使不是基于功利性的原因，从哲学角度出发，人们也明显渴望对复数有一个在逻辑上令人满意的解释。

With the advent of the Age of Reason, when mathematics was held up as a model to be followed not only in the natural sciences but also in philosophy as well as political and social thought, the inadequacy of a rational explanation of complex numbers was disturbing.

随着理性时代的到来，数学不仅被视为自然科学应遵循的模型，在哲学以及政治和社会思想领域也是如此，复数缺乏合理的解释这一情况令人不安 。

By 1831, the great German mathematician Karl Friedrich Gauss (1777-1855) had overcome his scruples concerning complex numbers (the phrase complex numbers is due to him) and, in connection with a work on number theory, published his results on the geometric representation of complex numbers as points in the plane.

到 1831 年，伟大的德国数学家卡尔・弗里德里希・高斯（1777-1855）克服了对复数的顾虑（“复数” 这个术语就是由他提出的），并且在一篇关于数论的著作中，发表了他将复数表示为平面上点的几何表示法的研究成果。

However, from Gauss’s diary, which was left among his papers, it is clear that he was already in possession of this interpretation by 1797.

然而，从他遗留在文件中的日记可以看出，高斯在 1797 年就已经有了这种解释。

Through this representation, Gauss clarified the “true metaphysics of imaginary numbers” and bestowed on them complete franchise in mathematics.

通过这种表示法，高斯阐明了 “虚数的真正本质”，并赋予了复数在数学领域的完全合法地位。

Similar representations by the Norwegian surveyor Casper Wessel (1745-1818) in 1797 and by the Swiss clerk Jean-Robert Argand (1768-1822) in 1806 went largely unnoticed.

挪威测量员卡斯珀・韦塞尔（1745-1818）在 1797 年、瑞士职员让-罗贝尔・阿尔冈（1768-1822）在 1806 年提出了类似的表示法，但在很大程度上未被注意到。

The concept modulus of complex numbers is due to Argand, and absolute value, for modulus, is due to Karl Theodor Wilhelm Weierstrass (1815-1897).

复数模的概念是由阿尔冈提出的，而用绝对值表示模则是卡尔・西奥多・威廉・魏尔斯特拉斯（1815-1897）的贡献。

The Cartesian coordinate system called the complex plane or Argand diagram is also named after the same Argand.

被称为复平面或阿尔冈图的笛卡尔坐标系也是以阿尔冈的名字命名的。

Mention should also be made of an excellent little treatise by C.V. Mourey (1828), in which the foundations for the theory of directional numbers are scientifically laid.

还值得一提的是 C.V. 穆雷在 1828 年撰写的一篇出色的小论文，在其中科学地奠定了有向数理论的基础。

The general acceptance of the theory is not a little due to the labors of Augustin Louis Cauchy (1789-1857) and Niels Henrik Abel (1802-1829), especially the latter, who was the first to boldly use complex numbers, with a success that is well-known.

这一理论能够被普遍接受，在很大程度上归功于奥古斯丁・路易・柯西（1789-1857）和尼尔斯・亨利克・阿贝尔（1802-1829）的努力，尤其是阿贝尔，他是首位大胆使用复数并取得显著成功的人。

Geometric applications of complex numbers appeared in several memoirs of prominent mathematicians such as August Ferdinand Möbius (1790-1868), George Peacock (1791-1858), Giusto Bellavitis (1803-1880), Augustus De Morgan (1806-1871), Ernst Kummer (1810-1893), and Leopold Kronecker (1823-1891).

复数在几何方面的应用出现在许多杰出数学家的论文中，比如奥古斯特・费迪南德・莫比乌斯（1790-1868）、乔治・皮科克（1791-1858）、朱斯托・贝拉维蒂斯（1803-1880）、奥古斯塔斯・德・摩根（1806-1871）、恩斯特・库默尔（1810-1893）以及利奥波德・克罗内克（1823-1891）。

In the next three decades, further development took place. Especially, in 1833 William Rowan Hamilton (1805-1865) gave an essentially rigorous algebraic definition of complex numbers as pairs of real numbers.

在接下来的三十年里，复数理论进一步发展。特别是在 1833 年，威廉・罗恩・哈密顿（1805-1865）给出了复数的一个基本严谨的代数定义，将其定义为实数对。

However, a lack of confidence in them persisted; for example, the English mathematician and astronomer George Airy (1801-1892) declared: “I have not the smallest confidence in any result which is essentially obtained by the use of imaginary symbols.”

然而，人们对复数仍然缺乏信心；例如，英国数学家和天文学家乔治・艾里（1801-1892）宣称：“对于任何本质上通过使用虚数符号得到的结果，我一点信心都没有。”

The English logician George Boole (1815-1864) in 1854 called $\sqrt{-1}$ an “uninterpretable symbol.”

英国逻辑学家乔治・布尔（1815-1864）在 1854 年称 $\sqrt{-1}$ 为 “无法解释的符号”。

The German mathematician Leopold Kronecker believed that mathematics should deal only with whole numbers and with a finite number of operations, and is credited with saying: “God made the natural numbers; all else is the work of man.”

