从三次方程到复平面：复数概念的奇妙演进（一）

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，此为第一篇。

生料，不同的文章不同的点。

机翻，未校。

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Reflections on the History of Complex Numbers

复数的历史回顾

The first occurrence of square roots of negative quantities in Western literature can be found in Ars Magna by Cardan (1545) in the solution of the classical problem: Find two numbers whose sum and product are given.

西方文献中首次出现负数的平方根，是在卡尔达诺（1545 年）的《大术》中，用于解决一个经典问题：已知两数之和与两数之积，求这两个数。

In the case the sum is 10 and the product 40, Cardan acknowledges that there is no solution but proposes to extend the calculation in the following way: share 10 into two quantities, $5+m$ and $5-m$, where $m$ is a mysterious quantity so that $(5+m)(5-m)=25-m^2=40$; therefore $m^2=-15$.

在两数之和为 10，两数之积为 40 的情况下，卡尔达诺承认没有实数解，但提议按如下方式扩展计算：将 10 分为两个数，$5+m$ 和 $5-m$，其中 $m$ 是一个神秘的数，使得 $(5+m)(5-m)=25-m^2=40$；因此 $m^2=-15$。

The emergence of square roots of negative numbers also occurred in examining solutions of equations of the form $x^3=px+q$. The Tartaglia-Cardan formula for solving such equations cannot be applied in the irreducible case when $4p^3-27q^2<0$. This case leads to the square root of a negative quantity.

负数平方根的出现也出现在研究形如 $x^3=px+q$ 的方程的解时。当 $4p^3-27q^2<0$ 时，求解这类方程的塔尔塔利亚-卡尔达诺公式在不可约情形下无法应用，这种情况会导致出现负数的平方根。

Bombelli, in Algebra (1572) makes a decisive step when in solving the equation: $x^3=15x+4$, he introduces “something” whose square is –1, which he names: “piu di meno” and formulates rules for calculating with it.

邦贝利在 1572 年的《代数学》中迈出了决定性的一步。在求解方程 $x^3=15x+4$ 时，他引入了一种平方为-1 的 “东西”，他将其命名为 “piu di meno”，并制定了用它进行计算的规则。

From that time, this “thing” was used in calculations, first in intermediate steps for finding real number solutions, then for expressing impossible solutions allowing the generalization of results and theorems in the theory of equations.

从那时起，这个 “东西” 就被用于计算，起初是在求实数解的中间步骤中使用，然后用于表示不可能的解，这使得方程理论中的结果和定理得以推广。

For instance, it allowed Albert Girard in 1629, to give the first formulation of the Fundamental Theorem of Algebra in the Invention nouvelle en l’algèbre: “Toutes les equations d’algèbre reçoivent autant de solutions que la denomination de la plus haute quantité le démontre” (in modern terms, all polynomial equations have as many roots as the degree of the polynomial).

例如，1629 年，阿尔贝・吉拉尔在《代数新发明》中首次阐述了代数基本定理：“Toutes les equations d’algèbre reçoivent autant de solutions que la denomination de la plus haute quantité le démontre”（用现代术语来说，所有多项式方程的根的个数与多项式的次数相同）。

In 1637, Descartes in the Geometry, introduces the expression “imaginary quantities” which was used until Gauss (1831) introduced the terminology “complex numbers” and clarified the status of these quantities.

1637 年，笛卡尔在《几何学》中引入了 “虚量” 这一表述，这个表述一直被使用，直到 1831 年高斯引入 “复数” 这一术语，并阐明了这些量的地位。

In much of the $18^{th}$ century, complex numbers were used in calculations but their meaning was the object of metaphysical discussions as was the case for infinitesimals. Many contradictions arose from applying the usual rules of calculations with square roots when using the notation $\sqrt{-a}$ with $a$ positive, Euler introduce the notation “$i$” in the Elements d’algèbre (1774), but this notation, which was also used by Gauss, was not automatically accepted.

在 18 世纪的大部分时间里，复数被用于计算，但它的含义如同无穷小量一样，是形而上学讨论的对象。当使用 $\sqrt{-a}$（$a$ 为正数）的符号并应用通常的平方根计算规则时，出现了许多矛盾。欧拉在 1774 年的《代数基础》中引入了符号 “$i$”，高斯也使用了这个符号，但它并没有被立即接受。

The question of the meaning of imaginary quantities was solved through their geometric interpretation at the beginning of the $19^{th}$ century. Geometric interpretations were independently proposed by different, though not necessarily well-known mathematicians: Argand, l’Abbé Buée in France, Wessel in Denmark, and then Gauss.

19 世纪初，通过对虚量的几何解释，解决了虚量的含义问题。不同的（虽然不一定是知名的）数学家独立地提出了几何解释，如法国的阿尔冈、布埃神父，丹麦的韦塞尔，还有高斯。

In a letter to Bessel in 1811, Gauss proposed to use the plane for representing all quantities, both reals and imaginaries, by attaching to the point of coordinates $(a,b)$ the quantity $a+bi$. The construction by Argand (1806) was different as he started from the determination of proportional means between two quantities ($x$ such that $a:x:x:b$, or as $\frac{a}{x}=\frac{x}{b}$).

