从三次方程到复平面：复数概念的奇妙演进（三）

注：本文为 "复数 | 历史 / 演进" 相关文章。

因 csdn 篇幅限制分篇连载，此为第三篇。

生料，不同的文章不同的点。

机翻，未校。

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Complex Numbers

History:

1. Complex numbers were first introduced by G. Cardano (1501-1576) in his Ars Magna , chapter 37 (published 1545) as a tool for finding (real!) roots of a cubic equation: x 3 + a x + b = 0 x^3 + ax + b = 0 x3+ax+b=0.

   复数最早由 G. 卡丹（1501-1576）在其《大术》第 37 章（1545 年出版）中引入，作为求解三次方程 x 3 + a x + b = 0 x^3 + ax + b = 0 x3+ax+b=0 的实根的工具。

   However, he had serious misgivings about such expressions (e.g. 5 + − 15 5 + \sqrt {-15} 5+−15 ). He referred to thinking about them as "mental torture".

   然而，他对这些表达式（例如 5 + − 15 5 + \sqrt {-15} 5+−15 ）感到非常困惑，甚至称思考它们是 "精神折磨"。

2. R. Bombelli (1572): In his three books on Algebra, he introduced the symbol i i i and established rules for calculating in C \mathbb {C} C.

   R. 邦贝利（1572）：在他的三本代数书中，他引入了符号 i i i，并建立了在复数域 C \mathbb {C} C 中的计算规则。

3. A. Girard (1629): called a + − b a + \sqrt {-b} a+−b "solutions impossibles".

   A. 吉拉德（1629）：称 a + − b a + \sqrt {-b} a+−b 为 "不可能的解"。

4. R. Descartes (1637): called them "imaginary numbers".

   R. 笛卡尔（1637）：称它们为 "虚数"。

5. After Descartes, many leading mathematicians made free use of complex numbers: Bernoulli, Moivre, Euler, . . . , Argand, Gauss, . . .

   在笛卡尔之后，许多著名数学家开始广泛使用复数：伯努利、德摩弗、欧拉、......、阿尔冈、高斯、......

6. The term "complex number" seems to have originated with C. F. Gauss (1831).

   "复数" 这一术语似乎是由 C. F. 高斯（1831）首次提出的。

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Discovery of Complex numbers

复数的发现

YangYi Zhao*

SaintMary's High School, Manhasset, United States

*Corresponding author:[email protected]

Abstract

摘要

A branch of mathematical analysis called complex analysis studies the functions of complex numbers. It is also sometimes referred to as the theory of functions of a complex variable. Complex analysis has been used extensively in mathematics, physics, and engineering over the years, particularly in the areas of algebraic geometry, fluid dynamics, quantum mechanics, and other related fields. It dates back to the 16th century, when Italian mathematicians Girolamo Cardano and Raphael Bombelli first noticed complex numbers while attempting to solve a particular algebra, and was later developed by Cauchy and Riemann in the 19th century. The development of complex numbers has a lengthy history. Mathematicians have advanced the discipline of mathematics significantly after thousands of years of development. During this time, mathematicians also found a great deal of previously unknown mathematical information and proved formulae and phenomena that had previously been impossible to verify. And it covers complex numbers as well as some of the mathematics related to them. In this paper, the complete discovery of complex numbers from cubic equation to topology of complex numbers is detailedly revealed.

数学分析的一个分支 ------ 复分析，研究复数的函数。它有时也被称为复变函数论。多年来，复分析在数学、物理学和工程学中有着广泛的应用，尤其是在代数几何、流体动力学、量子力学及其他相关领域。它的起源可以追溯到 16 世纪，当时意大利数学家吉罗拉莫・卡尔达诺（Girolamo Cardano）和拉斐尔・邦贝利（Raphael Bombelli）在试图求解一个特定的代数问题时首次注意到了复数，后来在 19 世纪，柯西（Cauchy）和黎曼（Riemann）对其进行了进一步发展。复数的发展有着漫长的历史。经过数千年的发展，数学家们在数学学科上取得了显著的进步。在此期间，数学家们还发现了大量以前未知的数学知识，并证明了以前无法验证的公式和现象。它涵盖了复数以及一些与之相关的数学知识。在本文中，详细揭示了从三次方程到复数拓扑的复数完整发现历程。

Keywords:

Complex numbers, cubic equation, complex set and topolgy.

关键词：复数；三次方程；复集与拓扑

1. Introduction

引言

The history of Complex Number goes back a long way. After thousands of years of development, mathematicians have made great progress in the study of mathematics. Mathematicians also discovered a lot of undiscovered mathematical knowledge during this period, and successively proved the formulas and phenomena that could not be proved before. And that includes complex numbers and some of the mathematics associated with complex numbers. So what is a complex number? When were complex numbers discovered and proved to exist? The square root of a negative number cannot, under normal circumstances, be specified by the definition of a square root. It wasn't until around 300 years ago that this inquiry received an answer [1-15]. These numbers appear often in the study of mathematical fields, whether or not they make sense in everyday life.

复数的历史源远流长。经过数千年的发展，数学家们在数学研究方面取得了巨大的进展。在此期间，数学家们还发现了许多未被发现的数学知识，并相继证明了以前无法证明的公式和现象。这其中就包括复数以及一些与复数相关的数学知识。那么，什么是复数呢？复数是何时被发现并被证明存在的呢？在正常情况下，负数的平方根无法根据平方根的定义来确定。直到大约 300 年前，这个问题才得到解答 [1 - 15]。这些数在数学领域的研究中经常出现，无论它们在日常生活中是否有意义。

Mathematicians examine complex numbers along the way. Research in fields like function roots, Cartesian coordinate systems, and intermediate algebraic equations is strengthened by starting with delayed consensus, showing the existence of complex systems, and concluding with the proof of complex numbers. Complex analysis is being studied by an increasing number of individuals since it is universally acknowledged as a mathematical reality. As a result, the field of complex analysis was created to investigate complex numbers. It is therefore obvious that the creation of complex numbers, made possible by the extension of rigorous proof, gave rise to new and enlarging viewpoints from which to approach the many fields of mathematics.

数学家们一直在研究复数。从达成共识较晚的情况出发，证明复数系统的存在，再到证明复数的存在，这一系列过程加强了在函数根、笛卡尔坐标系和中级代数方程等领域的研究。由于复数被普遍认为是一种数学事实，越来越多的人开始研究复分析。因此，复分析领域应运而生，用于研究复数。显然，通过严格证明的拓展使得复数得以创立，这为研究数学的众多领域带来了新的、更广阔的视角。

Italian cubic equations include the oldest references to imaginary numbers. Nicolo Tartaglia must be noted in this situation. Because of his linguistic difficulties as a youngster as a result of the murder in his community, he is a math prodigy and an intriguing figure who is frequently referred to as "the stammerer." One of his most notable contributions was the invention of a "secret" technique for resolving a certain kind of cubic problem. Despite being absolutely exemplary in style, Tartaglia's approach to algebra was revolutionary. Instead than using numerical examples, this strategy relies on abstract reasoning to answer the frequently asked question, "Does this work for any number?" In order to depict his scenario, Tartaglia created numerous equations using the Diophantine approach [115].

意大利的三次方程中最早提及了虚数。在这方面，尼科洛・塔尔塔利亚（Nicolo Tartaglia）值得关注。由于他小时候所在社区发生的谋杀案，他存在语言障碍，但他是一位数学天才，也是一个引人注目的人物，常被称为 "口吃者"。他最显著的贡献之一是发明了一种解决特定类型三次方程问题的 "秘密" 方法。塔尔塔利亚的代数方法在风格上堪称典范，同时具有革命性。这种方法不使用数值例子，而是依靠抽象推理来回答 "这个方法对任意数都适用吗？" 这一常见问题。为了描述他的情况，塔尔塔利亚使用丢番图方法创建了许多方程 [115]。

2. Main works

主要研究成果

2.1 Cubic equation and birth of complex numbers

三次方程与复数的诞生

Italian cubic equations include the oldest references to imaginary numbers. Nicolo Tartaglia must be noted in this situation. Because of his linguistic difficulties as a youngster as a result of the murder in his community, he is a math prodigy and an intriguing figure who is frequently referred to as "the stammerer." One of his most notable contributions was the invention of a "secret" technique for resolving a certain kind of cubic problem. Despite being absolutely exemplary in style, Tartaglia's approach to algebra was revolutionary. Instead than using numerical examples, this strategy relies on abstract reasoning to answer the frequently asked question, "Does this work for any number?" In order to depict his scenario, Tartaglia created numerous equations using the Diophantine method. In 1539, Tartaglia secretly revealed these procedures and formulas to his buddy, the Italian mathematician Girolamo Cardano. Cardano, however, violated Tartaglia's confidence by releasing this strategy in his own book Ars Magna. So, with the release of Ars Magna, Tartaglia's approach began to lose its memory. As a result, the innovation became known as Cardano's triple depression cure.