德国数学家利奥波德・克罗内克认为数学只应该处理整数和有限次运算，他有一句名言：“上帝创造了自然数，其余的都是人类的杰作。”

He felt that irrational, imaginary, and all other numbers excluding the positive integers were man’s work and therefore unreliable.

他认为无理数、虚数以及除正整数之外的所有其他数都是人为创造的，因此不可靠。

However, the French mathematician Jacques Salomon Hadamard (1865-1963) said the shortest path between two truths in the real domain passes through the complex domain.

然而，法国数学家雅克・所罗门・哈达玛（1865-1963）说过，在实数领域中，两个真理之间的最短路径要经过复数领域。

By the latter part of the 19th century, all vestiges of mystery and distrust of complex numbers could be said to have disappeared, although some resistance continued among a few textbook writers well into the 20th century.

到 19 世纪后期，可以说对复数的所有神秘感和不信任都已消失，尽管在 20 世纪，仍有一些教科书作者对复数持有抵触态度。

Nowadays, complex numbers are viewed in the following different ways:

如今，人们从以下不同角度看待复数：

1. points or vectors in the plane;

   平面上的点或向量；

2. ordered pairs of real numbers;

   实数的有序对；

3. operators (i.e., rotations of vectors in the plane);

   算子（即平面向量的旋转）；

4. numbers of the form $a+bi$, with $a$ and $b$ real numbers;

   形如 $a+bi$ 的数，其中 $a$ 和 $b$ 是实数；

5. polynomials with real coefficients modulo $x^2+1$;

   实系数多项式模 $x^2+1$；

6. matrices of the form $\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]$, with $a$ and $b$ real numbers;

   形如 $\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]$ 的矩阵，其中 $a$ 和 $b$ 是实数；

7. an algebraically closed complete field (a field is an algebraic structure that has the four operations of arithmetic).

   一个代数封闭的完备域（域是一种具有四则运算的代数结构）。

The foregoing descriptions of complex numbers are not the end of the story. Various developments in the 19th and 20th centuries enabled us to gain a deeper insight into the role of complex numbers in mathematics (algebra, analysis, geometry, and the most fundamental work of Peter Gustav Lejeune Dirichlet (1805-1859) in number theory); engineering (stresses and strains on beams, resonance phenomena in structures as different as tall buildings and suspension bridges, control theory, signal analysis, quantum mechanics, fluid dynamics, electric circuits, aircraft wings, and electromagnetic waves); and physics (relativity, fractals, and the Schrödinger equation).

上述对复数的描述并非故事的全部。19 世纪和 20 世纪的各种发展让我们更深入地了解了复数在数学（代数、分析、几何以及彼得・古斯塔夫・勒热纳・狄利克雷（1805-1859）在数论方面的基础工作）、工程学（梁的应力和应变、从高楼到悬索桥等不同结构中的共振现象、控制理论、信号分析、量子力学、流体动力学、电路、飞机机翼以及电磁波）和物理学（相对论、分形以及薛定谔方程）中的作用。

Although scholars who employ complex numbers in their work today do not think of them as mysterious, these quantities still have an aura for the mathematically naive.

尽管如今在工作中使用复数的学者并不觉得它们神秘，但对于数学知识欠缺的人来说，复数仍然带有一种神秘色彩。

For example, the famous 20th-century French intellectual and psychoanalyst Jacques Lacan (1901-1981) saw a sexual meaning in $\sqrt{-1}$.

例如，20 世纪著名的法国知识分子、精神分析学家雅克・拉康（1901-1981）从 $\sqrt{-1}$ 中看出了性的含义。

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Complex Numbers: Interpretations and Developments

复数：阐释与发展

Sarah Teklinski

萨拉・特克林斯基

Introduction

引言

The history of the study of complex numbers endured much evolution and controversy from some of the most renowned mathematicians of the time. Some great mathematicians, like Newton, completely turned down the notion of a complex number, while others were suspicious of their existence. In fact, it was only during the $16^{th}$ century that mathematicians even began to study these so-called imaginary numbers [1]. Before then, if a mathematician arrived at an answer with the square root of a negative number, he would automatically discard that solution [1]. The focus of this paper is on the various developments and interpretations on the square root of a negative number put forth by Hieronimo Cardano (1501-1576), Rafael Bombelli (1526-1572), Gottfried Wilhelm Leibniz (1646-1716), René Descartes (1596-1650), John Wallis (1612-1703), Caspar Wessel (1745-1818), Robert Argand (1768-1822), Abbé Adrien-Quentin Buée (1748-1826), William Rowan Hamilton (1805-1852), and Leonhard Euler (1707-1783). The paper will conclude with a brief excerpt about complex function theory and some of the most important theorems related to complex numbers, including Euler’s identity, the Cauchy-Riemann Equations, and Laplace’s equations.

