复数 | 存在意义 / 实际应用（篇 1 ）

注：本文为 “复数” 相关讨论合辑。
英文引文，机翻未校。
如有内容异常，请看原文。

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Do complex numbers really exist?

复数真的存在吗？

Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What’s the best way to explain to a non-mathematician that complex numbers are necessary and meaningful, in the same way that real numbers are?
复数涉及负一的平方根，大多数非数学专业人士难以接受这样的数具有实际意义。相比之下，他们认为实数具有明确且直观的意义。向非数学专业人士解释复数与实数同样必要且有意义的最佳方式是什么？

This is not a Platonic question about the reality of mathematics, or whether abstractions are as real as physical entities, but an attempt to bridge a comprehension gap that many people experience when encountering complex numbers for the first time. The wording, although provocative, is deliberately designed to match the way that many people actually ask this question.
这并非关于数学实在性的柏拉图式问题，也不是探讨抽象概念是否与物理实体同样真实的问题，而是试图填补许多人首次接触复数时面临的理解鸿沟。尽管表述带有挑衅性，但刻意采用这种方式是为了贴合许多人实际提出该问题的角度。

– goblin GONE
Commented Nov 20, 2013 at 4:44

The question isn’t whether or not complex numbers exist, but whether or not there exists a complex number system (intuitively, a number system isomorphic to $\mathbb{C}$). According to ZFC, it does. This is unsurprising, given the usual geometric interpretation of $\mathbb{C}$ as a complex plane. For a treatment of these kinds of issues (but not $\mathbb{C}$ in particular), check out Goldrei’s exceptional book.
问题不在于复数是否存在，而在于是否存在一个“复数系统”（直观上，即与 $\mathbb{C}$ 同构的数系）。根据策梅洛 - 弗兰克尔集合论（ZFC），这样的系统是存在的。考虑到 $\mathbb{C}$ 通常被几何解释为复平面，这一点并不令人意外。若要了解这类问题（而非特指 $\mathbb{C}$）的探讨，可参考戈德雷（Goldrei）的佳作。

– Shai Deshe
Commented Mar 31, 2015 at 14:50

I don’t know if this was mentioned before, but one way I like to tackle this is to explain that irrational numbers aren’t any more “real” than imaginary numbers. They are simply abstract answers to abstract questions.
我不知道之前是否有人提到过，但我喜欢用一种方式来解释：无理数并不比虚数更“真实”。它们都只是抽象问题的抽象答案。

– user123641
Commented Jan 30, 2018 at 22:01

Richard Dedekind asked a question which is relevant (although he asked it in a different context): “Was sind und sollen die Zahlen?” which translates as “What are the numbers… and just what SHOULD they be?” (English needs the extra length to encapsulate the same meaning.) Essentially, Dedekind asserts in his question that humans are free to define just what numbers “are”. What we decide numbers to be is usually driven by pragmatic needs and ends. Of course, once one realizes that complex numbers encapsulate rotation, applications abound and it becomes useful for $i\times i$ to be a “number”.
理查德·戴德金（Richard Dedekind）曾提出一个相关问题（尽管背景不同）：“Was sind und sollen die Zahlen?”，翻译成中文是“数是什么，又应当是什么？”（英文需要更长的表述才能涵盖相同含义）。本质上，戴德金在这个问题中主张，人类可以自由定义“数”是什么。我们对“数”的定义通常由实际需求和目标驱动。当然，一旦人们意识到复数可用来描述旋转，其应用便无处不在，此时 $i\times i$ 成为一个“数”也就具有了实际意义。

Answers

There are a few good answers to this question, depending on the audience. I’ve used all of these on occasion.
针对不同受众，这个问题有几种不错的解答方式，我在不同场合都用过。

A way to solve polynomials

解决多项式的工具

We came up with equations like $x-5=0$, what is $x$?, and the naturals solved them (easily). Then we asked, “wait, what about $x+5=0$?” So we invented negative numbers. Then we asked “wait, what about $2x=1$?” So we invented rational numbers. Then we asked “wait, what about $x^2=2$?” so we invented irrational numbers.
我们遇到过诸如 $x-5=0$ 这样的方程，问 $x$ 是多少？自然数可以轻松解决这类问题。接着我们会问：“那 $x+5=0$ 该怎么解？”于是我们发明了负数。然后又问：“那 $2x=1$ 呢？”于是发明了有理数。之后再问：“那 $x^2=2$ 呢？”于是又发明了无理数。

Finally, we asked, “wait, what about $x^2=-1$?” This is the only question that was left, so we decided to invent the “imaginary” numbers to solve it. All the other numbers, at some point, didn’t exist and didn’t seem “real”, but now they’re fine. Now that we have imaginary numbers, we can solve every polynomial, so it makes sense that that’s the last place to stop.
最后，我们问：“那 $x^2=-1$ 该怎么解？”这是仅剩的问题，于是我们决定发明“虚数”来解决它。其他所有数在某个阶段都曾“不存在”，且看似不“真实”，但如今已被广泛接受。有了虚数后，我们能解出所有多项式，因此将复数作为数系扩展的终点是合理的。

Pairs of numbers

数对视角

This explanation goes the route of redefinition. Tell the listener to forget everything he or she knows about imaginary numbers. You’re defining a new number system, only now there are always pairs of numbers. Why? For fun. Then go through explaining how addition/multiplication work. Try and find a good “realistic” use of pairs of numbers (many exist).
这种解释采用重新定义的思路。让听众暂时忘掉所有关于虚数的已有认知，我们要定义一个新的数系——只不过现在的数都是数对形式。为什么要这么做？纯粹为了探索。接着向听众解释这个数系中加法和乘法的运算规则，再找一个数对的“实际”应用场景（这样的场景有很多）。

Then, show that in this system, $(0,1)\ast (0,1)=(-1,0)$, in other words, we’ve defined a new system, under which it makes sense to say that $\sqrt{-1}=i$, when $i=(0,1)$. And that’s really all there is to imaginary numbers: a definition of a new number system, which makes sense to use in most places. And under that system, there is an answer to $\sqrt{-1}$.
然后向他们展示，在这个数系中，$(0,1)\ast (0,1)=(-1,0)$。也就是说，我们定义的这个新系统中，当 $i=(0,1)$ 时，$\sqrt{-1}=i$ 是成立的。这就是虚数的本质：它只是一个新数系的定义，且在大多数场景下使用都是合理的。在这个数系里，$\sqrt{-1}$ 确实有解。

The historical explanation

历史视角解释

Explain the history of the imaginary numbers. Showing that mathematicians also fought against them for a long time helps people understand the mathematical process, i.e., that it’s all definitions in the end.
向听众讲述虚数的历史。告诉他们，数学家们也曾长期抗拒虚数，这能帮助人们理解数学的发展过程——归根结底，数学概念都是人为定义的。

I’m a little rusty, but I think there were certain equations that kept having parts of them which used $\sqrt{-1}$, and the mathematicians kept throwing out the equations since there is no such thing.
我对细节记得不太清楚了，但印象中，过去有些方程总会出现含 $\sqrt{-1}$ 的项，当时的数学家们因为觉得不存在这样的数，就直接舍弃了这些方程。

Then, one mathematician decided to just “roll with it”, and kept working, and found out that all those square roots cancelled each other out.
后来有一位数学家决定“顺着这个思路走下去”，继续深入研究，结果发现所有含根号的项最终都抵消了。

Amazingly, the answer that was left was the correct answer (he was working on finding roots of polynomials, I think). Which lead him to think that there was a valid reason to use $\sqrt{-1}$, even if it took a long time to understand it.
令人惊讶的是，最后得到的结果恰好是“正确答案”（我记得他当时在研究多项式求根问题）。这让他意识到，即便需要很长时间去理解，使用 $\sqrt{-1}$ 也是有合理依据的。

– Noah Snyder
Commented Jul 20, 2010 at 22:57

I believe the historical context was in solving cubic equations. If you try to solve a cubic equation that has a real solution, sometimes you have to use complex numbers in intermediate steps in the calculation. But then when you get to the end you find a real number that really is a solution to a real cubic equation!
我认为这一历史背景与三次方程求解有关。当你试图解一个有实根的三次方程时，有时在计算的中间步骤必须用到复数，但最终得到的结果却是实数——而且这个实数确实是该三次方程的解！