意大利的三次方程中最早提及了虚数。在这方面，尼科洛・塔尔塔利亚（Nicolo Tartaglia）值得关注。由于他小时候所在社区发生的谋杀案，他出现了语言障碍，但他是一位数学天才，也是一个引人注目的人物，常被称为 "口吃者"。他最显著的贡献之一是发明了一种解决特定类型三次方程问题的 "秘密" 方法。塔尔塔利亚的代数方法在风格上堪称典范，同时具有革命性。这种方法不使用数值例子，而是依靠抽象推理来回答 "这个方法对任意数都适用吗？" 这一常见问题。为了描述他的情况，塔尔塔利亚使用丢番图方法创建了许多方程。1539 年，塔尔塔利亚将这些步骤和公式秘密透露给了他的朋友，意大利数学家吉罗拉莫・卡尔达诺（Girolamo Cardano）。然而，卡尔达诺在自己的著作《大术》（Ars Magna）中公布了这一方法，辜负了塔尔塔利亚的信任。因此，随着《大术》的出版，塔尔塔利亚的方法开始被人遗忘。结果，这项创新被称为卡尔达诺的三次降次解法。

One of the most ruthless mathematicians in history is said to be Cardano. There is a good reason why he is frequently referred to as the bad guy. He applied the Tartaglia technique. He devised a formula for the answer and included it in his own book, Ars Magna. When Cardano published the book, he almost rudely praised the Italian mathematician Ferro, saying he had also developed a crude form to solve the depression cube. While he actually got these ideas from Tartaglia, Cardano didn't give the credit to Tartaglia but chose to take it himself. In other words, Cardano stole the fame and credit that belonged to Tartaglia. When Tartaglia won the Mathematical Contest at the University of Bologna in 1535, he left a huge legacy in the history of mathematics, in which he showed the general algebraic formula for solving cubic equations, which at the time had been considered inappropriate. Possible, as it requires understanding the square root of negative numbers. In the competition, he defeated Scipione del Ferro, who coincidentally came up with a partial solution to the cubic problem earlier than Tartaglia. While del Ferro's solution predates Tartaglia's, it is more limited in content. Tartaglia is considered the first general solution. In the competitive and brutal environment of 16th - century Italy, Tartaglia even encoded his solution in poetry in an attempt to make it harder for other mathematicians to steal it [1-15].

据说卡尔达诺是历史上最无情的数学家之一。他常被视为坏人是有充分理由的。他应用了塔尔塔利亚的方法，设计出了答案的公式，并将其收录在自己的《大术》一书中。卡尔达诺在出版这本书时，几乎是无礼地称赞意大利数学家费罗（Ferro），称费罗也开发出了一种求解降次三次方程的粗略形式。实际上，他的这些想法来自塔尔塔利亚，但卡尔达诺没有将功劳归于塔尔塔利亚，而是据为己有。换句话说，卡尔达诺窃取了属于塔尔塔利亚的声誉和功劳。1535 年，塔尔塔利亚在博洛尼亚大学的数学竞赛中获胜，他在数学史上留下了巨大的遗产，他展示了求解三次方程的一般代数公式，而这个公式在当时被认为是不合适的。这是可能的，因为它需要理解负数的平方根。在比赛中，他击败了希皮奥内・德尔・费罗（Scipione del Ferro），巧合的是，费罗比塔尔塔利亚更早提出了三次方程问题的部分解法。虽然费罗的解法比塔尔塔利亚的更早，但内容上更有限。塔尔塔利亚被认为是第一个给出通用解法的人。在 16 世纪竞争激烈且残酷的意大利环境中，塔尔塔利亚甚至将他的解法编成诗歌，试图让其他数学家更难窃取 [1 - 15]。

Rafael Bombelli, a famous Renaissance European engineer. He was also a brilliant mathematician. He was the first to put forward the concept of complex numbers. His book Algebra, published in 1572, discussed the square root of negative numbers. The book had a wide influence in Europe [1-15].

拉斐尔・邦贝利（Rafael Bombelli）是文艺复兴时期欧洲著名的工程师，同时也是一位杰出的数学家。他第一个提出了复数的概念。他在 1572 年出版的《代数学》（Algebra）一书中讨论了负数的平方根。这本书在欧洲产生了广泛的影响 [1 - 15]。

Bombelli believed that the quantities appearing in Cardano's method were real. This is also a potential for new discovery for mathematics, not just a symbol of partially ineffective scholarship. Because he was the first to have this idea, he basically built it from scratch, so he decided to rename the content. Because symbolic notation was not widely used at the time to convey mathematics, especially since Pombelli's notation for imaginary numbers had not yet been created. Thus, at the time this method of representing mathematical expressions of the unknown by using the Bombelli language would be very different from the symbolic notation used today.

邦贝利认为，卡尔达诺方法中出现的量是真实存在的。这对数学来说是一个新发现的潜力，而不仅仅是部分无用学术的象征。因为他是第一个有这种想法的人，基本上是从零开始构建，所以他决定重新命名这些内容。由于当时符号表示法在数学表达中还没有广泛使用，尤其是庞贝利（Pombelli）的虚数符号还未发明。因此，当时用邦贝利的语言来表示未知量的数学表达式的方法与今天使用的符号表示法有很大不同。

Bombelli tackles the cube's irreducible situation before obtaining the complex number's cube root. Additionally, he created his own technique for resolving cubic problems. He bravely acknowledges the reality of negative square roots, or in other words he assumes that there are negative numbers in square roots. So Bombelli pioneered a new approach. Bombelli's method nicely marks the beginning of the plural. The claim is that Bombelli stopped calling them numbers. Instead, he sees them as "a new kind of connected radical." Bombelli considers the development of research on triple irreducible instances. Consequently, a cubic has three actual roots. He begins by demonstrating how Cardano's formula is irreducible according to Tartaglia and Cardano since it makes it difficult to identify these roots. He then filled in a gap in his study by demonstrating how a combination of fictitious roots may result in a real number. That is, to think differently and creatively. This experiment also supports Cardano's methodology. Here is Bombelli's explanation of how real numbers are created from complex numbers. x 3 = 15 x + 4 x^{3}=15 x+4 x3=15x+4 , x = − 121 ± 2 3 x=\sqrt [3]{\sqrt {-121} \pm 2} x=3−121 ±2 x = 2 ± − 121 3 x=\sqrt [3]{2 \pm \sqrt {-121}} x=32±−121

邦贝利在求复数的立方根之前，先处理了三次方程的不可约情形。此外，他还创造了自己的求解三次方程问题的方法。他勇敢地承认了负平方根的存在，换句话说，他假设平方根中存在负数。因此，邦贝利开创了一种新方法。邦贝利的方法很好地标志着复数研究的开端。据说邦贝利不再把它们称为数，而是将其视为 "一种新的相关根式"。邦贝利考虑了三次不可约情形的研究发展。因此，一个三次方程有三个实根。他首先证明了根据塔尔塔利亚和卡尔达诺的理论，卡尔达诺公式是不可约的，因为它很难确定这些根。然后，他通过证明虚根的组合如何能得到一个实数，填补了自己研究中的一个空白。也就是说，他以不同寻常且富有创造性的方式思考。这个实验也支持了卡尔达诺的方法。以下是邦贝利对如何从复数得到实数的解释。 x 3 = 15 x + 4 x^{3}=15x + 4 x3=15x+4， x = − 121 ± 2 3 x=\sqrt [3]{\sqrt {-121}\pm2} x=3−121 ±2 ， x = 2 ± − 121 3 x=\sqrt [3]{2\pm\sqrt {-121}} x=32±−121

Where there should be three roots, only one exists, according to Bombelli, and none of these roots. Bombelli started his own mathematical studies in 1560. He really demonstrated how to modify the fictitious phrase provided by Cardano technology to generate a real number. Here is an illustration from the source that shows how actual value can be drawn out. The two expressions in the next equations, according to Bombelli, merely change in sign. He initially applied his creative interpretation to these formulae by using the numbers A and B. As he puts it, there doesn't appear to be much of a difference between the two phrases on the surface alone - there is just one different sign. In common notation, there are 2 + − 121 3 = a + b − 1 \sqrt [3]{2+\sqrt {-121}}=a+b \sqrt {-1} 32+−121 =a+b−1 ; 2 − − 121 3 = a − b − 1 \sqrt [3]{2-\sqrt {-121}}=a-b \sqrt {-1} 32−−121 =a−b−1 . Now, check the first expression above. 2 + − 121 = ( a + b − 1 ) 3 = a 3 + 3 a 2 b − 1 + 3 a b 2 ( − 1 ) 2 + b 3 ( − 1 ) 3 = a ( a 2 − 3 b 2 ) + b ( 3 a 2 − b 2 ) − 1 2+\sqrt {-121}=(a+b \sqrt {-1})^{3}=a^{3}+3 a^{2} b \sqrt {-1}+3 a b^{2}(\sqrt {-1})^{2}+b^{3}(\sqrt {-1})^{3}=a (a^{2}-3 b^{2})+b (3 a^{2}-b^{2}) \sqrt {-1} 2+−121 =(a+b−1 )3=a3+3a2b−1 +3ab2(−1 )2+b3(−1 )3=a(a2−3b2)+b(3a2−b2)−1 . And then it can compute the solution that satisfies these two equations, u = 2 u = 2 u=2, v = 1 v = 1 v=1. When these amounts are added to the a and b equations above, The equation can be written as 2 + − 121 3 = 2 + − 1 \sqrt [3]{2+\sqrt {-121}}=2+\sqrt {-1} 32+−121 =2+−1 ; 2 − − 121 3 = 2 − − 1 \sqrt [3]{2-\sqrt {-121}}=2-\sqrt {-1} 32−−121 =2−−1 . Because the unintentionally revealed real numbers are likewise complicated, Bombelli's notion is incredibly original and startling. In essence, Bombelli provided more empirical support for the idea that any real number may be expressed as a complex number. A mathematical innovation in this area is Bombelli's theory. The organizational definition of the set of real numbers and the close connection between the set of real numbers and imaginary numbers provide significant depth to the development of set theory. Bombelli also gave later mathematicians a fresh perspective on how to approach this area of mathematics if they want to pursue it further.