复数的研究历史历经诸多演变，还引发了当时一些著名数学家的争议。像牛顿这样的伟大数学家，完全拒绝复数的概念，而其他人则对其存在表示怀疑。事实上，直到 16 世纪，数学家们才开始研究这些所谓的虚数 [1]。在此之前，如果一位数学家得到的答案中含有负数的平方根，他会自动舍弃这个解 [1]。本文重点探讨希罗尼莫・卡尔达诺（1501-1576）、拉斐尔・邦贝利（1526-1572）、戈特弗里德・威廉・莱布尼茨（1646-1716）、勒内・笛卡尔（1596-1650）、约翰・沃利斯（1612-1703）、卡斯帕・韦塞尔（1745-1818）、罗伯特・阿尔冈（1768-1822）、阿贝・阿德里安-昆廷・比埃（1748-1826）、威廉・罗恩・哈密顿（1805-1852）和莱昂哈德・欧拉（1707-1783）对负数平方根的各种发展和解释。本文最后将简要介绍复变函数理论以及一些与复数相关的最重要定理，包括欧拉恒等式、柯西-黎曼方程和拉普拉斯方程。

The Beginnings

起源

The first known problem involving complex numbers arose in the $1^{st}$ century AD when the Greek mathematician Heron of Alexandria (10 AD – 70 AD) attempted to calculate the volume of a frustum of a pyramid [1]. In one part of his solution he arrived at the quantity $\sqrt{81-144}$ , but he immediately threw aside this notion of the square root of a negative number. Mathematicians from then on followed in his footsteps, and any thoughts about the square root of a negative number were tossed aside. In 486 AD we find the Indian mathematician Bhaskara Acharya who claimed that the square root of a negative number does not exist since negative numbers cannot be a square [1]. The thoughts of Heron and Acharya seemed to be the trend for the next eleven centuries, for whenever mathematicians came across the square root of a negative number in their answer, like Heron, they disregarded these solutions, for they thought these types of solutions were not mathematically possible and thus were meaningless. It was not until the $16^{th}$ century that mathematicians started to ponder about what these concepts actually meant.

已知最早涉及复数的问题出现在公元 1 世纪，当时希腊数学家亚历山大港的希罗（公元 10 年-70 年）试图计算截头锥体的体积 [1]。在他的求解过程中，有一部分得出了 $\sqrt{81-144}$ 这个量，但他立刻摒弃了负数平方根这个概念。从那以后，数学家们都遵循他的做法，任何关于负数平方根的想法都被抛诸脑后。公元 486 年，印度数学家婆什迦罗・阿查里亚宣称负数的平方根不存在，因为负数不能是一个数的平方 [1]。在接下来的十一个世纪里，希罗和阿查里亚的观点似乎成为了主流，每当数学家在答案中遇到负数的平方根时，就像希罗一样，他们会忽略这些解，因为他们认为这类解在数学上是不可能的，因此毫无意义。直到 16 世纪，数学家们才开始思考这些概念究竟意味着什么。

Introduction to the Study of Complex Numbers

复数研究的开端

In the $16^{th}$ century we find the first person to think about the meaning of a complex number: Cardano (1501-1576). In 1545, he published his Ars Magna, which contains one of the most historic achievements in mathematics of the time discovered by Scipio Del Ferro (1465-1526): the general solution to a cubic equation on the form $x^3+px=q$ [5]. He attempted to solve the cubic equation $x^3=15x+4$ for $x$ , for which he found that $x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$ . He, like great mathematicians before him, claimed there was no solution, and he asserted a solution of this type was ridiculous, but nonetheless, he was not “afraid” of these types of solutions like others previously [4]. Although he thought this type of solution was bizarre, this was a historically significant result because it represented the first time that the square root of a negative number was written down [4]. Another turning point in the beginning of the study of complex numbers was when Cardano solved for $x$ and $y$ in $x+y=10$ and $xy=40$ [3], finding that $x=5+\sqrt{-15}$ and $y=5-\sqrt{-15}$ . After multiplying these solutions together, he concluded that the result was 40, and he was confused as to how solutions containing the square root of a negative number could represent a real number. It was Bombelli (1526-1572) who elaborated on this idea and proposed his interpretation of complex numbers [4].

16 世纪，我们迎来了首位思考复数意义的人：卡尔达诺（1501-1576）。1545 年，他出版了《大术》一书，书中包含了当时由希皮奥内・德尔・费罗（1465-1526）发现的数学领域最具历史性的成就之一：关于 $x^3+px=q$ 形式三次方程的通解 [5]。他试图求解三次方程 $x^3=15x+4$ 中的 $x$ ，得出 $x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$。和他之前的伟大数学家一样，他宣称这个方程无解，并断言这种类型的解很荒谬，但尽管如此，他不像之前的其他人那样 “惧怕” 这类解 [4]。虽然他认为这种解很怪异，但这一结果具有重要的历史意义，因为这是负数的平方根首次被写下来 [4]。复数研究开端的另一个转折点是，卡尔达诺在求解 $x+y=10$ 和 $xy=40$ 中的 $x$ 和 $y$ 时 [3]，得到 $x=5+\sqrt{-15}$ ，$y=5-\sqrt{-15}$。将这些解相乘后，他得出结果为 40，他感到困惑的是，包含负数平方根的解如何能表示一个实数。正是邦贝利（1526-1572）对这一想法进行了详细阐述，并提出了他对复数的解释 [4]。