– Edan Maor
Commented Jul 20, 2010 at 23:08

@Noah - Yep that sounds exactly like what I was thinking about.
@诺亚——对，这和我记忆中的内容完全一致。

– Neil Mayhew
Commented Jul 20, 2010 at 23:08

It was Gerolamo Cardano in 1545. For an enjoyable historical treatment see “An Imaginary Tale” by Paul J. Nahin.
这是杰罗拉莫·卡尔达诺（Gerolamo Cardano）在 1545 年提出的。若想了解有趣的历史细节，可参阅保罗·J·纳欣所著的《虚数的故事》。

– Giorgi
Commented Nov 29, 2010 at 18:52

@Edan - The nice thing about “Pairs of numbers” explanation is that it is extension of real numbers (all real numbers have the second number zero) so real numbers are subset of the pairs. All operations that you define on the pairs result in the same result for real numbers as if you are just manipulating a real number.
@伊丹——“数对视角”解释的优点在于，它是实数的扩展（所有实数都可表示为第二个数为 0 的数对），因此实数是数对的子集。你在数对上定义的所有运算，应用到实数（数对形式）上的结果，与直接对实数进行运算的结果完全一致。

– lafrasu
Commented Apr 19, 2011 at 12:38

The rationals, $\mathbb{Q}$, can also be thought of as constructed from pairs of numbers.
有理数 $\mathbb{Q}$ 其实也可以看作由数对构造而来。

The concept of mathematical numbers and “existing” is a tricky one. What actually “exists”?
数学中的“数”与“存在”的概念本身就很复杂。究竟什么东西才算“存在”？

Do negative numbers exist? Of course they do not. You can’t have a negative number of apples.
负数存在吗？当然不存在——你不可能拥有“负几个苹果”。

Yet, the beauty of negative numbers is that when we define them (rigorously), then all of a sudden we can use them to solve problems we were never ever able to solve before, or we can solve them in a much simpler way.
但负数的妙处在于，一旦我们（严格地）定义了它，突然之间就能用它解决以前无法解决的问题，或者用更简单的方法解决原有问题。

Imagine trying to do simple physics without the idea of negative numbers!
试想一下，如果没有负数的概念，简单的物理学问题该如何解决！

But are they “real”? Do they “exist”? No, they don’t. But they are just tools that help us solve real life problems.
但负数“真实”吗？它们“存在”吗？答案是否定的。但它们是帮助我们解决现实问题的工具。

To go back to your question about complex numbers, I would say that the idea that they exist or not has no bearing on whether they are actually useful in solving the problems of every day life, or making them many, many, many times more easy to solve.
回到你关于复数的问题：我认为复数是否“存在”，与它们能否用于解决日常生活中的问题、能否让问题变得简单得多，没有任何关系。

The math that makes your computer run involves the tool that is complex numbers, for instance.
例如，让你的电脑正常运行的数学原理中，就用到了复数这一工具。

– Andrew Grimm
Commented Nov 3, 2010 at 22:52

Negative numbers is an abstraction most people are familiar with - for example, a negative budget balance.
负数是大多数人都熟悉的抽象概念——比如预算赤字（负的预算余额）。

– David
Commented Nov 12, 2010 at 18:17
Negative numbers is the concept of owing. Having -5 apples can be interpreted as owing 5 apples.
负数代表“欠”的概念。拥有 -5 个苹果，可以理解为欠别人 5 个苹果。

– André Caron
Commented Dec 8, 2010 at 16:45

@David: there are other, more concrete instances of negative numbers that don’t involve other abstractions such as debt. Negative acceleration (a.k.a. slowing down) is one very concrete negative number. Any description that involves decreasing also inherently involves a negative number.
@大卫：除了债务这类抽象概念，负数还有更具体的实例。负加速度（即减速）就是一个非常具体的负数案例。任何描述“减少”的场景，本质上都涉及负数。

– Justin L.
Commented Dec 8, 2010 at 22:52

It’s really not as intuitive as you would like to believe, being raised your whole life with the concept of anti-numbers. Many prominent mathematicians had problems accepting the validity of negative numbers even up until the 18th century.
你从小到大接触“相反数”概念，可能觉得负数很直观，但事实并非如此。直到 18 世纪，许多著名数学家仍难以接受负数的合理性。

– JP19
Commented Dec 22, 2010 at 7:31

My father is a maths teacher, and I have seen that some kids found negative numbers a bit hard to grasp in the beginning.
我父亲是数学老师，我见过有些孩子一开始很难理解负数。

One need only consult the history of algebra to find many informal discussions on the existence (and consistence) of complex numbers. Any informal attempt to justify the existence of $\mathbb{C}$ will face the same obstacles that existed in earlier times. Namely, the lack of any rigorous (set-theoretic) foundation makes it difficult to be precise - both syntactically and semantically. Nowadays the set-theoretic foundation of algebraic structures is so subconscious that is is easy to overlook just how much power it provides for such purposes. But this oversight is easily remedied. One need only consult some of the older literature where even leading mathematicians struggled immensely to rigorously define complex numbers. For example, see the quote below by Cauchy and Hankel’s scathing critique - which may well make your jaw drop! (Below is an excerpt from my post on the notion of formal polynomial rings and their quotients).
只要查阅代数学史，就能发现许多关于复数存在性（及一致性）的非形式化讨论。任何试图非形式化证明 $\mathbb{C}$ 存在的尝试，都会面临过去同样的障碍：由于缺乏严格的（集合论）基础，无论是语法层面还是语义层面，都难以做到精确。如今，代数结构的集合论基础已深入人心，人们很容易忽视它在这类问题上的作用。但这种忽视很容易纠正——只需查阅一些早期文献，就能看到即使是顶尖数学家，也曾为严格定义复数而大费周折。例如，下面柯西（Cauchy）的引文以及汉克尔（Hankel）尖锐的批评，可能会让你大吃一惊！（以下内容摘自我关于形式多项式环及其商环概念的帖子）

A major accomplishment of the set-theoretical definition of algebraic structures was to eliminate imprecise syntax and semantics. Eliminating the syntactic polynomial term $a+b\cdot x+c\cdot x^2$ and replacing it by its rigorous set-theoretic semantic reduction $(a,b,c,0,0,\dots )$ eliminates many ambiguities. There can no longer be any doubt about the precise denotation of the symbols $x,+,\cdot$ or about the meaning of equality of polynomials, since, by set theoretic definition, tuples are equal iff their components are equal. The set-theoretic representation (“implementation”) of these algebraic objects gives them rigorous meaning, reducing their semantics to that of set-theory.
代数结构的集合论定义有一个重大贡献：消除了不精确的语法和语义。去掉多项式的语法项 $a+b\cdot x+c\cdot x^2$，用严格的集合论语义归约形式 $(a,b,c,0,0,\dots )$ 替代，可消除诸多歧义。此时，符号 $x,+,\cdot$ 的精确含义，以及多项式相等的定义，都不再存在疑问——根据集合论定义，元组相等当且仅当它们的分量相等。这些代数对象的集合论表示（“实现”）为它们赋予了严格的意义，将其语义归约为集合论的语义。

Similarly for complex numbers $a+b\cdot i$ and their set-theoretic representation by Hamilton as pairs of reals $(a,b)$. Before Hamilton gave this semantic reduction of $\mathbb{C}$ to ${\mathbb{R}}^2$, prior syntactic constructions (e.g. by Cauchy) as formal expressions or terms $a+b\cdot i$ were subject to heavy criticism regarding the precise denotation of their constituent symbols, e.g. precisely what is the meaning of the symbols $i,+,=$? In modern language, Cauchy’s construction of $\mathbb{C}$ is simply the quotient ring $\mathbb{R}[x]/(x^2+1)\cong \mathbb{R}[i]$, which he described essentially as real polynomial expressions modulo $x^2+1$. However, in Cauchy’s time mathematics lacked the necessary (e.g. set-theoretical) foundations to rigorously define the syntactic expressions comprising the polynomial ring term-algebra $\mathbb{R}[x]$ and its quotient ring $\mathrm{mod}{x}^2+1$. The best that Cauchy could do was to attempt to describe the constructions in terms of imprecise natural (human) language, e.g, in 1821 Cauchy wrote:
复数 $a+b\cdot i$ 的情况也是如此——汉密尔顿（Hamilton）将其集合论表示定义为实数对 $(a,b)$。在汉密尔顿将 $\mathbb{C}$ 语义归约为 ${\mathbb{R}}^2$ 之前，柯西等人将复数构造为形式表达式 $a+b\cdot i$ 的做法，因构成符号（如 $i,+,=$）的精确含义不明确而备受批评。用现代语言来说，柯西对 $\mathbb{C}$ 的构造本质上是商环 $\mathbb{R}[x]/(x^2+1)\cong \mathbb{R}[i]$，即实数多项式模 $x^2+1$ 的剩余类。但在柯西时代，数学缺乏必要的（如集合论）基础，无法严格定义多项式环项代数 $\mathbb{R}[x]$ 及其模 $x^2+1$ 商环的语法表达式。柯西所能做的，只是用不精确的自然语言来描述这些构造，例如他在 1821 年写道：