按照邦贝利的说法，本应有三个根，但实际上只存在一个根，而且这些根都并非普通意义上的根。邦贝利在 1560 年开始了自己的数学研究。他切实地展示了如何修改卡尔达诺方法中给出的虚数表达式来得到一个实数。这里有一个源自他研究的例子，展示了如何得出实际值。邦贝利认为，下面方程中的两个表达式仅仅是符号不同。他最初用 A 和 B 这两个数对这些公式进行了创造性的解释。正如他所说，仅从表面上看，这两个表达式似乎没有太大区别 ------ 只是符号不同。用常见的符号表示，有 2 + − 121 3 = a + b − 1 \sqrt [3]{2+\sqrt {-121}}=a + b\sqrt {-1} 32+−121 =a+b−1 ； 2 − − 121 3 = a − b − 1 \sqrt [3]{2-\sqrt {-121}}=a - b\sqrt {-1} 32−−121 =a−b−1 。现在，来看上面的第一个表达式。 2 + − 121 = ( a + b − 1 ) 3 = a 3 + 3 a 2 b − 1 + 3 a b 2 ( − 1 ) 2 + b 3 ( − 1 ) 3 = a ( a 2 − 3 b 2 ) + b ( 3 a 2 − b 2 ) − 1 2+\sqrt {-121}=(a + b\sqrt {-1})^{3}=a^{3}+3a^{2} b\sqrt {-1}+3ab^{2}(\sqrt {-1})^{2}+b^{3}(\sqrt {-1})^{3}=a (a^{2}-3b^{2})+b (3a^{2}-b^{2})\sqrt {-1} 2+−121 =(a+b−1 )3=a3+3a2b−1 +3ab2(−1 )2+b3(−1 )3=a(a2−3b2)+b(3a2−b2)−1 。然后可以计算出满足这两个方程的解， u = 2 u = 2 u=2， v = 1 v = 1 v=1 。将这些值代入上面关于 a 和 b 的方程中，该方程可写为 2 + − 121 3 = 2 + − 1 \sqrt [3]{2+\sqrt {-121}}=2+\sqrt {-1} 32+−121 =2+−1 ； 2 − − 121 3 = 2 − − 1 \sqrt [3]{2-\sqrt {-121}}=2-\sqrt {-1} 32−−121 =2−−1 。由于无意之中揭示出的实数同样复杂，邦贝利的观点极具创新性和震撼性。从本质上讲，邦贝利为任何实数都可以表示为复数这一观点提供了更多的实证支持。邦贝利的理论是该领域的一项数学创新。实数集的组织定义以及实数集与虚数集之间的紧密联系为集合论的发展提供了重要的深度。邦贝利还为后来想要进一步深入研究这一数学领域的数学家们提供了全新的研究视角。

Bombelli's research is the foundation of algebra, and he is considered the founder of new algebra. This is a great achievement, because Bombelli first recognized imaginary numbers, and was able to let those mathematicians in later generations understand and see algebra from a new perspective. As those mathematicians know. Later, this view was also confirmed and accepted, because his bold argument was more like a more concrete proof idea. Simply put, he takes something that is currently known to exist and shows why there is a misunderstanding of it, rather than formulating rules from calculations. His research and discovery of complex numbers also provided mathematicians with the opportunity to expand the thinking of cubic functions and study algebra from new aspects [1-15].

邦贝利的研究是代数学的基础，他被视为新代数学的奠基人。这是一项伟大的成就，因为邦贝利率先认识到了虚数，并且能够让后世的数学家从新的视角去理解和看待代数学。正如那些数学家所了解的那样。后来，这一观点也得到了证实和认可，因为他大胆的论证更像是一种更为具体的证明思路。简单来说，他从人们已知存在的事物出发，解释为何会对其存在误解，而不是通过计算来制定规则。他对复数的研究和发现也为数学家们提供了拓展三次函数思维、从新的角度研究代数学的机会 [1-15]。

2.2 Complex number and the complex plane.

复数与复平面

Complex numbers are defined as the set of C : = { a + i b : a , b ∈ all real number } C:=\{a + ib: a, b \in \text {all real number}\} C:={a+ib:a,b∈all real number}, where i = − 1 i = \sqrt {-1} i=−1 (that is, i 2 = − 1 i^{2}=-1 i2=−1). For z = x + i y ∈ C z=x + iy \in C z=x+iy∈C, the real part is Re ( z ) : = x \text {Re}(z):=x Re(z):=x. And the imaginary part is Im ( z ) : = y \text {Im}(z):=y Im(z):=y. The absolute value is ∣ z ∣ : = ( x 2 + y 2 ) 1 2 |z|:=(x^{2}+y^{2})^{\frac {1}{2}} ∣z∣:=(x2+y2)21. For the operations on C C C, C C C admits the operations of Additions: ( x 1 + i y 1 ) + ( x 2 + i y 2 ) = ( x 1 + x 2 ) + i ( y 1 + y 2 ) (x_1 + iy_1)+(x_2 + iy_2)=(x_1 + x_2)+i (y_1 + y_2) (x1+iy1)+(x2+iy2)=(x1+x2)+i(y1+y2); Multiplication: ( x 1 + i y 1 ) × ( x 2 + i y 2 ) = ( x 1 x 2 − y 1 y 2 ) + i ( x 1 y 2 + x 2 y 1 ) (x_1 + iy_1)\times (x_2 + iy_2)=(x_1x_2 - y_1y_2)+i (x_1y_2 + x_2y_1) (x1+iy1)×(x2+iy2)=(x1x2−y1y2)+i(x1y2+x2y1); Complex conjugation: x + i y ‾ = x − i y \overline {x + iy}=x - iy x+iy=x−iy.

复数被定义为集合 C : = { a + i b : a , b ∈ 所有实数 } C:=\{a + ib: a, b \in \text {所有实数}\} C:={a+ib:a,b∈所有实数}，其中 i = − 1 i = \sqrt {-1} i=−1 （即 i 2 = − 1 i^{2}=-1 i2=−1）。对于 z = x + i y ∈ C z=x + iy \in C z=x+iy∈C，其实部定义为 Re ( z ) : = x \text {Re}(z):=x Re(z):=x，虚部定义为 Im ( z ) : = y \text {Im}(z):=y Im(z):=y，绝对值定义为 ∣ z ∣ : = ( x 2 + y 2 ) 1 2 |z|:=(x^{2}+y^{2})^{\frac {1}{2}} ∣z∣:=(x2+y2)21。在集合 C C C 上，定义了以下运算：加法： ( x 1 + i y 1 ) + ( x 2 + i y 2 ) = ( x 1 + x 2 ) + i ( y 1 + y 2 ) (x_1 + iy_1)+(x_2 + iy_2)=(x_1 + x_2)+i (y_1 + y_2) (x1+iy1)+(x2+iy2)=(x1+x2)+i(y1+y2)；乘法： ( x 1 + i y 1 ) × ( x 2 + i y 2 ) = ( x 1 x 2 − y 1 y 2 ) + i ( x 1 y 2 + x 2 y 1 ) (x_1 + iy_1)\times (x_2 + iy_2)=(x_1x_2 - y_1y_2)+i (x_1y_2 + x_2y_1) (x1+iy1)×(x2+iy2)=(x1x2−y1y2)+i(x1y2+x2y1)；复共轭： x + i y ‾ = x − i y \overline {x + iy}=x - iy x+iy=x−iy。