1811 年，高斯在给贝塞尔的一封信中提议用平面来表示所有的量，包括实数和虚数，将量 $a+bi$ 与坐标为 $(a,b)$ 的点对应起来。阿尔冈在 1806 年的构造有所不同，他从确定两个量之间的比例中项（$x$ 满足 $a:x:x:b$，即 $\frac{a}{x}=\frac{x}{b}$）出发。

In the case where $a$ and $b$ have different signs, for instance $a=1$ and $b=-1$, he pointed out the impossibility of solving the problem while staying on the real line, and proposed to think in terms of directions in the plane as in Figure 1.

在 $a$ 和 $b$ 异号的情况下，例如 $a=1$，$b=-1$，他指出在实数范围内无法解决这个问题，并提议像图 1 那样从平面方向的角度来思考。

Figure 1

图 1

Thus, if the directed segments $\overrightarrow{KB}$ and $\overrightarrow{KC}$ are associated with the numbers 1 and –1, the directed segments $\overrightarrow{KB}$ and $\overrightarrow{KD}$ appear as the solutions of the equation $1:x::x:–1$, or $\frac{1}{x}=\frac{x}{-1}$. Argand pointed out that these solutions are usually denoted by $\sqrt{-1}$ and $-\sqrt{-1}$.

因此，如果有向线段 $\overrightarrow{KB}$ 和 $\overrightarrow{KC}$ 分别与数字 1 和-1 相关联，那么有向线段 $\overrightarrow{KB}$ 和 $\overrightarrow{KD}$ 就是方程 $1:x::x:–1$（即 $\frac{1}{x}=\frac{x}{-1}$）的解。阿尔冈指出，这些解通常表示为 $\sqrt{-1}$ 和 $-\sqrt{-1}$。

Then Argand defined operations between directed segments: addition through the parallelogram law, which corresponds to the addition of vectors associated with the directed segments, and multiplication as in Figure 2 for two unitary directed segments $\overrightarrow{KB}$ and $\overrightarrow{KC}$ as $\overrightarrow{KD}=\overrightarrow{KB}\cdot \overrightarrow{KC}$, $\angle AKB=\angle CKD$, and thus $KA:KB::KC:KD$.

然后阿尔冈定义了有向线段之间的运算：加法通过平行四边形法则，这与和有向线段相关联的向量加法相对应；对于两个单位有向线段 $\overrightarrow{KB}$ 和 $\overrightarrow{KC}$，乘法如图 2 所示，$\overrightarrow{KD}=\overrightarrow{KB}\cdot \overrightarrow{KC}$，$\angle AKB=\angle CKD$，因此 $KA:KB::KC:KD$。

Figure 2

图 2

For non-unitary directed segments, the lengths are multiplied. He thus developed a calculation on directed segments that provided a geometrical interpretation of imaginary quantities and of the calculations with these. In this construction, angles and the link between multiplication and rotation played a fundamental role.

对于非单位有向线段，长度相乘。这样他发展出了一套有向线段的计算方法，为虚量以及虚量的计算提供了几何解释。在这个构造中，角度以及乘法和旋转之间的联系起到了基础性的作用。

The $19^{th}$ century saw new constructions of complex numbers, this time of an algebraic nature. This was the case with Cauchy, whose ambition in his Mémoire sur la théorie des équivalences algébriques substituées à la théorie des imaginaires (1847) was to “se débarrasser complètement des expressions imaginaires, en réduisant la lettre $i$ à n’être plus qu’une quantité réelle [to completely banish imaginary expressions by reducing the letter $i$ to be only a real quantity].”

19 世纪出现了复数的新构造，这次是代数性质的。柯西就是如此，他在 1847 年的《用代数等价理论取代虚数理论的回忆录》中，目标是 “se débarrasser complètement des expressions imaginaires, en réduisant la lettre $i$ à n’être plus qu’une quantité réelle（通过将字母 $i$ 简化为仅仅是一个实数，从而完全摒弃虚数表达式）”。

For doing so, he referred to the theory of congruences by Gauss and considered the quotient of the set of real polynomials $\mathbb{R}[X]$ by the polynomial $X^2+1$. Each class in this quotient is a representative of the form $aX+b$ corresponding to the rest in its division by $X^2+1$. Cauchy extended the operations on $\mathbb{R}[X]$ to this quotient, showing that the sum of the class of $a+bX$ and $c+dX$ is the class of $(a+c)+(b+d)X$, and that the product of these classes is $(ac-bd)+(ad+bc)X$ as the polynomial $X^2$ is congruent to the polynomial –1. Thus in Cauchy’s theory, “$i$” becomes a real but undetermined quantity.