In analysis, we call a symbolic expression any combination of symbols or algebraic signs which means nothing by itself but which one attributes a value different from the one it should naturally be […] Similarly, we call symbolic equations those that, taken literally and interpreted according to conventions generally established, are inaccurate or have no meaning, but from which can be deduced accurate results, by changing and altering, according to fixed rules, the equations or symbols within […] Among the symbolic expressions and equations whose theory is of considerable importance in analysis, one distinguishes especially those that have been called imaginary. – Cauchy, Cours d’analyse,1821, S.7.1
在分析学中，我们将“符号表达式”定义为：由符号或代数记号组成的组合，其本身无意义，但我们会赋予它一个与其“自然含义”不同的价值……类似地，“符号方程”指的是：按字面意思和通用惯例解释时，要么不准确、要么无意义，但通过按固定规则变换方程或符号，可从中推导出精确结果的方程……在分析学中，理论意义重大的符号表达式和方程里，有一类被特别称为“虚数”相关的表达式和方程。——柯西，《分析教程》，1821 年，第 7.1 节

While nowadays, using set theory, we can rigorously interpret such “symbolic expressions” as terms of formal languages or term algebras, it was far too imprecise in Cauchy’s time to have any hope of making sense to his colleagues, e.g. Hankel replied scathingly:
如今，借助集合论，我们能将这类“符号表达式”严格解释为形式语言的项或项代数，但在柯西时代，这种表述极其不精确，他的同行根本无法理解。例如，汉克尔就尖锐地批评道：

If one were to give a critique of this reasoning, we can not actually see where to start. There must be something “which means nothing,” or “which is assigned a different value than it should naturally be” something that has “no sense” or is “incorrect”, coupled with another similar kind, producing something real. There must be “algebraic signs” - are these signs for quantities or what? as a sign must designate something combined with each other in a way that has “a meaning.” I do not think I’m exaggerating in calling this an unintelligible play on words, ill-becoming of mathematics, which is proud and rightly proud of the clarity and evidence of its concepts. – Hankel
若要批评这种推理，我甚至不知从何下手。必须存在某种“无意义的东西”，或“被赋予非自然价值的东西”，某种“无意义”或“错误”的东西，与另一种类似的东西结合，却能产生真实的结果。还必须存在“代数符号”——这些符号代表量，还是别的什么？符号必须指代某种东西，且组合方式需“有意义”。我认为称其为“难以理解的文字游戏”并不夸张，这与数学格格不入——数学向来以其概念的清晰性和明确性为荣，且这种自豪是理所应当的。——汉克尔

Thus it comes as no surprise that Hamilton’s elimination of such “meaningless” symbols - in favor of pairs of reals - served as a major step forward in placing complex numbers on a foundation more amenable to his contemporaries. Although there was not yet any theory of sets in which to rigorously axiomatize the notion of pairs, they were far easier to accept naively - esp. given the already known closely associated geometric interpretation of complex numbers. Hamilton introduced pairs as ‘couples’ in 1837:
因此，汉克尔用实数对替代这类“无意义”符号，为复数建立起更易被同时代人接受的基础，无疑是一大进步。尽管当时尚无集合论来严格公理化“数对”概念，但人们很容易从朴素层面接受它——尤其是考虑到复数早已存在相关的几何解释。1837 年，汉克尔将数对称为“couples”（偶），并提出：

p. 6: The author acknowledges with pleasure that he agrees with M. Cauchy, in considering every (so-called) Imaginary Equation as a symbolic representation of two separate Real Equations: but he differs from that excellent mathematician in his method generally, and especially in not introducing the sign $\sqrt{-1}$ until he has provided for it, by his Theory of Couples, a possible and real meaning, as a symbol of the couple $(0,1)$
第 6 页：笔者欣然认同柯西先生的观点，即所有（所谓的）虚方程都是两个独立实方程的符号表示；但笔者的方法与这位杰出数学家的方法存在整体差异，尤其是在引入 $\sqrt{-1}$ 符号前，笔者已通过“偶理论”为其赋予了可能且真实的意义——将其定义为偶 $(0,1)$ 的符号。

p. 111: But because Mr. Graves employed, in his reasoning, the usual principles respecting about Imaginary Quantities, and was content to prove the symbolical necessity without showing the interpretation, or inner meaning, of his formulae, the present Theory of Couples is published to make manifest that hidden meaning: and to show, by this remarkable instance, that expressions which seem according to common views to be merely symbolical, and quite incapable of being interpreted, may pass into the world of thoughts, and acquire reality and significance, if Algebra be viewed as not a mere Art or Language, but as the Science of Pure Time. – Hamilton, 1837
第 111 页：格雷夫斯（Graves）先生在推理中运用了关于虚数的常用原理，仅证明了符号的必要性，却未解释其公式的含义或内在意义。因此，笔者发表此“偶理论”，旨在揭示这一隐藏的意义；并通过这一显著案例表明：若将代数视为“纯粹时间的科学”，而非单纯的技艺或语言，那么在常人看来仅为符号、无法解释的表达式，也能进入思想领域，获得真实性与意义。——汉克尔，1837 年

Not until the much later development of set-theory was it explicitly realized that ordered pairs and, more generally, n-tuples, serve a fundamental foundational role, providing the raw materials necessary to construct composite (sum/product) structures - the raw materials required for the above constructions of polynomial rings and their quotients. Indeed, as Akihiro Kanamori wrote on p. 289 (17) of his very interesting paper on the history of set theory:
直到集合论后期发展，人们才明确认识到：有序对乃至更一般的 n 元组，在基础数学中扮演着核心角色——它们为构造复合（和/积）结构提供了必要原料，也是上述多项式环及其商环构造所需的基础。正如金丸修（Akihiro Kanamori）在其关于集合论史的精彩论文第 289 页（17）中所写：

In 1897 Peano explicitly formulated the ordered pair using $(x,y)$ and moreover raised the two main points about the ordered pair: First, equation 18 of his Definitions stated the instrumental property which is all that is required of the ordered pair:
1897 年，皮亚诺（Peano）明确用 $(x,y)$ 表示有序对，并提出了有序对的两个核心性质：首先，其《定义》中的第 18 条方程指出了有序对的关键工具性性质——这也是有序对所需满足的全部性质：

$(x,y)=(a,b)⟺x=a$ and $y=b$

Second, he broached the possibility of reducibility, writing: “The idea of a pair is fundamental, i.e., we do not know how to express it using the preceding symbols.”
其次，他提出了可归约性的问题，写道：“有序对的概念是基础的，即我们无法用此前的符号来表示它。”

Once set-theory was fully developed one had the raw materials (syntax and semantics) to provide rigorous constructions of algebraic structures and precise languages for term algebras. The polynomial ring $\mathbb{R}[x]$ is nowadays just a special case of much more general constructions of free algebras. Such equationally axiomatized algebras and their genesis via so-called ‘universal mapping properties’ are topics discussed at length in any course on Universal Algebra - e.g. see Bergman for a particularly lucid presentation.
集合论完全发展后，人们便拥有了（语法和语义层面的）原料，可对代数结构进行严格构造，并为项代数提供精确语言。如今，多项式环 $\mathbb{R}[x]$ 只是更一般的“自由代数”构造的特例。这类等式公理化代数及其通过“泛映射性质”产生的过程，是泛代数课程中详细讨论的主题——例如，伯格曼（Bergman）的著作对此有特别清晰的阐述。

• [1] William Rowan Hamilton. Theory of conjugate functions, or algebraic couples; with a preliminary and elementary essay on algebra as the science of pure time
  Trans. Royal Irish Academy, v.17, part 1 (1837), pp. 293-422.)
  https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/PureTime/PureTime.pdf