Bombelli considered the same cubic equation as Cardano, $x^3=15x+4$ [4], and by inspection he found that $x=4$ was a solution [4]. After long division and factoring, he found that $x=-2+\sqrt{3}$ and $x=-2-\sqrt{3}$ were the other two solutions [4]. Recall that solving this cubic using Cardano’s method yields a solution of $x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$ seemingly different than Bombelli’s solutions. The curious type of person would ask themselves what happened to Cardano’s solutions, and what is exactly what Bombelli did. He noted that the two terms of Cardano’s solution only differed in sign, so he denoted $\sqrt[3]{2+\sqrt{-121}}$ as $a+\sqrt{-b}$ and $\sqrt[3]{2-\sqrt{-121}}$ as $a-\sqrt{-b}$ [4], and after much manipulation of these two expressions, he eventually concluded that the sum of the two is 4. With this Bombelli had given his interpretation that the two complex numbers Cardano arrived at were in fact real, but they were denoted in an unfamiliar notion.

邦贝利考虑了与卡尔达诺相同的三次方程 $x^3=15x+4$ [4]，通过观察他发现 $x=4$ 是一个解 [4]。经过长除法和因式分解，他发现 $x=-2+\sqrt{3}$ 和 $x=-2-\sqrt{3}$ 是另外两个解 [4]。回想一下，用卡尔达诺的方法求解这个三次方程得到的解是 $x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$ ，这似乎与邦贝利的解不同。好奇的人会问自己，卡尔达诺的解怎么了，而邦贝利正是这么做的。他注意到卡尔达诺解中的两项仅符号不同，所以他将 $\sqrt[3]{2+\sqrt{-121}}$ 记为 $a+\sqrt{-b}$ ，将 $\sqrt[3]{2-\sqrt{-121}}$ 记为 $a-\sqrt{-b}$ [4]，经过对这两个表达式的大量运算，他最终得出这两项之和为 4。由此，邦贝利给出了他的解释：卡尔达诺得到的两个复数实际上是实数，只是用了一种不常见的表示方式。

This discovery revolutionized the mathematical world, for he had interpreted what a complex number meant, therefore creating the study of complex numbers. Bombelli published his results in L’Algebra, which contained all his notions about complex numbers [4]. Some mathematicians developed on Bombelli’s ideas, yet some were skeptical of this radical idea and thus were not satisfied with his arguments. The movement to dig deeper into the meaning of complex numbers did not end here, though. In fact, Bombelli’s discovery was only the beginning of a new branch of mathematics.

这一发现彻底改变了数学界，因为他解释了复数的含义，从而开创了复数的研究。邦贝利在《代数学》一书中发表了他的研究成果，书中包含了他关于复数的所有观点 [4]。一些数学家在邦贝利的思想基础上进行了拓展，但也有一些人对这个激进的想法持怀疑态度，因此对他的论证并不满意。然而，深入探究复数意义的脚步并未就此停止。事实上，邦贝利的发现仅仅是数学一个新分支的开端。

A Geometrical Interpretation

几何解释

About 100 years later, many mathematicians accepted Bombelli’s contributions, but the great Leibniz still was not convinced. Leibniz (1646-1716) was not satisfied with Bombelli’s treatment of Cardano’s formula, and Leibniz did not comprehend how adding two complex numbers would yield a real number [4].

大约 100 年后，许多数学家接受了邦贝利的贡献，但伟大的莱布尼茨仍然心存疑虑。莱布尼茨（1646-1716）对邦贝利对卡尔达诺公式的处理方式并不满意，他也不理解两个复数相加如何能得到一个实数 [4]。

In the $17^{th}$ century we find that mathematicians were not satisfied with just a general meaning of complex numbers. Mathematicians were baffled at what the geometric interpretation could be, and many played their hand at deducing such an interpretation. Descartes (1596-1650) was the first mathematician to try to find a geometric interpretation for imaginary numbers, but after much analysis, he concluded that imaginary numbers could not be associated with a geometric construction [4].

在 17 世纪，我们发现数学家们并不满足于复数的一般含义。他们对复数的几何解释感到困惑，许多人都试图推导出这样的解释。笛卡尔（1596-1650）是首位试图为虚数寻找几何解释的数学家，但经过大量分析后，他得出结论：虚数无法与几何构造联系起来 [4]。

The next person to attempt to construct a geometric interpretation of imaginary numbers was Wallis (1612-1703) [4]. The key word here is attempt – Wallis was able to deduce an argument of some sort, but his argument was not quite convincing enough for the mathematicians of the time, and he had nothing revolutionary to say about $\sqrt{-1}$ . He composed a complicated geometrical argument, which concluded with his construction of a diagram of this sort below and claiming that $A^{\prime }P=\sqrt{(B{A}^{\prime })(A^{\prime }D)}$ [4].

接下来尝试为虚数构建几何解释的是沃利斯（1612-1703）[4]。这里的关键词是 “尝试”—— 沃利斯能够给出某种论证，但他的论证对当时的数学家来说并不够有说服力，而且他对 $\sqrt{-1}$ 也没有提出什么革命性的观点。他构建了一个复杂的几何论证，最终得出了如下这样一个图表，并声称 $A^{\prime }P=\sqrt{(B{A}^{\prime })(A^{\prime }D)}$ [4]。

His argument that he had come up with some sort of interpretation for $\sqrt{-1}$ was based on his thought that $B{A}^{\prime }$ was a negative distance while $A^{\prime }D$ was positive. He believed that because a negative number can be thought of as starting at 0 on a number line and moving left, then $B{A}^{\prime }$ was a negative distance [4]. He therefore concluded that $(B{A}^{\prime })(A^{\prime }D)$ was negative and thus $A^{\prime }P$ represented the square root of a negative number.