为此，他参考了高斯的同余理论，考虑了实多项式集合 $\mathbb{R}[X]$ 关于多项式 $X^2+1$ 的商。这个商集中的每个类都是形如 $aX+b$ 的代表元，它对应于该多项式除以 $X^2+1$ 的余数。柯西将 $\mathbb{R}[X]$ 上的运算扩展到这个商集上，表明 $a+bX$ 的类与 $c+dX$ 的类之和是 $(a+c)+(b+d)X$ 的类，并且这些类的乘积是 $(ac-bd)+(ad+bc)X$，因为多项式 $X^2$ 与多项式-1 同余。因此在柯西的理论中，“$i$” 变成了一个实数但未确定的量。

Another algebraic construction was proposed by Hamilton in the Theory of Conjugate Functions and Algebraic Couples in 1833. Complex numbers were defined as couples of real numbers $(a,b)$, where each real number was associated with the couple of the form $(a,0)$; addition was defined as the ordinary addition of couples and multiplication such that there exists a couple whose square is –1. This led to the following definition for the product $(a,b)\cdot (c,d)=(ab-cd,ac+bd)$. In this system the number “$i$” that Hamilton named the secondary unit is the couple $(0,1)$.

另一种代数构造是哈密顿在 1833 年的《共轭函数与代数偶理论》中提出的。复数被定义为实数对 $(a,b)$，其中每个实数与形如 $(a,0)$ 的数对相关联；加法被定义为数对的普通加法，乘法使得存在一个数对，其平方为-1。这就引出了如下的乘积定义：$(a,b)\cdot (c,d)=(ab-cd,ac+bd)$。在这个系统中，哈密顿称为次要单位的数 “$i$” 是数对 $(0,1)$。

After many unsuccessful attempts to extend this construction to the three-dimensional space through the use of triplets, Hamilton eventually created the field of quaternions, a non-commutative field where geometrical notions such as scalar product and vector product find a natural interpretation.

在多次尝试通过使用三元组将这种构造扩展到三维空间失败后，哈密顿最终创建了四元数领域，这是一个非交换领域，在其中诸如标量积和向量积等几何概念找到了自然的解释。

This brief describes only a small part of the rich history of complex numbers and does not touch the theory of functions of complex variables also initiated by Cauchy, and the way this theory contributed to a number of mathematical domains including number theory, differential geometry, and dynamical systems.

这篇简述仅仅描述了复数丰富历史的一小部分，并没有涉及同样由柯西开创的复变函数理论，以及该理论对包括数论、微分几何和动力系统等众多数学领域的贡献。

This brief history illustrates how readers might approach and discuss many important issues from a mathematical and epistemological perspective including why and how new mathematical objects are introduced, the diverse ways they can become legitimate and meaningful, the potential and limitation of symbolic choices, the role of representations and connections between these, and the importance of connections between domains.

这段简史说明了读者可以如何从数学和认识论的角度来探讨和讨论许多重要问题，包括为什么以及如何引入新的数学对象、它们以何种多样的方式变得合理且有意义、符号选择的潜力和局限、各种表示形式的作用以及它们之间的联系，还有不同领域之间联系的重要性。

The many investigations and connections that can be considered using complex numbers suggests the study of complex numbers should have a place in teacher education. Furthermore, well-structured and cognitively rather simple ideas encapsulated in complex numbers form a persuasive argument to keep the rich intuitive, geometric and algebraic meanings of complex numbers as a part of high school mathematics teaching syllabi.

利用复数可以进行的众多研究和建立的联系表明，复数的学习在教师教育中应有一席之地。此外，复数中蕴含的结构良好且认知上相对简单的思想，有力地支持了将复数丰富的直观、几何和代数意义保留在高中数学教学大纲中的观点。

Reference

参考文献

Commission Inter-IREM Epistémologie et Histoire des Mathématiques. Images, Imaginaires, Imaginations-Une perspective historique pour l’introduction des nombres complexes. Paris: Editions Ellipses, 1998.

跨 IREM 数学认识论与历史委员会。《图像、虚数、想象 —— 引入复数的历史视角》。巴黎：Ellipses 出版社，1998 年。

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A Short History of Complex Numbers

复数简史

Orlando Merino

奥兰多・梅里诺

University of Rhode Island

罗德岛大学

January, 2006

Abstract

摘要

This is a compilation of historical information from various sources, about the number $i=\sqrt{-1}$. The information has been put together for students of Complex Analysis who are curious about the origins of the subject, since most books on Complex Variables have no historical information (one exception is Visual Complex Analysis, by T. Needham).

本文汇编了来自各种来源的关于数字 $i=\sqrt{-1}$ 的历史信息。这些信息是为对该学科起源感到好奇的复分析学生整理的，因为大多数关于复变函数的书籍都没有历史相关内容（唯一的例外是 T. 尼达姆所著的《直观复分析》）。

A fact that is surprising to many (at least to me!) is that complex numbers arose from the need to solve cubic equations, and not (as it is commonly believed) quadratic equations. These notes track the development of complex numbers in history, and give evidence that supports the above statement.