• [2] Akihiro Kanamori. The Empty Set, the Singleton, and the Ordered Pair
  The Bulletin of Symbolic Logic, Vol. 9, No. 3. (Sep., 2003), pp. 273-298.
  http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.95.9839
  PS https://www.math.ucla.edu/~asl/bsl/0903/0903-001.ps
  PDF https://ifile.it/b20c48j

• [3] George M. Bergman. An Invitation to General Algebra and Universal Constructions.
  PS https://math.berkeley.edu/~gbergman/245/
  PDF https://ifile.it/yquj5w1

– SamB
Commented Nov 27, 2010 at 20:36

Did you miss the part where he said “to a non-mathematician”?
你是不是没看到他说“面向非数学专业人士”这部分？

– Bill Dubuque
Commented Dec 1, 2010 at 17:04

@SamB: The above is meant to provide to the OP some historical, foundational and pedagogical advice on the difficulty of such an endeavor. Hopefully the OP will find it helpful in devising his explanations for non-mathematicians.
@萨姆B：以上内容旨在为提问者（OP）提供关于“向非数学专业人士解释复数”这一难题的历史、基础及教学建议。希望提问者能借此设计出更适合非数学专业人士的解释。

– SamB
Commented Dec 1, 2010 at 19:28

ah, I suppose it might ;
啊，我想或许确实有帮助；

No number does “really exist” the way trees or atoms exist. In physics people however have found use for complex numbers just as they have found use for real numbers.
没有任何数会像树木或原子那样“真实存在”。但在物理学中，人们发现复数和实数一样有用。

– Mauro
Commented Nov 22, 2010 at 15:20

I think this is a non-answer. Clearly the mathematical concept of numbers is not “real” and in the past concepts, as zero, that now we consider clear were misterious. Still people tend to understand and consider “real” positive, negative and fractional numbers because they see a link with reality. In this sense they “exist” and that’s why basic algebra is normally understood by a vaste majority.
我认为这等于没回答。显然，数学中的“数”概念并非“真实存在”，过去像“零”这样如今清晰的概念，也曾令人困惑。但人们仍倾向于认为正、负、分数是“真实”的，因为能看到它们与现实的联系。从这个意义上说，它们“存在”，这也是大多数人能理解基础代数的原因。

– Christian
Commented Nov 29, 2010 at 12:41

@Mauro: There a long philosophical debate about whether numbers can exist in some Platonian heaven of ideas. Things that physics people use in their formulas have per definition a link to reality.
@毛罗：关于数是否存在于柏拉图式的“理念世界”中，存在长期的哲学争论。但物理学家在公式中使用的概念，本质上都与现实存在联系。

– Jason Orendorff
Commented Dec 2, 2010 at 22:10

@Christian: I wonder what you mean by “the way trees or atoms exist”. What way is that, exactly? Any decent definition of “exist” has to at least admit that things like time, sounds, shadows, communication, and the color red exist, don’t you think? It’s hard to imagine any such definition would rule out the number six.
@克里斯蒂安：你说“像树木或原子那样存在”，具体是指什么方式？任何合理的“存在”定义，至少都得承认时间、声音、影子、交流、红色这些事物存在，不是吗？很难想象有哪种定义会把数字 6 排除在外。

– isomorphismes
Commented Dec 22, 2010 at 7:16

What about Quine’s insight to define a natural number $n\in \mathbb{N}$ as the equivalence class of all sets with cardinality $n$?
蒯因（Quine）提出将自然数 $n\in \mathbb{N}$ 定义为“所有基数为 $n$ 的集合构成的等价类”，这个观点怎么看？

– Christian
Commented Feb 19, 2011 at 14:16

@Jason: You can run very straightforward tests to see whether a particular tree exists. Thinking of a test that tests whether time exists is qualitatively different.
@贾森：要判断某棵树是否存在，只需做简单的检验。但要设计检验“时间是否存在”的方法，性质就完全不同了。

In my opinion, the most natural way to view complex number is as a class of maps from the plane to itself. Specifically, lets define $(R,\theta )$ to be the map which multiplies every point in the plane by the number $R$, and then rotates it by the angle $\theta$. We may call these maps “dilations with rotations.”
在我看来，理解复数最自然的方式是将其视为“从平面到自身的一类映射”。具体来说，定义 $(R,\theta )$ 为这样一种映射：先将平面上的每个点乘以 $R$（伸缩），再将其旋转 $\theta$ 角。我们可将这类映射称为“旋缩变换”。

Such maps can be added and composed (multiplied) in the obvious way, and its not hard to work out that the sum and product of two such mappings is another dilation with rotation.
这类映射可按直观方式进行加法和复合（乘法）运算，且不难证明：两个旋缩变换的和与积，仍是一个旋缩变换。

We can also identify the real number $x$ with the map $(x,0)$, i.e. the map which multiplies every point in the plane by $x$. Then we see that these maps have the magical property that $-1$ has a square root! Namely, if $P$ is the mapping $(1,\pi /2)$ (i.e. rotate every point by angle $\pi /2$), then applying $P$ twice is the same as multiplying every number by $-1$, i.e. $P^2=-1$!
我们还可将实数 $x$ 与映射 $(x,0)$ 等同——即“将平面上所有点乘以 $x$”的映射。此时会发现，这类映射有个奇妙性质：$-1$ 存在平方根！即若 $P$ 是映射 $(1,\pi /2)$（将每个点旋转 $\pi /2$ 角），则应用 $P$ 两次，等价于将所有点乘以 $-1$，即 $P^2=-1$！

As should be obvious by now, these maps are just complex numbers in disguise.
显然，这些映射本质上就是复数的另一种表现形式。

Unsurprisingly, they are singularly useful for solving polynomial equations. Indeed, the real number $x^{\prime }$ is a root of the polynomial equations $a_0{x}^n+a_1{x}^{n-1}+\cdots +a_n=0$ if and only if the mapping $(x^{\prime },0)$ satisfies the same equation. So viewing polynomial equations over the set of these mappings loses no solutions, while at the same time giving us additional freedom to do operations such as taking the square roots of negative numbers.
毫不意外，它们在多项式求解中极为有用。事实上，实数 $x^{\prime }$ 是多项式方程 $a_0{x}^n+a_1{x}^{n-1}+\cdots +a_n=0$ 的根，当且仅当映射 $(x^{\prime },0)$ 满足该方程。因此，在这类映射集合上研究多项式方程，不会丢失任何解，同时还能让我们获得额外运算自由（如对负数开平方）。

– Tom Stephens
Commented Jul 28, 2010 at 2:34

+1: This is a great reply to an unfortunately phrased question.
+1：这个回答很棒，很好地回应了这个表述欠佳的问题。

– Warren P
Commented May 22, 2011 at 20:17

This is the kind of “thinking” answer that makes me want to read and re-learn about all the mathematical stuff I learned a bit of and then promptly forgot all about. GREAT answer.
这是那种能让人想重新阅读、重新学习那些学过又很快遗忘的数学知识的“启发性”答案。太赞了！

– Hagen von Eitzen
Commented Sep 1, 2012 at 20:02

Very nice, but while multiplication is fine, with this setup addition becomes a bit unnatural.
非常好，但乘法虽然直观，在这个框架下，加法的定义却有些不自然。

Even though quite a few people have already contributed towards answering your question, I would like to offer some thoughts of mine that may help you.
尽管已有不少人回答了你的问题，我仍想分享一些看法，希望能对你有帮助。

I am not going to give you ways of showing that complex numbers are necessary and meaningful, rather an idea about why I think people find them meaningless and how this can be resolved.
我不会直接说明复数为何必要且有意义，而是想分析人们为何觉得复数“无意义”，以及如何解决这一认知问题。

I am prompted by the part of your question that says: “…most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning.” I have thought about this a lot in the past and have come to the conclusion that the problem in finding complex numbers meaningful lies in the meaning we have attached to the number systems “preceding” them in the hierachical chain: $\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R}$. The meaning we have attached is that of quantity! Numbers all the way up to and including the reals are scalar quantities: We use them for distance, area, volume, weight, speed, intensity and so on. However, when you get to the complex numbers all that has to go away. There is no such thing as $2+3i$ kilogrammes or $-4i$ dollars or $1-5i$ centimetres… Yet we ask the learner to call them numbers!
你的问题中提到“大多数非数学专业人士难以接受复数有意义，而认为实数有明确直观的意义”，这启发了我。过去我对此思考良多，最终得出结论：人们难以理解复数意义的核心，在于我们对复数之前的数系（$\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R}$）赋予的“数量”含义。从自然数到实数，我们都将其视为“标量”——用于表示距离、面积、体积、重量、速度、强度等。但到了复数，这种“数量”含义完全不适用了：不存在 $2+3i$ 千克、$-4i$ 美元或 $1-5i$ 厘米……可我们仍要求学习者将其称为“数”！