他认为自己对 $\sqrt{-1}$ 有了某种解释，其依据是他认为 $B{A}^{\prime }$ 是负距离，而 $A^{\prime }D$ 是正距离。他认为，因为负数可以被看作是在数轴上从 0 开始向左移动，所以 $B{A}^{\prime }$ 是负距离 [4]。因此，他得出 $(B{A}^{\prime })(A^{\prime }D)$ 是负数，从而 $A^{\prime }P$ 代表了负数的平方根。

Once we enter into the $18^{th}$ century, we find that people could not wrap their heads around a geometrical, or more generally, a logical, explanation of complex numbers. The notion of complex numbers even seemed to break all rational explanations of mathematics. For $a$, $b$ negative integers, is $\sqrt{ab}=\sqrt{a}\sqrt{b}$ ? Well, with complex numbers, an example of where this is not true is $70=\sqrt{(-100)(-49)}=\sqrt{-100}\sqrt{-49}=(10i)(7i)=(70)(-1)=-70$ . The search for a geometric interpretation of $\sqrt{-1}$ was not over, though, despite the seemingly illogical nature of $\sqrt{-1}$ . We find that Wessel (1745-1818), ironically not even a mathematician, but rather a surveyor, was able to crack the mystery behind $\sqrt{-1}$

进入 18 世纪，人们仍然无法理解复数的几何解释，或者更普遍地说，无法理解其逻辑解释。复数的概念似乎打破了数学所有合理的解释。对于负整数 $a$、$b$ ，$\sqrt{ab}=\sqrt{a}\sqrt{b}$ 成立吗？在复数的情况下，有一个反例：$70=\sqrt{(-100)(-49)}=\sqrt{-100}\sqrt{-49}=(10i)(7i)=(70)(-1)=-70$。尽管 $\sqrt{-1}$ 看似不合逻辑，但寻找 $\sqrt{-1}$ 几何解释的脚步并未停止。具有讽刺意味的是，韦塞尔（1745-1818）并非数学家，而是一名测量员，他却解开了 $\sqrt{-1}$ 背后的谜团。

Wessel presented his paper (1797) that simplified the geometric interpretations of complex numbers that Wallis originally suggested [4]. He considered the standard notion of the real number line, where starting at 0, moving to the right suggests the numbers are getting larger, and moving to the left suggests the numbers are getting smaller, thus becoming negative. He built upon this, creating a vertical “imaginary” axis perpendicular to the horizontal “real” axis, therefore creating the complex plane. A complex number $a+bi$ represented moving $a$ units right (if $a$ is positive) or $|a|$ units left (if $a$ is negative) and $b$ units up (if $b$ is positive) or $|b|$ units down (if $b$ is negative), and Wessel denoted this notation the rectangular, or Cartesian form [4]. He built upon this idea, later creating the polar form, which is as follows. He would denote $a+bi$ as a vector, and he let $\theta$ represent the angle formed from the positive part of the horizontal axis to the vector (a counterclockwise motion). Using the Pythagorean Theorem and a basic knowledge of trigonometry, he concluded that $a+bi=\sqrt{a^2+b^2}(\cos (\theta )+i\sin (\theta ))$ , writing that $\sqrt{a^2+b^2}$ is the modulus of the complex number $a+bi$ and $\theta$ is the argument of the complex number [4].

韦塞尔在 1797 年发表了一篇论文，简化了沃利斯最初提出的复数几何解释 [4]。他考虑了标准的实数轴概念，即从 0 开始，向右移动表示数字增大，向左移动表示数字减小，进而变为负数。他在此基础上进行拓展，创建了一条垂直于水平 “实轴” 的 “虚轴”，从而构建出了复平面。一个复数 $a+bi$ 表示在水平方向上向右移动 $a$ 个单位（若 $a$ 为正）或向左移动 $|a|$ 个单位（若 $a$ 为负），在垂直方向上向上移动 $b$ 个单位（若 $b$ 为正）或向下移动 $|b|$ 个单位（若 $b$ 为负），韦塞尔将这种表示法称为直角坐标形式或笛卡尔形式 [4]。他进一步拓展这一概念，随后创造出了极坐标形式，具体如下。他将 $a+bi$ 表示为一个向量，令 $\theta$ 表示从水平轴正半轴到该向量所形成的夹角（逆时针方向）。利用勾股定理和基本的三角函数知识，他得出 $a+bi=\sqrt{a^2+b^2}(\cos (\theta )+i\sin (\theta ))$ ，并指出 $\sqrt{a^2+b^2}$ 是复数 $a+bi$ 的模，$\theta$ 是复数的辐角 [4]。

Wessel was historically significant because he was the first to explicitly denote that the imaginary axis is perpendicular to the real axis, even though some mathematicians such as Henri Dominique Truel and Karl Friedrich Gauss (1777-1855) proposed the idea before but never published their results [4]. (Gauss was monumental in the sense that he was the first person to use the phrase complex number [2].) Another monumental discovery which stems from Wessel’s definition of the complex plane was that $i$ represents the vector with no movement on the real axis and 1 unit up on the imaginary axis [4], to which the reader can see that this represents a 90 degree rotation counterclockwise. More generally, if we multiply an arbitrary vector $a+bi$ by $i$, we find that the product $(a+bi)i=ai+b{i}^2=ai-b=-b+ai$. Essentially, $\sqrt{-1}$ can be thought of as a rotation operator by 90 degrees [4].