许多人（至少对我来说！）感到惊讶的是，复数源于求解三次方程的需要，而非（人们通常认为的）二次方程。这些笔记追溯了复数在历史上的发展，并给出了支持上述观点的证据。

1. Al - Khwarizmi (780 - 850) in his Algebra has solution to quadratic equations of various types. Solutions agree with is learned today at school, restricted to positive solutions [9] Proofs are geometric based. Sources seem to be greek and hindu mathematics. According to G. J. Toomer, quoted by Van der Waerden,Under the caliph al - Ma’mun (reigned 813 - 833) al - Khwarizmi became a member of the “House of Wisdom” (Dar al - Hikma), a kind of academy of scientists set up at Baghdad, probably by Caliph Harun al - Rashid, but owing its preeminence to the interest of al - Ma’mun, a great patron of learning and scientific investigation. It was for al - Ma’mun that Al - Khwarizmi composed his astronomical treatise, and his Algebra also is dedicated to that ruler.

   阿尔 - 花剌子米（780 - 850）在他的《代数学》中给出了各类二次方程的解法。其解法与如今学校里所学的一致，但仅限于正根 。证明基于几何方法。其来源似乎是希腊和印度数学。据范德瓦尔登引用 G. J. 图默的说法，在哈里发马蒙（813 - 833 年在位）统治时期，阿尔 - 花剌子米成为了 “智慧宫”（Dar al - Hikma）的一员，这是一个由哈里发哈伦・拉希德可能在巴格达建立的科学院，但它的卓越地位得益于马蒙的兴趣，马蒙是学术和科学研究的伟大赞助人。阿尔 - 花剌子米为马蒙撰写了他的天文学论著，他的《代数学》也是献给这位统治者的。

2. The methods of algebra known to the arabs were introduced in Italy by the Latin translation of the algebra of al - Khwarizmi by Gerard of Cremona (1114 - 1187), and by the work of Leonardo da Pisa (Fibonacci)(1170 - 1250).

   阿拉伯人所知晓的代数方法通过克雷莫纳的杰拉德（1114 - 1187）对阿尔 - 花剌子米《代数学》的拉丁文翻译，以及比萨的莱昂纳多（斐波那契）（1170 - 1250）的著作传入意大利。

   About 1225, when Frederick II held court in Sicily, Leonardo da Pisa was presented to the emperor. A local mathematician posed several problems, all of which were solved by Leonardo. One of the problems was the solution of the equation

   大约在 1225 年，当腓特烈二世在西西里岛举行宫廷活动时，比萨的莱昂纳多被引荐给皇帝。一位当地数学家提出了几个问题，莱昂纳多全部解答了出来。其中一个问题是求解方程

   $$x^3+2x^2+10x=20$$

3. The general cubic equation

   一般的三次方程

   $$x^3+a{x}^2+bx+c=0$$

   can be reduced to the simpler form

   可以简化为更简单的形式

   $$x^3+px+q=0$$

   through the change of variable $x^{\prime }=x+\frac{1}{3}a$. This change of variable appears for the first time in two anonymous florentine manuscripts near the end of the 14th century.

   通过变量替换 $x^{\prime }=x+\frac{1}{3}a$。这种变量替换首次出现在 14 世纪末的两份佛罗伦萨匿名手稿中。

   $$(a)x^3+px=q$$

   $$(b)x^3=px+q$$

   $$(c)x^3+q=px$$

   If only positive coefficients and positive values of x are admitted, there are three cases, all collectively known as depressed cubic:
   如果只允许正系数和正的 $x$ 值，那么有三种情况，统称为缺项三次方程：

   (a) $x^3+px=q$; (b) $x^3=px+q$; (c) $x^3+q=px$.

4. The first to solve equation (1) (and maybe (2) and (3)) was Scipione del Ferro, professor of U. of Bologna until 1526, when he died. In his deathbed, del Ferro confided the formula to his pupil Antonio Maria Fiore. Fiore challenged Tartaglia to a mathematical contest. The night before the contest, Tartaglia rediscovered the formula and won the contest. Tartaglia in turn told the formula (but not the proof) to Gerolamo Cardano, who signed an oath to secrecy. From knowledge of the formula, Cardano was able to reconstruct the proof. Later, Cardano learned that del Ferro had the formula and verified this by interviewing relatives who gave him access to del Ferro’s papers. Cardano then proceeded to publish the formula for all three cases in his Ars Magna (1545). It is noteworthy that Cardano mentioned del Ferro as first author, and Tartaglia as obtaining the formula later in independent manner.