Addition and multiplication are simply the natural extensions of the corresponding operations on all real numbers along with the rule that i 2 = − 1 i^{2}=-1 i2=−1. Furthermore, these operations make C C C a field: addition and multiplication are associative, commutative, admit identities (0 and 1, respectively), recognize inverses − z -z −z and 1 z \frac {1}{z} z1 (for z ≠ 0 z\neq0 z=0), and satisfy the distributive laws that exercise these factual persuasion. One also has h + k ‾ = h ‾ + k ‾ \overline {h + k}=\overline {h}+\overline {k} h+k=h+k, h k ‾ = h ‾ × k ‾ \overline {hk}=\overline {h}\times\overline {k} hk=h×k and Re ( h ) = 1 2 ( h + h ‾ ) \text {Re}(h)=\frac {1}{2}(h+\overline {h}) Re(h)=21(h+h); Im ( h ) = 1 2 i ( h − h ‾ ) \text {Im}(h)=\frac {1}{2i}(h-\overline {h}) Im(h)=2i1(h−h); ∣ h ∣ = ( h × h ‾ ) 1 2 |h|=(h\times\overline {h})^{\frac {1}{2}} ∣h∣=(h×h)21. For all h , w ∈ C h, w \in C h,w∈C. The last identity shows ∣ h ∣ = 0 |h| = 0 ∣h∣=0 if and only if h = 0 h = 0 h=0. This can also derived the tag "conjugating the denominator": 1 h = h ‾ ∣ h ∣ 2 \frac {1}{h}=\frac {\overline {h}}{|h|^{2}} h1=∣h∣2h (after checking h ≠ 0 h\neq0 h=0), for example, 1 s + i t = 1 s + i t × s − i t s − i t = s − i t s 2 + t 2 = s + i t ‾ ∣ s + i t ∣ 2 \frac {1}{s+it}=\frac {1}{s+it}\times\frac {s - it}{s - it}=\frac {s - it}{s^{2}+t^{2}}=\frac {\overline {s+it}}{|s+it|^{2}} s+it1=s+it1×s−its−it=s2+t2s−it=∣s+it∣2s+it. Using the above identities, it shows ∣ h k ∣ 2 = ( h k ) × ( h k ‾ ) = h k h ‾ × k ‾ = ∣ h ∣ 2 × ∣ k ∣ 2 → ∣ h k ∣ = ∣ h ∣ ∣ k ∣ |hk|^{2}=(hk)\times (\overline {hk})=h k\overline {h}\times\overline {k}=|h|^{2}\times|k|^{2}\to|hk|=|h||k| ∣hk∣2=(hk)×(hk)=hkh×k=∣h∣2×∣k∣2→∣hk∣=∣h∣∣k∣.

加法和乘法是实数相应运算的自然扩展，并遵循 i 2 = − 1 i^{2}=-1 i2=−1 这一规则。此外，这些运算使得 C C C 构成一个域：加法和乘法满足结合律、交换律，存在单位元（分别为 0 和 1），存在逆元 − z -z −z（对应加法）和 1 z \frac {1}{z} z1（ z ≠ 0 z\neq0 z=0 时，对应乘法），并且满足分配律。对于所有 h , w ∈ C h, w \in C h,w∈C，还有 h + k ‾ = h ‾ + k ‾ \overline {h + k}=\overline {h}+\overline {k} h+k=h+k， h k ‾ = h ‾ × k ‾ \overline {hk}=\overline {h}\times\overline {k} hk=h×k， Re ( h ) = 1 2 ( h + h ‾ ) \text {Re}(h)=\frac {1}{2}(h+\overline {h}) Re(h)=21(h+h)， Im ( h ) = 1 2 i ( h − h ‾ ) \text {Im}(h)=\frac {1}{2i}(h-\overline {h}) Im(h)=2i1(h−h)， ∣ h ∣ = ( h × h ‾ ) 1 2 |h|=(h\times\overline {h})^{\frac {1}{2}} ∣h∣=(h×h)21 。最后这个等式表明， ∣ h ∣ = 0 |h| = 0 ∣h∣=0 当且仅当 h = 0 h = 0 h=0。由此还可以推导出 "分母有理化" 的方法： 1 h = h ‾ ∣ h ∣ 2 \frac {1}{h}=\frac {\overline {h}}{|h|^{2}} h1=∣h∣2h（需检验 h ≠ 0 h\neq0 h=0），例如， 1 s + i t = 1 s + i t × s − i t s − i t = s − i t s 2 + t 2 = s + i t ‾ ∣ s + i t ∣ 2 \frac {1}{s+it}=\frac {1}{s+it}\times\frac {s - it}{s - it}=\frac {s - it}{s^{2}+t^{2}}=\frac {\overline {s+it}}{|s+it|^{2}} s+it1=s+it1×s−its−it=s2+t2s−it=∣s+it∣2s+it。利用上述等式，可以得到 ∣ h k ∣ 2 = ( h k ) × ( h k ‾ ) = h k h ‾ × k ‾ = ∣ h ∣ 2 × ∣ k ∣ 2 |hk|^{2}=(hk)\times (\overline {hk})=h k\overline {h}\times\overline {k}=|h|^{2}\times|k|^{2} ∣hk∣2=(hk)×(hk)=hkh×k=∣h∣2×∣k∣2，进而得出 ∣ h k ∣ = ∣ h ∣ ∣ k ∣ |hk|=|h||k| ∣hk∣=∣h∣∣k∣。

Also, h ∈ C h \in C h∈C is real if and only if h ‾ = h \overline {h}=h h=h, and h h h is imaginary if and only if h ‾ = − h \overline {h}=-h h=−h. Let α = a + b i \alpha=a + bi α=a+bi be a complex number, α ‾ = a − b i \overline {\alpha}=a - bi α=a−bi is called the conjugate of α \alpha α, and α α ‾ = a 2 + b 2 = ∣ α ∣ 2 \alpha\overline {\alpha}=a^{2}+b^{2}=|\alpha|^{2} αα=a2+b2=∣α∣2. Let α \alpha α, β \beta β be complex numbers, then α β ‾ = α ‾ β ‾ \overline {\alpha\beta}=\overline {\alpha}\overline {\beta} αβ=αβ, α + β ‾ = α ‾ + β ‾ \overline {\alpha+\beta}=\overline {\alpha}+\overline {\beta} α+β=α+β, α ‾ ‾ = α \overline {\overline {\alpha}}=\alpha α=α. What is Visualizing C C C? C C C should be visualized as a coordinate plane with the square of all real numbers, where a point a = p + i q a=p+iq a=p+iq is plotted as the pair ( p , q ) (p,q) (p,q). In this case, the coordinate plane is referred to as the complex plane, and the x x x-axis and y y y-axis are referred to as the real and imaginary axes, respectively. Using the square of all real integers as the distance formula, it is known that ∣ z ∣ |z| ∣z∣ is the distance from z z z to 0 on the complex plane. More generally, ∣ z − w ∣ |z - w| ∣z−w∣ is the distance from z z z to w w w.

此外， h ∈ C h \in C h∈C 为实数当且仅当 h ‾ = h \overline {h}=h h=h， h h h 为虚数当且仅当 h ‾ = − h \overline {h}=-h h=−h。设 α = a + b i \alpha=a + bi α=a+bi 为一个复数， α ‾ = a − b i \overline {\alpha}=a - bi α=a−bi 称为 α \alpha α 的共轭复数，且 α α ‾ = a 2 + b 2 = ∣ α ∣ 2 \alpha\overline {\alpha}=a^{2}+b^{2}=|\alpha|^{2} αα=a2+b2=∣α∣2。设 α \alpha α， β \beta β 为复数，则 α β ‾ = α ‾ β ‾ \overline {\alpha\beta}=\overline {\alpha}\overline {\beta} αβ=αβ， α + β ‾ = α ‾ + β ‾ \overline {\alpha+\beta}=\overline {\alpha}+\overline {\beta} α+β=α+β， α ‾ ‾ = α \overline {\overline {\alpha}}=\alpha α=α。如何直观理解集合 C C C 呢？可以将 C C C 看作是一个以所有实数的平方为基础的坐标平面，其中点 a = p + i q a=p+iq a=p+iq 被表示为坐标对 ( p , q ) (p,q) (p,q) 。在这种情况下，该坐标平面被称为复平面， x x x 轴和 y y y 轴分别被称为实轴和虚轴。根据以所有实数的平方定义的距离公式可知， ∣ z ∣ |z| ∣z∣ 是复平面上 z z z 到 0 的距离。更一般地， ∣ z − w ∣ |z - w| ∣z−w∣ 是复平面上 z z z 到 w w w 的距离。

Note that addition matches vector space addition on full real squares and can therefore be visualized in this way. To visualize the multiplication, recall for the first time that every point in the square of all real numbers can be described by polar coordinates and converted to C C C: for any z ∈ C ∖ { 0 } z \in C\setminus\{0\} z∈C∖{0}, ∣ z ∣ z ∣ ∣ = ∣ z ∣ × ∣ 1 ∣ z ∣ ∣ = ∣ z ∣ × 1 ∣ z ∣ = 1 |\frac {z}{|z|}|=|z|\times|\frac {1}{|z|}|=|z|\times\frac {1}{|z|}=1 ∣∣z∣z∣=∣z∣×∣∣z∣1∣=∣z∣×∣z∣1=1. Thus there exists θ ∈ all real number \theta\in\text {all real number} θ∈all real number so that z ∣ z ∣ = cos ⁡ θ + i sin ⁡ θ \frac {z}{|z|}=\cos\theta + i\sin\theta ∣z∣z=cosθ+isinθ. If we write e i θ = cos ⁡ θ + i sin ⁡ θ e^{i\theta}=\cos\theta + i\sin\theta eiθ=cosθ+isinθ, the definition of absolute value of complex number is given as for a given complex number α = α 1 + i α 2 ( α 1 , α 2 ∈ R ) \alpha=\alpha_1 + i\alpha_2 (\alpha_1,\alpha_2\in R) α=α1+iα2(α1,α2∈R), ∣ α ∣ = α 1 2 + α 2 2 |\alpha|=\sqrt {\alpha_1^{2}+\alpha_2^{2}} ∣α∣=α12+α22 . This notation will be justified by discussion of power series in sections above, then z = ∣ z ∣ e i θ z = |z|e^{i\theta} z=∣z∣eiθ.