韦塞尔具有重要的历史意义，因为他是第一个明确指出虚轴与实轴垂直的人，尽管像亨利・多米尼克・特吕埃和卡尔・弗里德里希・高斯（1777-1855）等一些数学家之前提出过这个想法，但从未发表他们的成果 [4]。（高斯的重要性还在于他是第一个使用 “复数” 这个术语的人 [2]。）从韦塞尔对复平面的定义中还衍生出了另一个重大发现，即 $i$ 表示在实轴上没有移动，在虚轴上向上移动 1 个单位的向量 [4]，读者可以看到这代表着逆时针旋转 90 度。更一般地说，如果我们将任意向量 $a+bi$ 乘以 $i$ ，会得到乘积 $(a+bi)i=ai+b{i}^2=ai-b=-b+ai$。从本质上讲，$\sqrt{-1}$ 可以被看作是一个逆时针旋转 90 度的旋转算子 [4]。

Even though Wessel introduced and published quite a few revolutionary arguments, many mathematicians were still not convinced of his discoveries. However, the two mathematicians Argand (1768-1822) and Buée (1748-1826) rediscovered and supported his ideas [4]. Argand’s approach (1806) is quite similar to Wessel’s argument and will not be repeated here [2]. In fact, Argand is quite often mistakenly credited as the first person to offer the geometric approach that Wessel made because his ideas did not receive as much fame as Argand’s. Buée’s argument, however, is different from Wessel’s but still results in his same conclusions.

尽管韦塞尔提出并发表了许多具有革命性的论点，但许多数学家仍然不相信他的发现。然而，数学家阿尔冈（1768-1822）和比埃（1748-1826）重新发现并支持了他的观点 [4]。阿尔冈在 1806 年提出的方法与韦塞尔的论证非常相似，这里不再赘述 [2]。事实上，阿尔冈常常被错误地认为是第一个提出韦塞尔那种几何方法的人，因为韦塞尔的观点没有阿尔冈的那么出名。不过，比埃的论证与韦塞尔的不同，但得出的结论是一样的。

Buée’s original goal was to answer the questions posed in Lazare Carnot’s (1753-1823) publication Géométrie de Position (1803) [4]. Carnot considered the division of a line segment of length $a$ into two segments such that when the two lengths are multiplied together, the product is equal to $\frac{1}{2}$ the original side squared. He attempted to solve the equation $x(a-x)=\frac{1}{2}{a}^2$ for $x$ of which the answer is found to be $x=\frac{1}{2}a\pm (i)(\frac{1}{2}a)$ . Like any mathematician before Cardano, Carnot was puzzled by the $\sqrt{-1}$ in his answer and thus assumed his initial condition was not possible. Buée, however, challenged this and offered a response. He claimed that the imaginary part of the answer, $\frac{1}{2}a$, represents that the point, which would divide $a$ into two segments according to the original condition, is located $\frac{1}{2}a$ units away perpendicular to the line segment. He did not justify his claim, though, making him lose credibility in the eyes of some mathematicians of the time.

比埃最初的目标是回答拉扎尔・卡诺（1753-1823）在 1803 年出版的《位置几何学》一书中提出的问题 [4]。卡诺考虑将长度为 $a$ 的线段分成两段，使得这两段长度相乘的结果等于原线段长度平方的一半。他试图求解方程 $x(a-x)=\frac{1}{2}{a}^2$ 中的 $x$ ，结果发现 $x=\frac{1}{2}a\pm (i)(\frac{1}{2}a)$。和卡尔达诺之前的任何数学家一样，卡诺对答案中的 $\sqrt{-1}$ 感到困惑，因此认为他最初的条件是不可能实现的。然而，比埃对此提出了质疑并给出了回应。他声称答案中的虚数部分 $\frac{1}{2}a$ 表示，根据原始条件将 $a$ 分成两段的那个点，位于垂直于该线段且距离为 $\frac{1}{2}a$ 个单位的位置。不过，他没有对自己的说法进行论证，这使得他在当时一些数学家眼中失去了可信度。

Looking back at the $17^{th}$ and $18^{th}$ centuries, it is clear that Wessel was the winner in terms of a geometrical argument that many mathematicians seemed to be fond of. Because of Wessel, mathematicians were able to find the geometrical approach they were longing for. As we enter the $19^{th}$ century, we find some mathematicians dissatisfied with the approaches put forth, and some looking to further expand upon the theory of complex numbers.