So, the learner is faced with an apparent contradiction in terms: They are asked to think of these strange entities as numbers while at the same time these new additions to the realm don’t behave as the numbers of old. And you should take into account that the older numbers have been around for a long time. That’s where I think the problem lies.
因此，学习者会面临一个明显的矛盾：既要将这些陌生的实体视为“数”，又发现它们的性质与以往的“数”完全不同——而且以往的数系已长期深入人心。这就是问题的核心所在。

To circumvent this, I think that what one needs to do is explain to the learner that - from now on - numbers are going to take on a much wider role than before: They are going to be used not only to denote quantities, but also to denote directions, which is precisely what complex numbers do on the two-dimensional plane. Every complex number can be represented by a vector and this automatically suggests a direction. The “older” meaning is not lost, since a complex number has a modulus in addition to its real and imaginary parts and these three are quantities.
要解决这个问题，我认为需要向学习者解释：从现在开始，“数”的作用将比以往更广泛——它们不仅用于表示“数量”，还用于表示“方向”。而复数在二维平面上的作用，正是表示方向。每个复数都可表示为一个向量，向量天然带有方向属性。同时，以往的“数量”含义并未消失：复数除了实部和虚部，还有模长，这三者都是“数量”。

Of course, one might say “But real numbers denote directions as well: $+1$ denotes a unit displacement to the right of $0$ along the x-axis while $-1$ denotes a displacement to the left”. This is indeed true, however, I sincerely doubt that the untrained mind of a learner, who encounters complex numbers for the first time, will delve in that direction. And, if it does, then that’s fine, because it shows that the direction concept was already lurking in the number hierarchy ever since negative integers were introduced!!
当然，有人可能会说“实数也能表示方向：$+1$ 表示沿 x 轴正方向离原点 0 移动 1 单位，$-1$ 表示沿负方向移动”。这确实没错，但我真心认为，首次接触复数的学习者（若无相关训练）不会想到这一点。即便想到了，也无妨——这恰好说明，从负整数引入开始，“方向”概念就已隐藏在数系中了！

I hope all this is of help to you.
希望这些看法能对你有帮助。

– Andrew Marshall
Commented Dec 19, 2010 at 18:59

+1 spot on! and it doesn’t help that the word “imaginary” is also used. The same non-mathematicians may very well have heard of groups, and likely have not questioned whether they are “real.” “Groups” is not already used and understood with technical meaning, so there is no overloading.
+1 说得太对了！而且“虚数”（imaginary）这个词本身也加剧了误解。这些非数学专业人士可能听说过“群”（group）的概念，却不会质疑“群是否真实存在”——因为“群”没有先验的非技术含义，不会造成概念混淆。

– sequence
Commented Nov 20, 2016 at 6:21

What about directions in 3-space, for instance? Can complex numbers be generalized to $n$ dimensions?
比如三维空间中的方向呢？复数能否推广到 $n$ 维？

– electronpusher
Commented Apr 24, 2017 at 18:05

@sequence I think you’re looking for Quaternions. Knightofmathematics, I think your answer is excellent. I think that interpreting complex numbers (and reals) in terms of “direction” is the best way to introduce them to non-mathematicians, which is why I think your answer may be the best solution to OP’s actual question (as opposed to the misleading title).
@sequence 我想你要找的是“四元数”（Quaternions）。Knightofmathematics（答主），你的回答非常棒！我认为从“方向”角度解释复数（乃至实数），是向非数学专业人士介绍复数的最佳方式——这也是你的回答能最好地解决提问者实际问题（而非误导性标题）的原因。

– Arturo don Juan
Commented Apr 13, 2018 at 2:11

This is such an insightful answer.
这是一个极具洞察力的回答。

– Will Orrick
Commented Apr 4, 2020 at 10:12

I agree that this is the one answer the addresses the psychological issue most people have. All numbers a typical student encounters prior to complex numbers occur in successions: natural numbers as points on a line, rational numbers fitting in between, irrationals filling holes that still remain, zero and negative numbers extending the pattern to the left. When complex numbers are introduced, there’s no place left in this scheme for them to go. Maybe if we had called them “plane numbers” instead…
我同意这是唯一解决了大多数人心理障碍的回答。学生在接触复数前遇到的所有数，都可在“数轴”上依次排列：自然数是数轴上的点，有理数填充其间，无理数填补剩余空隙，零和负数向左延伸。但复数引入后，在这个“数轴框架”里找不到位置。要是当初把复数叫做“平面数”就好了……

To the degree that anything actually “exists” in math, yes complex number exist.
从数学中“存在”的定义来看，复数确实存在。

Once you accept that groups, rings and fields exist, and that isomorphism of rings makes sense, complex numbers can be recognized as (isomorphic to) the subring (which happens to be a field) of the ring of $2\times 2$ real matrices.
一旦你认可群、环、域的存在，且环同构的概念有意义，就能发现：复数（同构于）$2\times 2$ 实矩阵环的一个子环——而这个子环恰好是一个域。

Generators of this subring are the following matrices
这个子环的生成元是以下两个矩阵：

$\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|$ and $\left|\begin{array}{cc}0& 1\\ -1& 0\end{array}\right|$

which correspond to $1$ and $i$ in the normal notation of complex numbers.
它们分别对应复数常用表示中的 $1$ 和 $i$。

As people tend to accept that matrices exist, this may be a convincing argument.
由于人们通常认可矩阵的存在，这个论证可能更有说服力。

– cobbal
Commented Jul 21, 2010 at 5:19

Interesting answer, but good luck convincing the people who claim they don’t exist with this argument :
回答很有趣，但想用这个论证说服那些认为复数不存在的人，祝你好运：

– donroby
Commented Jul 21, 2010 at 20:15

Well, first you ask if matrices exist.
嗯，首先你得问他们“矩阵存在吗”。

– SamB
Commented Nov 27, 2010 at 20:34

donroby: well, they certainly do exist in video game worlds (to an approximation – rounding errors throw a major wrench in the works…)
donroby（答主）：嗯，至少在电子游戏世界里，矩阵肯定是存在的（大致来说——舍入误差会带来不少麻烦……）

I think some confusion comes from calling the numbers that only have imaginary part “imaginary numbers”. I believe if mathematicians had other name to give to them (but I am really not thinking on a better name), there would be less confusion.
我认为部分困惑源于将“只有虚部的数”称为“虚数”（imaginary numbers）。如果数学家当初给它们起了别的名字（尽管我想不出更好的名字），困惑应该会少很多。

UPDATE: Even a person brilliant as Gauss thinks “imaginary” is a terrible naming:
更新：即便像高斯（Gauss）这样的天才，也认为“虚数”是个糟糕的命名：

Imaginary Numbers Are Real [Part 1: Introduction]

 
“That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question.”
-Gauss
 
“虚数这一学科迄今为止被神秘的晦涩性所笼罩，很大程度上要归因于其不合适的记法。例如，若将 + 1、-1 和 √-1 分别称为‘正向单位’‘反向单位’和‘侧向单位’，而非‘正单位’‘负单位’和‘虚（甚至不可能）单位’，那么这种晦涩性就绝不会存在。”
—— 高斯

– Beska
Commented Dec 1, 2010 at 16:44

I think this hits the nail on the head.
我认为这说到了点子上。

– Hagen von Eitzen
Commented Sep 1, 2012 at 20:09

And other types of numbers were born with such a name as well: think of “negative” and “irrational”. Which just reminds us how much difficulty these numbers had as well historically to be accepted.
其他类型的数也有类似的“负面”命名：比如“负数”（negative）和“无理数”（irrational）。这也提醒我们，这些数在历史上被接受也经历了诸多困难。

– FNH
Commented Mar 14, 2013 at 3:44

@sergiol , you can call thenm " complex numbers " !
@sergiol（答主），你可以叫它们“复数”（complex numbers）啊！