注意，复数的加法与全实平方空间上的向量加法一致，因此可以从向量加法的角度直观理解。为了直观理解复数乘法，首先要知道所有实数平方空间中的每一个点都可以用极坐标表示，并转换到复数域：对于任意 z ∈ C ∖ { 0 } z \in C\setminus\{0\} z∈C∖{0}， ∣ z ∣ z ∣ ∣ = ∣ z ∣ × ∣ 1 ∣ z ∣ ∣ = ∣ z ∣ × 1 ∣ z ∣ = 1 |\frac {z}{|z|}|=|z|\times|\frac {1}{|z|}|=|z|\times\frac {1}{|z|}=1 ∣∣z∣z∣=∣z∣×∣∣z∣1∣=∣z∣×∣z∣1=1。因此，存在 θ ∈ 所有实数 \theta\in\text {所有实数} θ∈所有实数，使得 z ∣ z ∣ = cos ⁡ θ + i sin ⁡ θ \frac {z}{|z|}=\cos\theta + i\sin\theta ∣z∣z=cosθ+isinθ。如果我们记 e i θ = cos ⁡ θ + i sin ⁡ θ e^{i\theta}=\cos\theta + i\sin\theta eiθ=cosθ+isinθ，对于给定的复数 α = α 1 + i α 2 ( α 1 , α 2 ∈ R ) \alpha=\alpha_1 + i\alpha_2 (\alpha_1,\alpha_2\in R) α=α1+iα2(α1,α2∈R)，其绝对值定义为 ∣ α ∣ = α 1 2 + α 2 2 |\alpha|=\sqrt {\alpha_1^{2}+\alpha_2^{2}} ∣α∣=α12+α22 。上述关于幂级数的讨论将证明这种表示法的合理性，此时 z = ∣ z ∣ e i θ z = |z|e^{i\theta} z=∣z∣eiθ。

Define for z ∈ C z \in C z∈C, z = ∣ z ∣ e i θ z = |z|e^{i\theta} z=∣z∣eiθ is said to be the polar form of z z z. The angle θ ∈ all real number \theta\in\text {all real number} θ∈all real number is said to be the argument of z z z, and is denoted arg ( z ) \text {arg}(z) arg(z). Using trigonometric identities, one can check that e i θ 1 e i θ 2 = e i ( θ 1 + θ 2 ) e^{i\theta_1} e^{i\theta_2}=e^{i (\theta_1+\theta_2)} eiθ1eiθ2=ei(θ1+θ2). Thus for z z z, w w w, z w = ∣ z ∣ e i arg ( z ) × ∣ w ∣ e i arg ( w ) = ∣ z w ∣ e i ( arg ( z ) + arg ( w ) ) zw=|z|e^{i\text {arg}(z)}\times|w|e^{i\text {arg}(w)}=|zw|e^{i (\text {arg}(z)+\text {arg}(w))} zw=∣z∣eiarg(z)×∣w∣eiarg(w)=∣zw∣ei(arg(z)+arg(w)). Complex conjugation x + i y ‾ = x − i y \overline {x + iy}=x - iy x+iy=x−iy corresponds to reflection over the real axis. Since the absolute value corresponds to distance in the complex plane, one obtains the triangle inequality: ∣ h + k ∣ ≤ ∣ h ∣ + ∣ k ∣ |h + k|\leq|h|+|k| ∣h+k∣≤∣h∣+∣k∣. This further implies the triangle inequality: ∣ ∣ h ∣ − ∣ k ∣ ∣ ≤ ∣ h − k ∣ ||h|-|k||\leq|h - k| ∣∣h∣−∣k∣∣≤∣h−k∣.

对于 z ∈ C z \in C z∈C，定义 z = ∣ z ∣ e i θ z = |z|e^{i\theta} z=∣z∣eiθ 为 z z z 的极坐标形式。其中， θ ∈ 所有实数 \theta\in\text {所有实数} θ∈所有实数 称为 z z z 的辐角，记作 arg ( z ) \text {arg}(z) arg(z)。利用三角函数恒等式可以验证 e i θ 1 e i θ 2 = e i ( θ 1 + θ 2 ) e^{i\theta_1} e^{i\theta_2}=e^{i (\theta_1+\theta_2)} eiθ1eiθ2=ei(θ1+θ2) 。因此，对于 z z z 和 w w w，有 z w = ∣ z ∣ e i arg ( z ) × ∣ w ∣ e i arg ( w ) = ∣ z w ∣ e i ( arg ( z ) + arg ( w ) ) zw=|z|e^{i\text {arg}(z)}\times|w|e^{i\text {arg}(w)}=|zw|e^{i (\text {arg}(z)+\text {arg}(w))} zw=∣z∣eiarg(z)×∣w∣eiarg(w)=∣zw∣ei(arg(z)+arg(w))。复共轭 x + i y ‾ = x − i y \overline {x + iy}=x - iy x+iy=x−iy 对应于关于实轴的反射。由于绝对值在复平面中表示距离，由此可得三角不等式： ∣ h + k ∣ ≤ ∣ h ∣ + ∣ k ∣ |h + k|\leq|h|+|k| ∣h+k∣≤∣h∣+∣k∣。这进一步推出另一个三角不等式： ∣ ∣ h ∣ − ∣ k ∣ ∣ ≤ ∣ h − k ∣ ||h|-|k||\leq|h - k| ∣∣h∣−∣k∣∣≤∣h−k∣。

Indeed, from the triangle inequality we have ∣ h ∣ − ∣ k ∣ ≤ ∣ h − k ∣ |h|-|k|\leq|h - k| ∣h∣−∣k∣≤∣h−k∣. Also, ∣ k ∣ = ∣ k − h + h ∣ ≤ ∣ k − h ∣ + ∣ h ∣ |k|=|k - h+h|\leq|k - h|+|h| ∣k∣=∣k−h+h∣≤∣k−h∣+∣h∣, so that − ( ∣ h ∣ − ∣ k ∣ ) = ∣ k ∣ − ∣ h ∣ ≤ ∣ k − h ∣ = ∣ h − k ∣ -(|h|-|k|)=|k|-|h|\leq|k - h|=|h - k| −(∣h∣−∣k∣)=∣k∣−∣h∣≤∣k−h∣=∣h−k∣. Hence it can get solutions ∣ R e ( h ) ∣ ≤ ∣ h ∣ |Re (h)|\leq|h| ∣Re(h)∣≤∣h∣ and ∣ I m ( h ) ∣ ≤ ∣ h ∣ |Im (h)|\leq|h| ∣Im(h)∣≤∣h∣.

确实，由三角不等式可得 ∣ h ∣ − ∣ k ∣ ≤ ∣ h − k ∣ |h| - |k| \leq |h - k| ∣h∣−∣k∣≤∣h−k∣。同时， ∣ k ∣ = ∣ k − h + h ∣ ≤ ∣ k − h ∣ + ∣ h ∣ |k| = |k - h + h| \leq |k - h| + |h| ∣k∣=∣k−h+h∣≤∣k−h∣+∣h∣，所以 − ( ∣ h ∣ − ∣ k ∣ ) = ∣ k ∣ − ∣ h ∣ ≤ ∣ k − h ∣ = ∣ h − k ∣ -(|h| - |k|) = |k| - |h| \leq |k - h| = |h - k| −(∣h∣−∣k∣)=∣k∣−∣h∣≤∣k−h∣=∣h−k∣。由此可得 ∣ R e ( h ) ∣ ≤ ∣ h ∣ |Re (h)| \leq |h| ∣Re(h)∣≤∣h∣ 且 ∣ I m ( h ) ∣ ≤ ∣ h ∣ |Im (h)| \leq |h| ∣Im(h)∣≤∣h∣。

The definition of the convergence is a sequence ( z n ) n ∈ ω ∈ C (z_{n}){n \in \omega} \in C (zn)n∈ω∈C converges to ω ∈ C \omega \in C ω∈C if lim ⁡ n → ∞ ∣ z n − ω ∣ = 0 \lim {n \to \infty} | z{n}-\omega |=0 limn→∞∣zn−ω∣=0, in which case it shows lim ⁡ n → ∞ z n = ω \lim {n \to \infty} z{n}=\omega limn→∞zn=ω and call ω \omega ω the limit of the sequence. More explicitly, ( z n ) n ∈ ω (z{n}){n \in \omega} (zn)n∈ω converges to ω \omega ω if ∀ ε > 0 \forall \varepsilon>0 ∀ε>0; ∃ N ∈ N \exists N \in \mathbb {N} ∃N∈N, so that ∀ n ≥ N , ∣ z n − ω ∣ < ε \forall{n \geq N},|z_{n}-\omega|<\varepsilon ∀n≥N,∣zn−ω∣<ε.