   第一个求解方程 (1)（可能还有方程 (2) 和 (3)）的是西皮奥内・德尔・费罗，他是博洛尼亚大学的教授，一直任职到 1526 年去世。在临终前，德尔・费罗将公式透露给了他的学生安东尼奥・玛丽亚・菲奥雷。菲奥雷向塔尔塔利亚发起数学竞赛挑战。在竞赛前一晚，塔尔塔利亚重新发现了该公式，并赢得了竞赛。塔尔塔利亚随后将公式（但没有证明）告诉了杰罗拉莫・卡尔达诺，卡尔达诺宣誓保密。基于对公式的了解，卡尔达诺得以重构证明过程。后来，卡尔达诺得知德尔・费罗早就有这个公式，并通过采访德尔・费罗的亲属查阅其论文证实了这一点。然后，卡尔达诺在他 1545 年出版的《大术》一书中公布了这三种情况的公式。值得注意的是，卡尔达诺提到德尔・费罗是首位发现者，塔尔塔利亚是后来独立获得该公式的。

5. A difficulty in case (2) that was not present in the solution to (1) is the possibility of having the square root of a negative number appear in the numerical expression given by the formula. Here is the derivation: Substitute $x=u+v$ into $x^3=px+q$ to obtain

   情况 (2) 存在一个在情况 (1) 的解法中没有出现的难点，即公式给出的数值表达式中可能会出现负数的平方根。以下是推导过程：将 $x=u+v$ 代入 $x^3=px+q$，得到

$$x^3-px=u^3+v^3+3uv(u+v)-p(u+v)=q$$

Set $3uv=p$ above to obtain $u^3+v^3=q$ and also $u^3{v}^3=(\frac{p}{3}{)}^3$. That is, the sum and the product of two cubes is known. This is used to form a quadratic equation which is readily solved:

令 $3uv=p$，则 $u^3+v^3=q$，且 $u^3{v}^3=(\frac{p}{3}{)}^3$。也就是说，两个立方数的和与积是已知的。由此构造一个二次方程并容易求解：

$$x=u+v=\sqrt[3]{\frac{1}{2}q+w}+\sqrt[3]{\frac{1}{2}q-w}$$

where

其中

$$w=\sqrt{(\frac{1}{2}q{)}^2-(\frac{1}{3}p{)}^3}$$

The so - called casus irreducibilis is when the expression under the radical symbol in $w$ is negative. Cardano avoids discussing this case in Ars Magna. Perhaps, in his mind, avoiding it was justified by the (incorrect) correspondence between the casus irreducibilis and the lack of a real, positive solution for the cubic.

当 $w$ 中根号下的表达式为负数时，就是所谓的不可约情形。卡尔达诺在《大术》中避免讨论这种情况。也许在他看来，不可约情形与三次方程不存在正实数解之间（错误的）对应关系，使他有理由避开这个问题。

6. According to [9], “Cardano was the first to introduce complex numbers $a+\sqrt{-b}$ into algebra, but had misgivings about it.” In Chapter 37 of Ars Magna the following problem is posed: “To divide 10 in two parts, the product of which is 40”.

   根据 [9] 的记载，“卡尔达诺是首位将复数 $a+\sqrt{-b}$ 引入代数学的人，但他对此心存疑虑”。在《大术》第 37 章中提出了以下问题：“将 10 分成两部分，使它们的乘积为 40”。

   It is clear that this case is impossible. Nevertheless, we shall work thus: We divide 10 into two equal parts, making each 5. These we square, making 25. Subtract 40, if you will, from the 25 thus produced, as I showed you in the chapter on operations in the sixth book leaving a remainder of - 15, the square root of which added to or subtracted from 5 gives parts the product of which is 40. These will be $5+\sqrt{-15}$ and $5-\sqrt{-15}$.

   显然，这种情况是不可能的。尽管如此，我们还是这样计算：将 10 分成两个相等的部分，每个部分为 5。对 5 进行平方，得到 25。从 25 中减去 40，就像我在第六本书关于运算的章节中教你的那样，得到余数 - 15。将 - 15 的平方根加到 5 上或从 5 中减去，就得到乘积为 40 的两部分。这两部分分别是 $5+\sqrt{-15}$ 和 $5-\sqrt{-15}$。

   Putting aside the mental tortures involved, multiply $5+\sqrt{-15}$ and $5-\sqrt{-15}$ making 25−(−15) which is +15. Hence this product is 40.

   暂且抛开其中复杂的思考过程，计算 $(5+\sqrt{-15})(5-\sqrt{-15})$，结果为 $25-(-15)=25+15=40$。

7. Rafael Bombelli authored l’Algebra (1572, and 1579), a set of three books. Bombelli introduces a notation for $\sqrt{-1}$, and calls it “pi´u di meno”.