回顾 17 和 18 世纪，显然在几何论证方面，韦塞尔的观点受到了许多数学家的喜爱。由于韦塞尔的贡献，数学家们找到了他们一直渴望的几何方法。进入 19 世纪，我们发现一些数学家对已提出的方法并不满意，还有一些人希望进一步拓展复数理论。

An Algebraic Approach

代数方法

As we enter the $19^{th}$ century, we find Hamilton (1805-1852) unhappy with the geometric interpretation of $\sqrt{-1}$; in fact, he did not even think $\sqrt{-1}$ should have a geometric interpretation, for he believed there should only be an algebraic interpretation [4]. In his publication Theory of Conjugate Functions or Algebraic Couples: with a Preliminary Essay on Algebra as a Science of Pure Time (1835), he considered what he called an ordered pair (couple) of real numbers $(a,b)$ [4]. We can see that Hamilton based this notation off of Wessel’s notation of $a+bi$. Hamilton defined addition as $(a,b)+(c,d)=(a+c,b+d)$ and multiplication as $(a,b)(c,d)=(ac-bd,bc+ad)$ [4].

进入 19 世纪，哈密顿（1805-1852）对 $\sqrt{-1}$ 的几何解释并不满意；事实上，他甚至认为 $\sqrt{-1}$ 不应该有几何解释，因为他认为只应该有代数解释 [4]。在他 1835 年出版的《共轭函数理论或代数偶：兼论作为纯时间科学的代数学》一书中，他考虑了他所称的实数的有序对（偶）$(a,b)$ [4]。我们可以看到，哈密顿的这种表示法是基于韦塞尔对 $a+bi$ 的表示法。哈密顿将加法定义为 $(a,b)+(c,d)=(a+c,b+d)$ ，乘法定义为 $(a,b)(c,d)=(ac-bd,bc+ad)$ [4]。

Start of the Theory of Complex Functions and Some Monumental Theorems

复变函数理论的开端及一些重要定理

In the $19^{th}$ century it seemed as though that mathematicians had learned the basics of complex numbers. Eventually, they began to consider functions of complex variables. Denoting $z=x+yi$ as our complex variable, they denoted the complex function to be $f(z)=f(x+yi)$ Another way to write this is $f(z)=u(x,y)+iv(x,y)$, which will come back into play in the “Cauchy-Riemann Equations and Lagrange’s Equations” section.

在 19 世纪，数学家们似乎已经掌握了复数的基础知识。最终，他们开始考虑复变量的函数。将 $z=x+yi$ 表示为我们的复变量，他们将复函数表示为 $f(z)=f(x+yi)$ 。另一种写法是 $f(z)=u(x,y)+iv(x,y)$ ，这在 “柯西-黎曼方程和拉普拉斯方程” 这一部分还会用到。

Euler and Euler’s Identity

欧拉与欧拉恒等式

Although Wessel is generally credited with many monumental contributions to $\sqrt{-1}$, Euler (1707-1783) knew the notions behind these contributions before Wessel [4]. In 1748, when Wessel was just three years old, Euler published the identity $e^{\pm ix}=\cos (x)\pm i\sin (x)$ Substituting in $x=-\frac{\pi }{2}$ in for $e^{\pm ix}=\cos (x)\pm i\sin (x)$, we arrive at $e^{-\frac{\pi }{2}i}=i^i$ . What we have just shown is that an imaginary number raised to an imaginary number is a real number, what mathematicians including Argand once thought was not possible [4]. Additionally, substituting in $x=\pi$, we arrive at Euler’s identity: $e^{i\pi }+1=0$ . Some call this the most beautiful theorem in all of mathematics, as it so elegantly combines five of the most fascinating numbers [4]. Furthermore, not only is it such a beautiful theorem, it holds many practical applications in differential equations and engineering.

虽然韦塞尔通常因对 $\sqrt{-1}$ 的许多重大贡献而受到赞誉，但欧拉（1707-1783）在韦塞尔之前就已经知晓这些贡献背后的理念 [4]。1748 年，韦塞尔年仅 3 岁时，欧拉发表了恒等式 $e^{\pm ix}=\cos (x)\pm i\sin (x)$ 。将 $x=-\frac{\pi }{2}$ 代入 $e^{\pm ix}=\cos (x)\pm i\sin (x)$ ，我们得到 $e^{-\frac{\pi }{2}i}=i^i$ 。我们刚刚展示的是，一个虚数的虚数次幂是一个实数，这是包括阿尔冈在内的许多数学家曾经认为不可能的事情 [4]。此外，将 $x=\pi$ 代入，我们得到欧拉恒等式：$e^{i\pi }+1=0$ 。一些人将其称为数学中最优美的定理，因为它如此优雅地将五个最迷人的数字结合在了一起 [4]。此外，它不仅是一个非常优美的定理，在微分方程和工程领域也有许多实际应用。

Cauchy-Riemann Equations and Lagrange’s Equations

柯西-黎曼方程和拉普拉斯方程

The works of Augustin-Louis Cauchy (1789-1857) and Bernhard Riemann (1826-1866) paved their way for the start of the theory of complex functions [2]. Let us revisit $f(z)=u(x,y)+iv(x,y)$. In order to study the theory of complex functions, one must be able to understand how to find $f^{\prime }(z)$. The Cauchy-Riemann Equations so elegantly define $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$ as $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$ [4]. These equations form the basis for complex function theory [2], and taking the derivative of each of the equations yields $\frac{{\partial }^{2}u}{\partial y\partial x}=\frac{{\partial }^{2}v}{\partial y^2},\frac{{\partial }^{2}v}{\partial y\partial x}=-\frac{{\partial }^{2}u}{\partial y^2},\frac{{\partial }^{2}u}{\partial x^2}=\frac{{\partial }^{2}v}{\partial x\partial y}$, and $\frac{{\partial }^{2}v}{\partial x^2}=-\frac{{\partial }^{2}u}{\partial x\partial y}$.