收敛的定义是：对于复数序列 ( z n ) n ∈ ω ∈ C (z_{n}){n \in \omega} \in C (zn)n∈ω∈C，如果 lim ⁡ n → ∞ ∣ z n − ω ∣ = 0 \lim {n \to \infty} | z{n}-\omega |=0 limn→∞∣zn−ω∣=0，则称该序列收敛于 ω ∈ C \omega \in C ω∈C ，此时记 lim ⁡ n → ∞ z n = ω \lim {n \to \infty} z{n}=\omega limn→∞zn=ω，并称 ω \omega ω 为该序列的极限。更明确地说，如果对于任意 ε > 0 \varepsilon>0 ε>0，存在 N ∈ N N \in \mathbb {N} N∈N，使得对于所有 n ≥ N n \geq N n≥N，都有 ∣ z n − ω ∣ < ε |z{n}-\omega|<\varepsilon ∣zn−ω∣<ε，那么 ( z n ) n ∈ ω (z_{n})_{n \in \omega} (zn)n∈ω 收敛于 ω \omega ω。

Since ∣ R e ( z ) ∣ , ∣ I m ( z ) ∣ ≤ ∣ z ∣ = ( R e ( z ) 2 + I m ( z ) 2 ) 1 2 |Re (z)|,|Im (z)| \leq|z|=(Re (z)^{2}+Im (z)^{2})^{\frac {1}{2}} ∣Re(z)∣,∣Im(z)∣≤∣z∣=(Re(z)2+Im(z)2)21, one has lim ⁡ n → ∞ z n = ω ↔ lim ⁡ n → ∞ R e ( z n ) = ω \lim {n \to \infty} z{n}=\omega \leftrightarrow \lim {n \to \infty} Re (z{n})=\omega limn→∞zn=ω↔limn→∞Re(zn)=ω and lim ⁡ n → ∞ I m ( z n ) = ω \lim {n \to \infty} Im (z{n})=\omega limn→∞Im(zn)=ω. Thus this notion of convergence agrees with the usual one on the square of all real number. The other definition is ( z n ) n ∈ ω < C (z_{n}){n \in \omega}<C (zn)n∈ω<C is a Cauchy sequence if ∀ ε > 0 ∃ N ∈ N \forall{\varepsilon>0} \exists N \in \mathbb {N} ∀ε>0∃N∈N So that ∀ n , m ≥ 0 \forall n, m \geq 0 ∀n,m≥0 are has ∣ z n − z m ∣ < ε |z_{n}-z_{m}|<\varepsilon ∣zn−zm∣<ε. To put it another way, the terms in a Cauchy sequence finally get as near as they are willing to. Cauchy sequences are those that gradually approach both the limit and one another. It turns out the converge is true: C C C is complete: all Cauchy sequences converge. This illustrates that all real number is complete, so the real and imaginary parts converge to some x , y ∈ x, y \in x,y∈ all real number, respectively. Define ω : = x + i y \omega :=x + iy ω:=x+iy, then lim ⁡ n → ∞ ∣ z n − ω ∣ = lim ⁡ n → ∞ ( ( R e ( z n ) − x ) 2 + ( I m ( z n ) − y ) 2 ) 1 2 = 0 \lim {n \to \infty}|z{n}-\omega|=\lim {n \to \infty}((Re (z{n})-x)^{2}+(Im (z_{n})-y)^{2})^{\frac {1}{2}}=0 limn→∞∣zn−ω∣=limn→∞((Re(zn)−x)2+(Im(zn)−y)2)21=0.

由于 ∣ R e ( z ) ∣ , ∣ I m ( z ) ∣ ≤ ∣ z ∣ = ( R e ( z ) 2 + I m ( z ) 2 ) 1 2 |Re (z)|,|Im (z)| \leq|z|=(Re (z)^{2}+Im (z)^{2})^{\frac {1}{2}} ∣Re(z)∣,∣Im(z)∣≤∣z∣=(Re(z)2+Im(z)2)21，所以 lim ⁡ n → ∞ z n = ω \lim {n \to \infty} z{n}=\omega limn→∞zn=ω 等价于 lim ⁡ n → ∞ R e ( z n ) = ω \lim {n \to \infty} Re (z{n})=\omega limn→∞Re(zn)=ω 且 lim ⁡ n → ∞ I m ( z n ) = ω \lim {n \to \infty} Im (z{n})=\omega limn→∞Im(zn)=ω。因此，这种收敛的概念与所有实数平方空间上通常的收敛概念是一致的。另一个定义是：如果对于任意 ε > 0 \varepsilon>0 ε>0，存在 N ∈ N N \in \mathbb {N} N∈N，使得对于所有 n , m ≥ 0 n, m \geq 0 n,m≥0，都有 ∣ z n − z m ∣ < ε |z_{n}-z_{m}|<\varepsilon ∣zn−zm∣<ε，那么序列 ( z n ) n ∈ ω ∈ C (z_{n}){n \in \omega} \in C (zn)n∈ω∈C 是一个柯西序列。换句话说，柯西序列中的项最终会变得任意接近。柯西序列是那些逐渐趋近于极限且彼此之间也逐渐趋近的序列。事实证明收敛性成立： C C C 是完备的，即所有柯西序列都收敛。这表明所有实数都是完备的，所以实部和虚部分别收敛到某个 x , y ∈ x, y \in x,y∈ 所有实数。定义 ω : = x + i y \omega :=x + iy ω:=x+iy，那么 lim ⁡ n → ∞ ∣ z n − ω ∣ = lim ⁡ n → ∞ ( ( R e ( z n ) − x ) 2 + ( I m ( z n ) − y ) 2 ) 1 2 = 0 \lim {n \to \infty}|z{n}-\omega|=\lim {n \to \infty}((Re (z{n})-x)^{2}+(Im (z{n})-y)^{2})^{\frac {1}{2}}=0 limn→∞∣zn−ω∣=limn→∞((Re(zn)−x)2+(Im(zn)−y)2)21=0。

Thus ( z n ) n ∈ N (z_{n})_{n \in \mathbb {N}} (zn)n∈N converges to ω \omega ω.

因此，序列 ( z n ) n ∈ N (z_{n})_{n \in \mathbb {N}} (zn)n∈N 收敛于 ω \omega ω。

2.3 Complex sets and Topology

复集与拓扑

Complex sets and Topology establish some notation and terminology for subsets C C C.

复集与拓扑为复数集 C C C 的子集建立了一些符号和术语。

Define for z 0 ∈ C z_{0} \in C z0∈C and r > 0 r>0 r>0 the open disc of radius r r r centered at z 0 z_{0} z0 is D r ( z 0 ) : = { z ∈ C : ∣ z − z 0 ∣ < r } \mathbb {D}{r}(z{0}):=\{z \in C: |z - z_{0}|<r\} Dr(z0):={z∈C:∣z−z0∣<r}. The closed disc of radius r r r centered at z 0 z_{0} z0 is D ‾ r ( z 0 ) : = { z ∈ C : ∣ z − z 0 ∣ ≤ r } \overline {\mathbb {D}}{r}(z{0}):=\{z \in C:|z - z_{0}| \leq r\} Dr(z0):={z∈C:∣z−z0∣≤r}. The circle of radius r r r centered at z 0 z_{0} z0 is C r ( z 0 ) : = { z ∈ C : ∣ z − z 0 ∣ = r } C_{r}(z_{0}):=\{z \in C:|z - z_{0}|=r\} Cr(z0):={z∈C:∣z−z0∣=r}. Finally, it can be reserved the following notation for the unit disc all real number ∶= D : = D 1 ( 0 ) = { Z ∈ C : ∣ z ∣ < 1 } \mathbb {D}:=\mathbb {D}_{1}(0)=\{Z \in C:|z|<1\} D:=D1(0)={Z∈C:∣z∣<1}

对于 z 0 ∈ C z_{0} \in C z0∈C 且 r > 0 r > 0 r>0，定义以 z 0 z_{0} z0 为圆心、 r r r 为半径的开圆盘为 D r ( z 0 ) : = { z ∈ C : ∣ z − z 0 ∣ < r } \mathbb {D}{r}(z{0}):=\{z \in C: |z - z_{0}|<r\} Dr(z0):={z∈C:∣z−z0∣<r}。以 z 0 z_{0} z0 为圆心、 r r r 为半径的闭圆盘为 D ‾ r ( z 0 ) : = { z ∈ C : ∣ z − z 0 ∣ ≤ r } \overline {\mathbb {D}}{r}(z{0}):=\{z \in C:|z - z_{0}| \leq r\} Dr(z0):={z∈C:∣z−z0∣≤r}。以 z 0 z_{0} z0 为圆心、 r r r 为半径的圆为 C r ( z 0 ) : = { z ∈ C : ∣ z − z 0 ∣ = r } C_{r}(z_{0}):=\{z \in C:|z - z_{0}|=r\} Cr(z0):={z∈C:∣z−z0∣=r}。最后，对于单位圆盘保留以下符号： D : = D 1 ( 0 ) = { Z ∈ C : ∣ z ∣ < 1 } \mathbb {D}:=\mathbb {D}_{1}(0)=\{Z \in C:|z|<1\} D:=D1(0)={Z∈C:∣z∣<1}

Let's discuss the topological characteristics of complex numbers now. These characteristics are only a translation from the square of all real numbers to complex numbers since the complex plane may be associated with the square of all real numbers and includes certain notations of distance and convergence.