   拉斐尔・邦贝利著有《代数学》（1572 年和 1579 年），共三卷。邦贝利引入了 $\sqrt{-1}$ 的符号，并称之为 “pi´u di meno”。

   The discussion of cubics in l’Algebra follows Cardano, but now the casus irreducibilis is fully discussed. Bombelli considered the equation

   《代数学》中对三次方程的讨论沿用了卡尔达诺的方法，但现在对不可约情形进行了全面讨论。邦贝利考虑方程

   $$x^3=15x+4$$

   for which the Cardan formula gives

   根据卡尔达诺公式，该方程的解为

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

   Bombelli observes that the cubic has $x=4$ as a solution, and then proceeds to explain the expression given by the Cardan formula as another expression for $x=4$ as follows. He sets

   邦贝利注意到该三次方程有一个解 $x=4$，然后他将卡尔达诺公式给出的表达式解释为 $x=4$ 的另一种表示形式，过程如下。他设

   $$\sqrt[3]{2-\sqrt{-121}}=a-bi$$

   from which he deduces and obtains,

   经过代数运算，

   $$\sqrt[3]{2+\sqrt{-121}}=a+bi$$

after algebraic manipulations, $a=2$ and $b=1$.Thus

他推导出 $a=2$，$b=1$。因此

$$x=a+bi+a-bi=2a=4$$

After doing this, Bombelli commented:

完成这些之后，邦贝利评论道：

“ At first, the thing seemed to me to be based more on sophism than on truth, but I searched until I found the proof.”

“起初，这件事在我看来似乎更多地基于诡辩而非真理，但我不断探索，直到找到了证明。”

8. Ren´e Descartes (1596 - 1650) was a philosopher whose work, La G´eom´etrie, includes his application of algebra to geometry from which we now have Cartesian geometry. Descartes was pressed by his friends to publish his ideas, and he wrote a treatise on science under the title “Discours de la m´ethod pour bien conduire sa raison et chercher la v´erit´e dans les sciences”. Three appendices to this work were La Dioptrique, Les M´et´eores, and La G´eom´etrie. The treatise was published at Leiden in 1637. Descartes associated imaginary numbers with geometric impossibility. This can be seen from the geometric construction he used to solve the equation $z^2=az-b^2$, with $a$ and $b^2$ both positive. According to [1], Descartes coined the term imaginary:

   勒内・笛卡尔（1596 - 1650）是一位哲学家，他的著作《几何学》包括了他将代数应用于几何的成果，由此我们如今有了笛卡尔几何。笛卡尔在朋友的敦促下发表他的思想，他写了一篇关于科学的论文，题为《谈谈正确引导理性在各门科学中寻求真理的方法》。这篇论文有三个附录，分别是《屈光学》《气象学》和《几何学》。该论文于 1637 年在莱顿出版。笛卡尔将虚数与几何上的不可能性联系起来。从他用于求解方程 $z^2=az-b^2$（其中 $a$ 和 $b^2$ 均为正数）的几何构造中可以看出这一点。根据 [1] 的记载，笛卡尔创造了 “虚数” 这个术语：

   “For any equation one can imagine as many roots [as its degree would suggest], but in many cases no quantity exists which corresponds to what one imagines.”

   “对于任何一个方程，人们都可以想象出与其次数相符数量的根，但在许多情况下，并不存在与人们所想象的根相对应的量。

9. John Wallis (1616 - 1703) notes in his Algebra that negative numbers, so long viewed with suspicion by mathematicians, had a perfectly good physical explanation, based on a line with a zero mark, and positive numbers being numbers at a distance from the zero point to the right, where negative numbers are a distance to the left of zero. Also, he made some progress at giving a geometric interpretation to $\sqrt{-1}$.

   约翰・沃利斯（1616 - 1703）在他的《代数学》中指出，长期以来被数学家们怀疑的负数有一个非常合理的物理解释，基于一条带有零点标记的直线，正数是从零点向右一定距离的数，而负数是从零点向左一定距离的数。此外，他在对 $\sqrt{-1}$ 进行几何解释方面取得了一些进展。

10. Abraham de Moivre (1667 - 1754) left France to seek religious refuge in London at eighteen years of age. There he befriended Newton. In 1698 he mentions that Newton knew, as early as 1676 of an equivalent expression to what is today known as de Moivre’s theorem:

    亚伯拉罕・棣莫弗（1667 - 1754）18 岁时离开法国，前往伦敦寻求宗教庇护。在那里，他与牛顿成为朋友。1698 年，他提到牛顿早在 1676 年就知道一个与如今被称为棣莫弗定理等价的表达式：

    $$(\cos (\theta )+i\sin (\theta ){)}^n=\cos (n\theta )+i\sin (n\theta )$$

    where $n$ is an integer. Apparently Newton used this formula to compute the cubic roots that appear in Cardan formulas, in the irreducible case. de Moivre knew and used the formula that bears his name, as it is clear from his writings - although he did not write it out explicitly.