– sergiol
Commented Feb 3, 2016 at 13:43

May be composite numbers would be what I would call to them.
或许我会叫它们“合成数”（composite numbers）。

– asmaier
Commented Oct 7, 2018 at 19:59

I like the name “Duonions” .
我喜欢“双元数”（Duonions）这个名字。

You may be interested to read the MathOverflow question “Demystifying Complex Numbers,” here. A teacher is asking how to motivate complex numbers to students taking complex analysis.
你可能会对 MathOverflow 上的“揭开复数的神秘面纱”问题感兴趣。提问者是一位教师，想知道如何向学习复分析的学生介绍复数的必要性。

– Neil Mayhew
Commented Jul 20, 2010 at 23:19

Thanks, very interesting. Not quite what I was looking for, but helpful all the same.
谢谢，内容很有趣。虽然不是我要找的，但还是有帮助。

– M. Winter
Commented Sep 17 at 14:02

This answer should have been comment. It just forwards to another post and has no content beyond that.
这个回答本该是评论，因为它只是转发了另一个帖子，没有额外内容。

Here is an example of a physical quantity which comes naturally as a complex number.
下面举一个“天然以复数形式存在的物理量”例子。

The impedance between two nodes of a linear electrical circuit working in alternate current, which is the analogous of the resistance of a resistor and is also measured in Ohms, is a complex quantity. For instance, the impedance of an (ideal) capacitor is imaginary.
线性交流电路中两个节点间的“阻抗”（impedance），就是一个复数量。它与电阻器的电阻类似，单位也是欧姆（Ohms）。例如，理想电容器的阻抗就是纯虚数。

– M. Winter
Commented Sep 17 at 14:05

Whether physical phenomena can be modeled by complex numbers makes no statement about whether they “really exist”. The quantum amplitude would have been another example.
物理现象可用复数建模，并不代表复数“真实存在”。量子振幅（quantum amplitude）是另一个例子。

Quantum mechanics, and hence physics and everything around us, fundamentally involves complex numbers.
量子力学——进而整个物理学、我们周围的一切——从根本上都涉及复数。

– Américo Tavares
Commented Sep 11, 2010 at 16:40

And electrical engineering as well.
电气工程也离不开复数。

– Cheerful Parsnip
Commented Dec 10, 2010 at 15:16

Indeed one could argue complex numbers are more fundamental than real numbers since quantum mechanics is all about unitary operators on Hilbert spaces.
确实有人认为复数比实数更基础，因为量子力学的核心就是希尔伯特空间上的幺正算子——而这离不开复数。

– M. Winter
Commented Sep 17 at 14:06

Whether physical phenomena can be modeled by complex numbers makes no statement about whether they “really exist”.
物理现象可用复数建模，并不代表复数“真实存在”。

– M. Winter
Commented Sep 17 at 14:13

@CheerfulParsnip One should be careful attributing reality to complex numbers based on QM. The ontological status of complex numbers in physics is fiercely debated. It is likely best to treat them as a really convenient tool for notation and computation rather than fundamental.
@CheerfulParsnip（评论者）：基于量子力学将“实在性”赋予复数，需要谨慎。复数在物理学中的本体论地位存在激烈争论。或许最好将其视为notation和计算的便捷工具，而非基础实体。

– Cheerful Parsnip
Commented Sep 18 at 1:41

@M.Winter the “reality” of mathematical objects is indeed a thorny philosophical issue, but I stand by my claim that complex numbers are more fundamental than real numbers from the perspective of QM. The ontology of real numbers of course is also suspect, so one could interpret this as not saying much. But if you look at things like the monster group, it seems difficult to say that it doesn’t have some sort of independent existence.
@M.Winter（评论者）：数学对象的“实在性”确实是棘手的哲学问题，但我仍坚持：从量子力学视角看，复数比实数更基础。当然，实数的本体论地位也存疑，因此这个观点或许说服力有限。但像“魔群”（monster group）这样的概念，很难说它不具有某种独立的“存在性”。

I’ll start by pointing out that a whole host of things that people think of as ‘real’ are on shakier ground than imaginary numbers. Given that quantum mechanics predicts a fundamental limit to how granular reality is, the whole concept of real numbers is on very shakey ground, yet people accept those as fine. I’d therefore suggest that it is merely a case of familiarity - people are less familiar with complex numbers than with some other mathematical constructs.
首先我要指出：人们认为“真实”的很多事物，其“存在基础”比虚数更薄弱。量子力学预测现实存在“最小粒度”的基本极限，这使得实数的整个概念基础都很不稳定，但人们仍坦然接受实数。因此我认为，这只是“熟悉度”问题——人们对复数的熟悉度，低于对其他一些数学概念的熟悉度。

As for an actual existence outside the realms of pure maths… your best bet is to look at quantum mechanics again. This area has some fascinating results that are only possible through the use of imaginary numbers. Incidentally, fundamental particles are the place in nature that gave a ‘physicality’ to negative numbers (the charge of an electron is negative) well after they were accepted as normal by most people.
至于复数在纯数学领域之外是否“真实存在”……最好的例子还是量子力学。该领域有一些迷人的结果，只有通过虚数才能得到。顺便提一句，基本粒子为负数赋予了“物理实在性”（如电子电荷为负）——而这是在负数被大多数人接受为“正常数”之后很久才实现的。

– Eric O. Korman
Commented Jul 20, 2010 at 22:39

the ``granularness" of quantum mechanics has nothing to do with putting the real numbers on shakey ground-- quite the contrary, quantum mechanics relies heavily on the real numbers. For example, in the standard versions of quantum mechanics the position of most particles can take any real value (not some discrete set of points).
量子力学的“粒度”与实数基础是否薄弱无关——恰恰相反，量子力学严重依赖实数。例如，在标准量子力学中，大多数粒子的位置可取任意实数值（而非离散点）。

– workmad3
Commented Jul 20, 2010 at 22:54
the maths used to calculate results in quantum mechanics may make use of reals, but that’s really due to convention and the fact that we (humans) have great difficulty with the hugeness of numbers that would be used if we calculated things in plank lengths and plank masses. It’s a convenient abstraction rather than an affirmation of any ‘existence’ of real numbers.
量子力学计算中可能用到实数，但这本质上是出于惯例——而且如果用普朗克长度、普朗克质量来计算，数值会极大，人类难以处理。因此实数只是便捷的抽象工具，而非其“存在”的证明。

– Neil Mayhew
Commented Jul 20, 2010 at 22:57

I think electrical circuit theory is a more approachable use of complex numbers, with the imaginary component being used to express phase. Most people think quantum mechanics is pretty weird and have trouble believing it (as did Einstein, so they’re in good company) so using quantum mechanics as a justification further convinces them that complex numbers are merely nonsense.
我认为电路理论中复数的应用更易理解：虚部用于表示相位。大多数人觉得量子力学很怪异，难以相信（爱因斯坦也如此，所以他们并不孤单），因此用量子力学证明复数的合理性，反而会让他们更觉得复数是无稽之谈。

– Eric O. Korman
Commented Jul 20, 2010 at 23:04

of course every physical theory we have is just an approximation. there are no widely accepted theories that say that spacetime is a lattice (which you seem to believe).
当然，我们现有的所有物理理论都只是近似。目前没有被广泛接受的理论认为时空是“晶格结构”（这似乎是你所主张的）。

– isomorphismes
Commented Mar 18, 2011 at 11:03

@Eric I think @workmad3 means QFT, not QM. Spin foams don’t use a Cauchy-complete number system for instance. And it probably is reasonable to dispute the “physical” existence of transcendental numbers. Is there a reason to believe that all series converge in real life? Dense is one thing, complete is another.
@埃里克：我想 @workmad3 指的是量子场论（QFT），而非量子力学（QM）。例如，自旋泡沫理论（spin foams）并不使用柯西完备的数系。而且质疑超越数的“物理存在性”或许是合理的——有理由相信现实中所有级数都收敛吗？“稠密”是一回事，“完备”是另一回事。

Yes, as much as any number “exists”. (We could say a mathematical “object” exists if it models something empirical.)
是的，复数和其他数一样“存在”。（我们可以说：若一个数学“对象”能建模经验现象，它就“存在”。）