现在来讨论复数的拓扑特征。由于复平面可以与所有实数的平方空间相关联，并且包含了一些距离和收敛的概念，所以这些特征只是从所有实数的平方空间到复数的一种转换。

The absolute value of a complex number satisfies the following properties. If α , β \alpha, \beta α,β are complex numbers, then ∣ α β ∣ = ∣ α ∣ × ∣ β ∣ |\alpha \beta|=|\alpha|\times|\beta| ∣αβ∣=∣α∣×∣β∣; ∣ α + β ∣ ≤ ∣ α ∣ + ∣ β ∣ |\alpha+\beta| \leq|\alpha|+|\beta| ∣α+β∣≤∣α∣+∣β∣ (triangle inequality). Let θ , φ \theta, \varphi θ,φ be complex numbers, then e i ( θ + φ ) = e i θ e i φ e^{i (\theta+\varphi)}=e^{i \theta} e^{i \varphi} ei(θ+φ)=eiθeiφ. Let α , β \alpha,\beta α,β be complex numbers, then e α + β = e α e β e^{\alpha+\beta}=e^{\alpha} e^{\beta} eα+β=eαeβ. The expression r e i θ r e^{i \theta} reiθ is called the polar form of the complex number x + i y x + iy x+iy. The number θ \theta θ is called the angle, or argument of z z z, and we write: θ = arg ⁡ z \theta=\arg z θ=argz. Let α ∈ S \alpha \in S α∈S, we say that f f f is continuous at z 0 z_{0} z0 if lim ⁡ z → z 0 f ( z ) = f ( z 0 ) = w 0 \lim {z \to z{0}} f (z)=f (z_{0})=w_{0} limz→z0f(z)=f(z0)=w0. More explicit: For any ε > 0 \varepsilon>0 ε>0 there exists δ = δ ( ε ) \delta=\delta (\varepsilon) δ=δ(ε) such that. If ∣ z − z 0 ∣ < δ ( ε ) |z - z_{0}|<\delta (\varepsilon) ∣z−z0∣<δ(ε), then ∣ f ( z ) − f ( z 0 ) ∣ < ε |f (z)-f (z_{0})|< \varepsilon ∣f(z)−f(z0)∣<ε. We say that f f f is differentiable at z 0 z_{0} z0 if lim ⁡ z → z 0 f ( z ) − f ( z 0 ) z − z 0 \lim {z \to z{0}} \frac {f (z)-f (z_{0})}{z - z_{0}} limz→z0z−z0f(z)−f(z0) exists. We denote the limit by f ′ ( z 0 ) f'(z_{0}) f′(z0). More explicit: For any ε > 0 \varepsilon>0 ε>0 there exists δ = δ ( ε ) \delta=\delta (\varepsilon) δ=δ(ε) such that. If 0 < ∣ z − z 0 ∣ < δ ( ε ) 0<|z - z_{0}|<\delta (\varepsilon) 0<∣z−z0∣<δ(ε), then ∣ f ( z ) − f ( z 0 ) z − z 0 − f ′ ( z 0 ) ∣ < ε |\frac {f (z)-f (z_{0})}{z - z_{0}}-f'(z_{0})|<\varepsilon ∣z−z0f(z)−f(z0)−f′(z0)∣<ε. Let S ⊆ C S \subseteq C S⊆C subset, if a point α \alpha α (not necessarily contained in S S S) such that ∀ D ( α , − ) \forall D (\alpha, -) ∀D(α,−) centered at α \alpha α contained both points of S S S and points are not in S S S.

复数的绝对值满足以下性质。如果 α , β \alpha, \beta α,β 是复数，那么 ∣ α β ∣ = ∣ α ∣ × ∣ β ∣ |\alpha \beta|=|\alpha|\times|\beta| ∣αβ∣=∣α∣×∣β∣； ∣ α + β ∣ ≤ ∣ α ∣ + ∣ β ∣ |\alpha+\beta| \leq|\alpha|+|\beta| ∣α+β∣≤∣α∣+∣β∣（三角不等式）。设 θ , φ \theta, \varphi θ,φ 是复数，则 e i ( θ + φ ) = e i θ e i φ e^{i (\theta+\varphi)}=e^{i \theta} e^{i \varphi} ei(θ+φ)=eiθeiφ。设 α , β \alpha,\beta α,β 是复数，则 e α + β = e α e β e^{\alpha+\beta}=e^{\alpha} e^{\beta} eα+β=eαeβ。表达式 r e i θ r e^{i \theta} reiθ 称为复数 x + i y x + iy x+iy 的极坐标形式。 θ \theta θ 称为 z z z 的辐角，记作 θ = arg ⁡ z \theta=\arg z θ=argz。设 α ∈ S \alpha \in S α∈S，如果 lim ⁡ z → z 0 f ( z ) = f ( z 0 ) = w 0 \lim {z \to z{0}} f (z)=f (z_{0})=w_{0} limz→z0f(z)=f(z0)=w0，则称 f f f 在 z 0 z_{0} z0 处连续。更明确地说：对于任意 ε > 0 \varepsilon>0 ε>0，存在 δ = δ ( ε ) \delta=\delta (\varepsilon) δ=δ(ε)，使得如果 ∣ z − z 0 ∣ < δ ( ε ) |z - z_{0}|<\delta (\varepsilon) ∣z−z0∣<δ(ε)，那么 ∣ f ( z ) − f ( z 0 ) ∣ < ε |f (z)-f (z_{0})|< \varepsilon ∣f(z)−f(z0)∣<ε。如果 lim ⁡ z → z 0 f ( z ) − f ( z 0 ) z − z 0 \lim {z \to z{0}} \frac {f (z)-f (z_{0})}{z - z_{0}} limz→z0z−z0f(z)−f(z0) 存在，则称 f f f 在 z 0 z_{0} z0 处可微，我们将这个极限记为 f ′ ( z 0 ) f'(z_{0}) f′(z0)。更明确地说：对于任意 ε > 0 \varepsilon>0 ε>0，存在 δ = δ ( ε ) \delta=\delta (\varepsilon) δ=δ(ε)，使得如果 0 < ∣ z − z 0 ∣ < δ ( ε ) 0<|z - z_{0}|<\delta (\varepsilon) 0<∣z−z0∣<δ(ε)，那么 ∣ f ( z ) − f ( z 0 ) z − z 0 − f ′ ( z 0 ) ∣ < ε |\frac {f (z)-f (z_{0})}{z - z_{0}}-f'(z_{0})|<\varepsilon ∣z−z0f(z)−f(z0)−f′(z0)∣<ε。设 S ⊆ C S \subseteq C S⊆C 是一个子集，如果存在一点 α \alpha α（不一定包含在 S S S 中），使得以 α \alpha α 为中心的任意 D ( α , − ) D (\alpha, -) D(α,−) 既包含 S S S 中的点，也包含不在 S S S 中的点。

Let U ⊆ C U \subseteq C U⊆C subset, U U U is called open if for ∀ α ∈ U \forall \alpha \in U ∀α∈U, there exists r ∈ R + r \in \mathbb {R}^{+} r∈R+ and such that D ( α , r ) ⊆ U D (\alpha, r) \subseteq U D(α,r)⊆U. Let U ⊆ C U \subseteq C U⊆C subset is called closed subsets if Cl ( U ) \text {Cl}(U) Cl(U) is open in C C C. Or Let U ⊆ C U \subseteq C U⊆C subset is called closed if U U U contains all its boundary points. Let α ⊆ C \alpha \subseteq C α⊆C is called an interior point of S S S if ∃ D ( α , r ) \exists D (\alpha, r) ∃D(α,r) with r ∈ R + r \in \mathbb {R}^{+} r∈R+ and D ( α , r ) ⊆ S D (\alpha, r) \subseteq S D(α,r)⊆S. At last, in order to prove power of complex numbers, two questions solved by complex numbers techniques are given.

设 U ⊆ C U \subseteq C U⊆C 是一个子集，如果对于任意 α ∈ U \alpha \in U α∈U，存在 r ∈ R + r \in \mathbb {R}^{+} r∈R+，使得 D ( α , r ) ⊆ U D (\alpha, r) \subseteq U D(α,r)⊆U，则称 U U U 是开集。设 U ⊆ C U \subseteq C U⊆C 是一个子集，如果 Cl ( U ) \text {Cl}(U) Cl(U)（ U U U 的闭包）在 C C C 中是开集，则称 U U U 是闭集。或者说，如果 U U U 包含其所有边界点，则称 U U U 是闭集。设 α ⊆ C \alpha \subseteq C α⊆C，如果存在 r ∈ R + r \in \mathbb {R}^{+} r∈R+ 的 D ( α , r ) D (\alpha, r) D(α,r)，使得 D ( α , r ) ⊆ S D (\alpha, r) \subseteq S D(α,r)⊆S，则称 α \alpha α 是 S S S 的内点。最后，为了证明复数的作用，给出了两个用复数技术解决的问题。

Take z = 1 z = 1 z=1 and n ∈ Z + n \in \mathbb {Z}^{+} n∈Z+ fixed, there are exactly n n n different numbers such that w n = z = 1 w^{n}=z = 1 wn=z=1.

Prove: Let θ = 2 π n \theta=\frac {2 \pi}{n} θ=n2π, w 1 = 1 × e i 2 π n = e i θ w_{1}=1\times e^{i \frac {2 \pi}{n}}=e^{i \theta} w1=1×ein2π=eiθ, w 2 = 1 × e i 4 π n = e 2 i θ w_{2}=1\times e^{i \frac {4 \pi}{n}}=e^{2i \theta} w2=1×ein4π=e2iθ, ..., w n = 1 × e i 2 π = 1 w_{n}=1\times e^{i 2 \pi}=1 wn=1×ei2π=1 are all satisfy w i n = 1 w_{i}^{n}=1 win=1 for i = 1 , ⋯   , n i = 1,\cdots,n i=1,⋯,n.