By combining $\frac{{\partial }^{2}u}{\partial y\partial x}=\frac{{\partial }^{2}v}{\partial y^2}$ and $\frac{{\partial }^{2}v}{\partial x^2}=-\frac{{\partial }^{2}u}{\partial x\partial y}$, one gets $\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}=0$, and by combining $\frac{{\partial }^{2}v}{\partial y\partial x}=-\frac{{\partial }^{2}u}{\partial y^2}$ and $\frac{{\partial }^{2}u}{\partial x^2}=\frac{{\partial }^{2}v}{\partial x\partial y}$, one gets $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0$. These two equations are known as Laplace’s equations [4], which have immense applications in engineering.

奥古斯丁-路易・柯西（1789-1857）和伯恩哈德・黎曼（1826-1866）的工作为复变函数理论的开端奠定了基础 [2]。让我们回顾一下 $f(z)=u(x,y)+iv(x,y)$ 。为了研究复变函数理论，必须理解如何求 $f^{\prime }(z)$ 。柯西-黎曼方程巧妙地将 $\frac{\partial u}{\partial x}$ 和 $\frac{\partial u}{\partial y}$ 定义为 $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ 以及 $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$ [4]。这些方程构成了复变函数理论的基础 [2]，对每个方程求导可得 $\frac{{\partial }^{2}u}{\partial y\partial x}=\frac{{\partial }^{2}v}{\partial y^2}$ ，$\frac{{\partial }^{2}v}{\partial y\partial x}=-\frac{{\partial }^{2}u}{\partial y^2}$ ，$\frac{{\partial }^{2}u}{\partial x^2}=\frac{{\partial }^{2}v}{\partial x\partial y}$ ，以及 $\frac{{\partial }^{2}v}{\partial x^2}=-\frac{{\partial }^{2}u}{\partial x\partial y}$ 。将 $\frac{{\partial }^{2}u}{\partial y\partial x}=\frac{{\partial }^{2}v}{\partial y^2}$ 与 $\frac{{\partial }^{2}v}{\partial x^2}=-\frac{{\partial }^{2}u}{\partial x\partial y}$ 相结合，可得 $\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}=0$；将 $\frac{{\partial }^{2}v}{\partial y\partial x}=-\frac{{\partial }^{2}u}{\partial y^2}$ 与 $\frac{{\partial }^{2}u}{\partial x^2}=\frac{{\partial }^{2}v}{\partial x\partial y}$ 相结合，可得 $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0$。这两个方程被称为拉普拉斯方程 [4]，在工程领域有着广泛的应用。

Conclusion

结论

The study of the developments and interpretations of complex numbers has had quite an arduous history. From considering what the square root of a negative number even is to delving into the theory of complex functions, there is so much to learn if trying to have a comprehensive understanding on all these topics, and we have Bombelli, Wessel, Hamilton, Euler, and more to thank for their revolutionary contributions to this field.

复数的发展和解释的研究历史颇为艰辛。从思考负数的平方根究竟是什么，到深入研究复变函数理论，如果想要全面理解所有这些主题，有太多内容需要学习。我们要感谢邦贝利、韦塞尔、哈密顿、欧拉等众多学者，感谢他们对这一领域做出的具有革命性的贡献。

References

参考文献

[1] Agarwal, R. P., Perera, K., Pinelas, S. (2010). An introduction to complex analysis. New York, NY: Springer.

[2] Diamond, L. E. (1957). Introduction to complex numbers. Mathematical Association of America, 30, 233-249.

[3] Kleiner, I. (1988). Thinking the unthinkable: The story of complex numbers (with a moral). National Council of Teachers of Mathematics, 81, 583-592.

[4] Nahin, P. J. (2010). An imaginary tale: The story of square root minus one. Princeton, NJ: Princeton University Press.

[5] Struik, D. J. (1954). A concise history of mathematics. London, UK: G. Bell and Sons LTD.

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via：

• History of Complex Numbers | SpringerLink
  https://link.springer.com/chapter/10.1007/978-1-4614-0195-7_50

• Complex Numbers: Interpretations and Developments - Sarah Teklinski
  https://sites.math.rutgers.edu/~zeilberg/math436/projects/TeklinksiP.pdf

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• 从三次方程到复平面：复数概念的奇妙演进（一）-CSDN博客
  https://blog.csdn.net/u013669912/article/details/147179204

• 从三次方程到复平面：复数概念的奇妙演进（三）-CSDN博客
  https://blog.csdn.net/u013669912/article/details/147193352

• 从三次方程到复平面：复数概念的奇妙演进（四）-CSDN博客
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