    其中 $n$ 为整数。显然，牛顿在不可约情形下，使用这个公式来计算卡尔达诺公式中出现的立方根。棣莫弗知晓并使用了以他名字命名的公式，从他的著作中可以明显看出 —— 尽管他没有明确写出这个公式。

11. L. Euler (1707 - 1783) introduced the notation $i=\sqrt{-1}[3]$, and visualized complex numbers as points with rectangular coordinates, but did not give a satisfactory foundation for complex numbers. Euler used the formula $x+iy=r(\cos \theta +i\sin \theta )$, and visualized the roots of $z^n=1$ as vertices of a regular polygon. He defined the complex exponential, and proved the identity $e^{i\theta }=\cos \theta +i\sin \theta$.

    莱昂哈德・欧拉（1707 - 1783）引入了符号 $i=\sqrt{-1}$ ，并将复数可视化为具有直角坐标的点，但他并没有为复数提供令人满意的基础。欧拉使用公式 $x+iy=r(\cos \theta +i\sin \theta )$ ，并将 $z^n=1$ 的根可视化为正多边形的顶点。他定义了复指数，并证明了等式 $e^{i\theta }=\cos \theta +i\sin \theta$。

12. Caspar Wessel (1745 - 1818), a Norwegian, was the first one to obtain and publish a suitable presentation of complex numbers. On march 10, 1797, Wessel presented his paper “On the Analytic Representation of Direction: An Attempt” to the Royal Danish Academy of Sciences. The paper was published in the Academy’s Memoires of 1799. Its quality was judged to be so high that it was the first paper to be accepted for publication by a non - member of the academy.

    卡斯帕・韦塞尔（1745 - 1818）是挪威人，他是首位获得并发表复数恰当表示方法的人。1797 年 3 月 10 日，韦塞尔向丹麦皇家科学院提交了他的论文《关于方向的解析表示：一次尝试》。该论文于 1799 年发表在科学院的《回忆录》上。因其质量极高，这篇论文成为非科学院成员发表的首篇被接受的论文。

    Wessel’s paper, written in danish, went unnoticed until 1897, when it was unearthed by an antiquarian, and its significance recognized by the Danish mathematician Sophus Christian Juel.

    韦塞尔的论文是用丹麦语撰写的，一直未被注意，直到 1897 年被一位古物研究者发现，丹麦数学家索菲斯・克里斯蒂安・尤尔认识到了它的重要性。

    Wessel’s approach used what we today call vectors. He uses the geometric addition of vectors (parallelogram law) and defined multiplication of vectors in terms of what we call today adding the polar angles and multiplying the magnitudes.

    韦塞尔的方法使用了我们如今所说的向量。他使用向量的几何加法（平行四边形法则），并根据我们如今所说的极角相加和模长相乘来定义向量的乘法。

13. Jean - Robert Argand (1768 - 1822) was a Parisian bookkeeper. It is not known whether he had mathematical training. Argand produced a pamphlet in 1806, run by a private press in small print. He failed to include his name in the title page. The title was “Essay on the Geometrical Interpretation of Imaginary Quantities”. One copy ended up in the hands of the mathematician A. Legendre (1752 - 1833) who in turn mentioned it in a letter to Francois Francais, a professor of mathematics. When Francais died, he inherited his papers to his brother Jaques who was a professor of military art and a mathematician. He found Legendre’s letter describing Argand’s mathematical results, but Legendre failed to mention Argand. Jaques published an article in 1813 in the Annales de Mathémathiques giving the basics of complex numbers. In the last paragraph of the paper, Jaques acknowledged his debt to Legendre’s letter, and urged the unknown author to come forward. Argand learned of this and his reply appeared in the next issue of the journal.

    让 - 罗贝尔・阿尔冈（1768 - 1822）是巴黎的一名簿记员。目前尚不清楚他是否接受过数学训练。1806 年，阿尔冈自费出版了一本小册子，字体很小。他没有在标题页上署名。书名是《虚数的几何解释随笔》。其中一份副本落到了数学家 A. 勒让德（1752 - 1833）手中，勒让德在给数学教授弗朗索瓦・弗朗索瓦的一封信中提到了这本书。弗朗索瓦去世后，他的论文由他的兄弟雅克继承，雅克是一位军事艺术教授兼数学家。雅克发现了勒让德描述阿尔冈数学成果的信件，但勒让德没有提及阿尔冈的名字。1813 年，雅克在《数学年鉴》上发表了一篇文章，介绍了复数的基础知识。在论文的最后一段，雅克承认他得益于勒让德的信件，并敦促这位不知名的作者站出来。阿尔冈得知此事后，他的回应发表在了该期刊的下一期上。

14. William Rowan Hamilton (1805 - 65) in an 1831 memoir defined ordered pairs of real numbers $((a,b))$ to be a couple. He defined addition and multiplication of couples: $((a,b)+(c,d)=(a+c,b+d)$ and $(a,b)(c,d)=(ac-bd,bc+ad))$. This is in fact an algebraic definition of complex numbers.