Here’s a way to visualize a vector “pointing” in the $i$ direction. If electrical power is traveling 100% efficiently from power generator to your home, then it’s pointing in the direction $⟨1⟩$ (or flip-flopping $⟨1⟩$ and $⟨-1⟩$ for A/C). If the power points in the direction of $⟨\sqrt{-1}⟩$ then the wire will heat up but transmit no useful electricity.
有一种可视化“沿 $i$ 方向的向量”的方法：若电能从发电机到你家的传输效率是 100%，则电能向量沿 $⟨1⟩$ 方向（交流电中则在 $⟨1⟩$ 和 $⟨-1⟩$ 之间交替）；若电能向量沿 $⟨\sqrt{-1}⟩$ 方向，则导线会发热，但不会传输有用电能。

– J. M. ain’t a mathematician
Commented Aug 24, 2010 at 3:19

I liked the way an old electrical engineering book put it: “there’s nothing imaginary about an electrical shock from j500 V!”
我喜欢一本老电工教材里的说法：“被 j500 伏电压电击，可一点都不‘虚’！”（注：电工领域常用 j 表示虚数单位，避免与电流符号 i 混淆）

– SamB
Commented Nov 27, 2010 at 20:42

That’s even better than the way poles in the left half-plane can lead to catastrophe (since left/right is on the real axis and all).
这比“左半平面的极点会导致灾难”的说法更形象（毕竟左右是在实轴上的概念）。

Are real numbers “real”? It’s not even computationally possible to compare two real numbers for equality!
实数真的“真实”吗？从计算角度看，甚至无法判断两个实数是否相等！

Interestingly enough, it is shown in Abstract Algebra courses that the idea of complex numbers arises naturally from the idea of real numbers - you could not say, for instance, that the real numbers are valid but the complex numbers aren’t (whatever your definition of valid is…)
有趣的是，抽象代数课程中会证明：复数的概念是从实数概念自然延伸而来的——例如，你不能说实数“有效”而复数“无效”（无论你对“有效”的定义是什么……）

There are geometric interpretations of imaginary numbers where they are thought of as parallelograms with a front and back, or oriented parallelograms. That interpretation requires geometric algebra but only uses real numbers.
虚数有几种几何解释：可将其视为“有正反的平行四边形”，或“有向平行四边形”。这种解释需要用到几何代数，但仅依赖实数。

Here is a link:
链接在此：
Geometric_algebra#Complex_numbers

That doesn’t have any pictures so it is admittedly not intuitive, but the answer is yes. Whether you think of imaginary numbers as square root of negative 1 or as parallelogram with a front and back, they exist.
该链接没有图示，因此确实不够直观，但答案是肯定的：无论将虚数视为负一的平方根，还是有向平行四边形，它们都是“存在”的。

– Neil Mayhew
Commented Jul 20, 2010 at 22:54

When I’ve used this explanation with people, they want to know why you can’t just use normal geometry, which doesn’t involve the concept of a negative square root.
我用这个解释时，人们会问：为什么不能直接用“普通几何”——即不涉及负平方根概念的几何——来解释？

– Jonathan Fischoff
Commented Jul 20, 2010 at 23:08

You can use normal geometry, you just have to add the idea of front and back, or alternatively clockwise and counter-clockwise. One way to prime the pump of understanding, is think how rotating something by 90 degrees twice gives -1. So rotate^2 = -1. i is like a 90 degree rotation.
可以用普通几何，只是需要加入“正反”概念，或“顺时针/逆时针”方向。启发理解的一个方法是：思考“将某物旋转 90 度两次，结果等同于反向（即乘以 -1）”，即旋转² = -1。而 i 就相当于一次 90 度旋转。

Complex numbers are the final step in a sequence of increasingly “unreal” extensions to the number system that humans have found it necessary to add over the centuries in order to express significant numerical concepts.
几个世纪以来，人类为表达重要的数值概念，不断扩展数系——每一步都更“抽象”，而复数是这一扩展过程的终点。

The first such “unreal number” was zero, back in the mists of time. It seems obvious to us now, but it must have seemed strange at first. How can the number of sheep I have be zero, when I don’t actually have any sheep?
第一个这样的“抽象数”是零，其起源可追溯到远古时期。如今我们觉得零很直观，但最初它肯定很怪异：我没有羊，怎么能用“零只羊”来表示呢？

Negative numbers are the next most obvious addition to the family of numbers. But what does it mean to have -2 apples? If I have 3 apples and you have 5, it’s convenient to be able to say that I have -2 more apples than you. Even so, during the middle ages many mathematicians were very uncomfortable with the idea of negative numbers and tried to arrange their equations so that they didn’t occur.
接下来最明显的扩展是负数。但“有 -2 个苹果”是什么意思？若我有 3 个苹果、你有 5 个，用“我比你多 -2 个苹果”来表述会很方便。即便如此，中世纪的许多数学家仍对负数感到不适，试图调整方程以避免出现负数。

Rationals (fractions) seem real enough, since I’m happy to have 2/5 of a pizza. However, this is not the number of pizzas that I have, just a ratio between 0 and 1 pizzas, and so is further removed from the concept of counting.
有理数（分数）看似很“真实”——比如我很乐意拥有 2/5 个披萨。但这并非“披萨的个数”，只是 0 到 1 个披萨之间的比例，因此仍偏离了“计数”的原始概念。

More significant philosophical problems problems arose when irrational numbers were first discovered by the classical Greeks. They were astonished when Euclid (or one of his predecessors) proved that the diagonal of a unit square could not be represented by ratio of two integers. This was such an outrageous idea to them that they called these numbers “irrational”. However, they couldn’t easily deny their existence since they have a direct geometric representation.
古希腊人首次发现无理数时，引发了更严重的哲学争议。当欧几里得（或其前辈）证明“单位正方形的对角线长度无法表示为两个整数的比”时，他们震惊不已——这个想法太离谱，因此将这类数称为“无理数”（irrational）。但由于无理数有直接的几何表示，他们又无法轻易否认其存在。

Irrational numbers were first understood as the solutions to algebraic equations such as $x^2=2$ but this still doesn’t cover all the possible numbers that we need. For example, the ratio of a circle’s diameter to its circumference is $\pi$ and so has a direct geometric representation. However, in 1882 $\pi$ was proved to be transcendental, meaning that it can’t be defined as the solution to a specific algebraic equation. This begins to seem a lot less “real”, especially when you consider that there are many important transcendental numbers that, unlike $\pi$, don’t have a geometric interpretation.
无理数最初被理解为代数方程（如 $x^2=2$）的解，但这仍无法涵盖我们所需的所有数。例如，圆的周长与直径之比 $\pi$ 有直接几何意义，但 1882 年人们证明 $\pi$ 是“超越数”——即无法表示为某个代数方程的解。这使得超越数看起来更不“真实”，尤其是很多重要的超越数（不像 $\pi$）甚至没有几何解释。

There are of course many algebraic equations that don’t have a solution even among the irrational numbers, and in some ways there’s no reason why they should. However, when 16th century mathematicians like Gerolamo Cardano began working on solutions to cubic equations they found that the square roots of negative numbers began cropping up very naturally in their procedures, even though the solutions themselves were purely real. This eventually led people to explore the arithmetic of complex numbers and they were surprised to find that it produced a consistent and beautiful theory.
当然，仍有很多代数方程即便在无理数范围内也无解，而且从某种意义上说，它们本就无需有解。但 16 世纪的数学家（如杰罗拉莫·卡尔达诺）研究三次方程求解时发现：即便方程的解是纯实数，计算过程中也会自然出现负平方根。这最终促使人们探索复数的运算规律，且惊讶地发现：复数理论既自洽又优美。

However, complex numbers don’t have such an intuitively obvious geometrical meaning as the numbers that came before. They are typically represented graphically as points in the 2D plane, and the rules of addition and multiplication are equivalent to certain operations on lengths and angles, but those operations aren’t driven by geometrical necessity in quite the way that squares and circles are. Even so, complex numbers are a perfect representation for various physical phenomena such as the state of particles in quantum mechanics and the behaviour of varying currents in electrical circuits. They are also very useful for reducing the cost of computation in 3D computer graphics.
但复数的几何意义，不像之前的数系那样直观。它们通常被表示为二维平面上的点，加法和乘法规则等价于对长度和角度的特定运算——但这些运算不像正方形、圆形那样，有“几何必要性”驱动。即便如此，复数仍是诸多物理现象的完美表示：如量子力学中的粒子状态、交流电路中的电流变化。在 3D 计算机图形学中，复数也能大幅降低计算成本。