Let n n n be a prime number. Then ( 1 + 2 cos ⁡ 2 π n ) ( 1 + 2 cos ⁡ 4 π n ) ( 1 + 2 cos ⁡ 6 π n ) ⋯ ( 1 + 2 cos ⁡ 2 k π n ) = 3 (1 + 2\cos\frac {2 \pi}{n})(1 + 2\cos\frac {4 \pi}{n})(1 + 2\cos\frac {6 \pi}{n})\cdots (1 + 2\cos\frac {2k \pi}{n}) = 3 (1+2cosn2π)(1+2cosn4π)(1+2cosn6π)⋯(1+2cosn2kπ)=3.

Let W = e 2 π i n W = e^{\frac {2 \pi i}{n}} W=en2πi, that is W n = 1 W^{n}=1 Wn=1, W − n 2 = e − π i = − 1 W^{-\frac {n}{2}}=e^{-\pi i}=-1 W−2n=e−πi=−1, 2 cos ⁡ 2 k π n = W k + W − k 2\cos\frac {2k \pi}{n}=W^{k}+W^{-k} 2cosn2kπ=Wk+W−k.

∏ k = 1 n ( 1 + 2 cos ⁡ 2 k π n ) = ∏ k = 1 n ( 1 + W k + W − k ) = ∏ k = 1 n W − k ( W 2 k + W k + 1 ) = W − n ( n + 1 ) 2 × 3 ∏ k = 1 n − 1 1 − W 3 k 1 − W k = ( − 1 ) n + 1 × 3 ∏ k = 1 n − 1 1 − W 3 k 1 − W k \prod_{k = 1}^{n}(1 + 2\cos\frac {2k \pi}{n})=\prod_{k = 1}^{n}(1 + W^{k}+W^{-k})=\prod_{k = 1}^{n} W^{-k}(W^{2k}+W^{k}+1)=W^{-\frac {n (n + 1)}{2}}\times3\prod_{k = 1}^{n - 1}\frac {1 - W^{3k}}{1 - W^{k}}=(-1)^{n + 1}\times3\prod_{k = 1}^{n - 1}\frac {1 - W^{3k}}{1 - W^{k}} ∏k=1n(1+2cosn2kπ)=∏k=1n(1+Wk+W−k)=∏k=1nW−k(W2k+Wk+1)=W−2n(n+1)×3∏k=1n−11−Wk1−W3k=(−1)n+1×3∏k=1n−11−Wk1−W3k.

Take n = 5 n = 5 n=5 as an example to illustrate an elegant identity. ( − 1 ) n + 1 × 3 ∏ k = 1 n − 1 1 − W 3 k 1 − W k = ( − 1 ) 6 × 3 ∏ k = 1 4 1 − W 3 k 1 − W k (-1)^{n + 1}\times3\prod_{k = 1}^{n - 1}\frac {1 - W^{3k}}{1 - W^{k}}=(-1)^{6}\times3\prod_{k = 1}^{4}\frac {1 - W^{3k}}{1 - W^{k}} (−1)n+1×3∏k=1n−11−Wk1−W3k=(−1)6×3∏k=141−Wk1−W3k.

取 z = 1 z = 1 z=1 且固定 n ∈ Z + n\in\mathbb {Z}^{+} n∈Z+（正整数集），恰好有 n n n 个不同的数 w w w 满足 w n = z = 1 w^{n}=z = 1 wn=z=1 。

证明：令 θ = 2 π n \theta=\frac {2\pi}{n} θ=n2π， w 1 = 1 × e i 2 π n = e i θ w_1 = 1\times e^{i\frac {2\pi}{n}} = e^{i\theta} w1=1×ein2π=eiθ， w 2 = 1 × e i 4 π n = e 2 i θ w_2 = 1\times e^{i\frac {4\pi}{n}} = e^{2i\theta} w2=1×ein4π=e2iθ，......， w n = 1 × e i 2 π = 1 w_n = 1\times e^{i2\pi}=1 wn=1×ei2π=1，对于 i = 1 , ⋯   , n i = 1,\cdots,n i=1,⋯,n，都有 w i n = 1 w_{i}^{n}=1 win=1 。

设 n n n 为质数，那么 ( 1 + 2 cos ⁡ 2 π n ) ( 1 + 2 cos ⁡ 4 π n ) ( 1 + 2 cos ⁡ 6 π n ) ⋯ ( 1 + 2 cos ⁡ 2 k π n ) = 3 (1 + 2\cos\frac {2\pi}{n})(1 + 2\cos\frac {4\pi}{n})(1 + 2\cos\frac {6\pi}{n})\cdots (1 + 2\cos\frac {2k\pi}{n}) = 3 (1+2cosn2π)(1+2cosn4π)(1+2cosn6π)⋯(1+2cosn2kπ)=3。

令 W = e 2 π i n W = e^{\frac {2\pi i}{n}} W=en2πi，即 W n = 1 W^{n}=1 Wn=1， W − n 2 = e − π i = − 1 W^{-\frac {n}{2}} = e^{-\pi i}=-1 W−2n=e−πi=−1， 2 cos ⁡ 2 k π n = W k + W − k 2\cos\frac {2k\pi}{n}=W^{k}+W^{-k} 2cosn2kπ=Wk+W−k。

∏ k = 1 n ( 1 + 2 cos ⁡ 2 k π n ) = ∏ k = 1 n ( 1 + W k + W − k ) = ∏ k = 1 n W − k ( W 2 k + W k + 1 ) = W − n ( n + 1 ) 2 × 3 ∏ k = 1 n − 1 1 − W 3 k 1 − W k = ( − 1 ) n + 1 × 3 ∏ k = 1 n − 1 1 − W 3 k 1 − W k \prod_{k = 1}^{n}(1 + 2\cos\frac {2k\pi}{n})=\prod_{k = 1}^{n}(1 + W^{k}+W^{-k})=\prod_{k = 1}^{n} W^{-k}(W^{2k}+W^{k}+1)=W^{-\frac {n (n + 1)}{2}}\times3\prod_{k = 1}^{n - 1}\frac {1 - W^{3k}}{1 - W^{k}}=(-1)^{n + 1}\times3\prod_{k = 1}^{n - 1}\frac {1 - W^{3k}}{1 - W^{k}} ∏k=1n(1+2cosn2kπ)=∏k=1n(1+Wk+W−k)=∏k=1nW−k(W2k+Wk+1)=W−2n(n+1)×3∏k=1n−11−Wk1−W3k=(−1)n+1×3∏k=1n−11−Wk1−W3k。

以 n = 5 n = 5 n=5 为例来阐释一个优美的恒等式。 ( − 1 ) n + 1 × 3 ∏ k = 1 n − 1 1 − W 3 k 1 − W k = ( − 1 ) 6 × 3 ∏ k = 1 4 1 − W 3 k 1 − W k (-1)^{n + 1}\times3\prod_{k = 1}^{n - 1}\frac {1 - W^{3k}}{1 - W^{k}}=(-1)^{6}\times3\prod_{k = 1}^{4}\frac {1 - W^{3k}}{1 - W^{k}} (−1)n+1×3∏k=1n−11−Wk1−W3k=(−1)6×3∏k=141−Wk1−W3k。

3. Summary

总结

Complex analysis is a subfield of mathematical analysis that focuses on the operations of complex numbers. The theory of functions of a complex variable is another name for it. In particular, algebraic geometry, fluid dynamics, quantum mechanics, and other related sciences have made substantial use of complex analysis in mathematics, physics, and engineering over the years. Girolamo Cardano and Raphael Bombelli, two Italian mathematicians, initially recognized complex numbers while attempting to solve a particular algebra in the 16th century, while Cauchy and Riemann further improved it in the 19th century. Complex numbers have a long history of development. After thousands of years of growth, mathematicians have made enormous advancements in the field. In addition, a considerable lot of previously undiscovered mathematical knowledge was discovered during this time, and previously unprovable formulas and phenomena were proved. Additionally, some of the mathematics associated with complex numbers and their use are covered. This paper presents a comprehensive account of the discovery of complex numbers, including everything from the cubic equation to their topology.

复分析是数学分析的一个子领域，主要研究复数的运算。它也被称为复变函数论。多年来，复分析在数学、物理学和工程学领域有着广泛应用，尤其在代数几何、流体动力学、量子力学等相关学科中发挥了重要作用。16 世纪，意大利数学家吉罗拉莫・卡尔达诺和拉斐尔・邦贝利在求解特定代数问题时首次发现了复数，19 世纪柯西和黎曼进一步推动了复数理论的发展。复数的发展历程源远流长，经过数千年的演进，数学家们在该领域取得了巨大的进步。在此期间，大量此前未知的数学知识被发掘出来，许多曾经无法证明的公式和现象也得到了论证。本文还涵盖了一些与复数相关的数学知识及其应用。全面阐述了复数的发现过程，包括从三次方程到复数拓扑的各个方面。

References

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