    威廉・罗恩・哈密顿（1805 - 1865）在 1831 年的一篇回忆录中，将实数的有序对 $((a,b))$ 定义为一对数。他定义了对数的加法和乘法：$((a,b)+(c,d)=(a+c,b+d)$ ，$(a,b)(c,d)=(ac-bd,bc+ad)$ 。这实际上是复数的一种代数定义。

15. Carl Friedrich Gauss (1777 - 1855). There are indications that Gauss had been in possession of the geometric representation of complex numbers since 1796, but it went unpublished until 1831, when he submitted his ideas to the Royal Society of Gottingen. Gauss introduced the term complex number.

    卡尔・弗里德里希・高斯（1777 - 1855）。有迹象表明，高斯自 1796 年起就掌握了复数的几何表示方法，但直到 1831 年他将自己的想法提交给哥廷根皇家学会时才发表。高斯引入了 “复数” 这个术语。

    “If this subjet has hitherto been considered from the wrong viewpoint and thus enveloped in mystery and surrounded by darkness, it is largely an unsuitable terminology which should be blamed. Had +1, - 1 and $\sqrt{-1}$, instead of being called positive, negative and imaginary (or worse still, impossible) unity, been given the names say,of direct, inverse and lateral unity, there would hardly have been any scope for such obscurity.”

    “如果到目前为止，这个主题一直从错误的角度被看待，从而笼罩在神秘和黑暗之中，那么在很大程度上应该归咎于不合适的术语。如果 +1、 - 1 和 $\sqrt{-1}$ ，不是被称为正、负和虚（或者更糟，不可能）单位，而是被赋予比如说正向、反向和侧向单位这样的名称，那么就几乎不会有这种晦涩难懂的情况了。”

    In a 1811 letter to Bessel, Gauss mentions the theorem that was to be known later as Cauchy’s theorem. This went unpublished, and was later rediscovered by Cauchy and by Weierstrass.

    在 1811 年给贝塞尔的一封信中，高斯提到了后来被称为柯西定理的内容。这一内容当时未发表，后来被柯西和魏尔斯特拉斯重新发现。

16. Augustin - Louis Cauchy (1789 - 1857) initiated complex function theory in an 1814 memoir submitted to the French Académie des Sciences. The term analytic function was not mentioned in his memoir, but the concept is there. The memoir was published in 1825. Contour integrals appear in the memoir, but this is not a first, apparently Poisson had a 1820 paper with a path not on the real line. Cauchy constructed the set of complex numbers in 1847 as $R[x]/(x^2+1)$.

    奥古斯丁 - 路易・柯西（1789 - 1857）在 1814 年提交给法国科学院的一篇回忆录中开创了复变函数理论。在他的回忆录中没有提及 “解析函数” 这个术语，但相关概念已经存在。这篇回忆录于 1825 年发表。回忆录中出现了围道积分，但这并非首创，显然泊松在 1820 年的一篇论文中就使用了不在实轴上的路径。1847 年，柯西将复数集构造为 $R[x]/(x^2+1)$。

    “We completely repudiate the symbol $\sqrt{-1}$, abandoning it without regret because we do not know what this alleged symbolism signifies nor what meaning to give to it.”

    “我们完全摒弃符号 $\sqrt{-1}$ ，毫无遗憾地抛弃它，因为我们不知道这个所谓的符号意味着什么，也不知道该赋予它什么含义。”

References

[1] B. Blank, An Imaginary Tale Book Review, in Notices of the AMS Volume 46, Number 10, November 1999, pp. 1233 - 1236.

[2] T. Needham, Visual Complex Analysis. New York: Oxford University Press, 1997.

[3] W. Dunham, Euler, The Master of Us All, The Dolciani Mathematical Expositions, Number 22, Mathematical Association of America, 1999.

[4] P. Nahin, An Imaginary Tale, Princeton U. Press, NJ 1998.

[5] R. Argand, Essai sur une Manière de Représenter Les Quantités Imaginaires dans Les Constructions Géométriques, Reprinted 1971 A. Blanchard.

[6] The MacTutor History of Mathematics archive, at www.gap-system.org/history/

[7] R. Descartes, La Géométrie, translated from French and Latin by D. E. Smith and M. Latham. Open Court Publishing Company, La Salle, Illinois, 1952.

[8] M. Crowe, A History of Vector Analysis, U. of Notre Dame Press, Notre Dame, 1967.

[9] B. L. van der Waerden, A History of Algebra, Springer Verlag, NY 1985.

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via：

• Reflections on the History of Complex Numbers
  http://projects.ias.edu/pcmi/briefs/HistoryBriefFinal.pdf

• A Short History of Complex Numbers - 2006.
  https://docslib.org/doc/2339313/a-short-history-of-complex-numbers

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• 从三次方程到复平面：复数概念的奇妙演进（二）-CSDN博客
  https://blog.csdn.net/u013669912/article/details/147194451

• 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
  https://blog.csdn.net/u013669912/article/details/147193576