The really special thing about complex numbers, though, is that they are the end of this journey that has been going on for millenia. There is no need to invent further number systems to provide solutions to problems expressed in terms of complex numbers because now every non-contradictory algebraic equation—no matter its degree—has a complete set of solutions within the complex number field. This is formalized by the Fundamental Theorem of Algebra, which states that any non-constant polynomial $P(z)=a_n{z}^n+a_{n-1}{z}^{n-1}+\cdots +a_{1}z+a_0$ (where $a_n\ne 0$ and $a_k$ are complex numbers) can be factored completely into linear terms of the form $(z-r_k)$, where each $r_k$ is a complex root. In other words, the complex numbers “close” the algebraic system—there are no gaps left to fill with new types of numbers, as there were when we lacked negatives, rationals, or irrationals.
不过，复数真正特别之处在于，它们是人类数千年数系拓展历程的终点。我们无需再发明新的数系来求解以复数表述的问题，因为如今所有无矛盾的代数方程 —— 无论次数高低 —— 在复数域内都存在完整的解集合。这一点通过代数基本定理得到形式化表述：任何非常数多项式 $P(z)=a_n{z}^n+a_{n-1}{z}^{n-1}+\cdots +a_{1}z+a_0$（其中 $a_n\ne 0$，且所有 $a_k$ 均为复数），都可完全因式分解为 $(z-r_k)$ 形式的一次项，其中每个 $r_k$ 都是该多项式的复根。换句话说，复数 “使代数体系闭合”—— 不再存在需要用新数系填补的空缺，而过去当我们缺乏负数、有理数或无理数时，这样的空缺是存在的。

Physicists have spent centuries breaking down matter into smaller components: molecules, atoms, protons, quarks, and beyond. Each new layer raises the question of whether there’s an even more fundamental particle beneath it—and because physics relies on experimental observation, we can never definitively declare an end to this chain. Mathematics, by contrast, allows for proof of completeness. We can rigorously show that the complex number field is algebraically closed; there is no mathematical “gap” that demands a new number system to resolve. Every other number system—natural numbers, integers, rationals, reals—exists as a subset of the complex numbers, a fragment of this complete framework. In this sense, complex numbers aren’t just another step in the journey of number systems—they are the destination.数百年来，物理学家一直将物质分解为更小的组成部分：分子、原子、质子、夸克，以及更微观的粒子。每一个新的层级都会引发新的疑问：其之下是否存在更基本的粒子？而由于物理学依赖实验观测，我们永远无法明确宣告这一分解链条的终点。相比之下，数学领域允许对 “完备性” 进行证明。我们可以严格证明，复数域是代数闭合的；不存在需要新数系来解决的数学 “空缺”。其他所有数系 —— 自然数、整数、有理数、实数 —— 都只是复数的子集，是这一完整体系的片段。从这个意义上说，复数并非数系发展历程中的又一步 —— 它们就是终点。

– BlueRaja - Danny Pflughoeft
Commented Jul 27, 2010 at 23:02

every non-contradictory equation has a solution within the complex numbers - surely you mean every polynomial equation?
“所有无矛盾的方程在复数域内都有解”——你其实是指所有多项式方程吧？

– KCd
Commented Aug 8, 2010 at 3:30
It is false that every other number system is a subset of the complex numbers. For example, modular arithmetic (say, adding and multiplying mod 5) can not be viewed as a subset of the complex numbers. In modular arithmetic 1 added to itself enough times gives you 0 and that doesn’t happen in $\mathbb{C}$. There are also many other number systems (p-adic numbers) which are not inside of $\mathbb{C}$ in any natural way. Moral: there is no “ultimate” number system.
“所有其他数系都是复数的子集”这一说法是错误的。例如，模算术（如模 5 的加减乘运算）不能被视为复数的子集——在模算术中，1 累加足够多次会得到 0，而这在 $\mathbb{C}$ 中不可能发生。还有许多其他数系（如 p 进数）也无法以“自然”的方式包含在 $\mathbb{C}$ 中。结论是：不存在“终极”数系。

– Andrew Marshall
Commented Dec 19, 2010 at 19:11
a very nice answer. But the last step is too idealistic in my view. Complex arithmetic with exponentiation is clock-arithmetic modulo $2\pi$. We have to introduce multivaluedness; the square root of $c^2$ is not necessarily $\pm c$, branch-cuts; recently I came across the concept of a “winding-number” which should provide uniqueness for complex exponentiation (in an article of Corless & al on the Lambert-w), which might be assigned as additional component to a complex number. I’m not sure that complex numbers are the final step…
回答非常精彩，但在我看来最后一步过于理想化。复数的指数运算本质上是模 $2\pi$ 的“时钟算术”，我们必须引入多值性——$c^2$ 的平方根未必是 $\pm c$，还需要考虑分支切割；最近我在科莱斯（Corless）等人关于朗伯 W 函数的文章中看到“绕数”概念，它可为复数指数运算提供唯一性，而这可能需要为复数增加额外的分量。因此我不确定复数就是数系扩展的最终步骤……

– Gottfried Helms
Commented Aug 26, 2011 at 18:47

As phrased, your question invites a philosophical answer but you said that’s not what you want. One way to approach this question is to ask if complex numbers correspond to anything in the real world. What can you count or measure with them?
从你的表述来看，这个问题很容易引出哲学层面的回答，但你说你不想要这样的答案。有一种切入角度是：复数是否对应现实世界中的某种事物？我们能用复数计数或测量什么？

As far as I can tell, complex numbers are most directly useful for measuring things that rotate or oscillate. They’re used by electrical engineers because voltages and currents can oscillate, and could be used for measuring springs or pendulums, or for anything that behaves like a wave. There are not that many situations in day-to-day life where you’d use complex numbers, but they’re used extensively in physics because waves and oscillations show up everywhere.
在我看来，复数最直接的用途是描述旋转或振荡的事物。电气工程师会用到复数，因为电压和电流会振荡；复数也可用于描述弹簧、钟摆或任何波动现象。日常生活中用到复数的场景不多，但在物理学中，由于波动和振荡无处不在，复数的应用极为广泛。

The argument isn’t worth having, as you disagree about what it means for something to ‘exist’. There are many interesting mathematical objects which don’t have an obvious physical counterpart. What does it mean for the Monster group to exist?
这场争论不值得继续，因为你们对“存在”的定义本身就有分歧。有许多有趣的数学对象并没有明显的物理对应物——比如“魔群”（Monster group）的“存在”又该如何定义呢？

– Neil Mayhew
Commented Jul 20, 2010 at 22:51

I know that complex numbers are a useful abstraction, and in my own mind I’m quite happy that they “exist”, in the same that math in general 'exists". However, I tagged the question as “teaching” because I want to hear other people’s ideas about how to help non-technicians grasp the wonder of complex numbers. I think your answer is trite and unhelpful so I’m downvoting it.
我知道复数是有用的抽象概念，在我看来，它们和数学本身一样“存在”。但我给这个问题打上了“教学”标签，是想听听其他人如何帮助非专业人士理解复数的奇妙之处。我认为你的回答空洞且无用，所以给你投了反对票。

– Dan Piponi
Commented Jul 21, 2010 at 22:58

I agree that the argument isn’t worth having. But I think you need to reach a certain state of enlightenment before you can appreciate why it’s not worth having. The entire mathematical community struggled with these kinds of issues for a long time so it’s clearly a non-trivial issue that can’t simply be dismissed in one sentence.
我同意这场争论不值得继续，但你需要达到一定的认知水平，才能理解为什么它不值得。整个数学界曾长期为这类问题争论不休，可见这绝非一个能靠一句话轻易打发的简单问题。

– SamB
Commented Nov 27, 2010 at 20:38

Is there a Monster group under my bed?
我的床底下有魔群吗？

The question you and your students are asking is whether the concept deserves to exist, i.e., is it really a useful concept? The best answer to offer non-mathematical people is that it is at the heart of many, many applications which are perhaps simpler to understand with this abstraction. You can draw historical parallels, for example to surds (such as $\sqrt{2}$), which were thought not to exist from the ancient Greek rational-geometric constructive perspective. But at the least, I think, you must point out the beauty and utility of Euler’s formula
你和你的学生真正想问的是：复数这个概念“值得”存在吗？也就是它是否真的有用？给非数学人士的最佳答案是：复数是无数应用的核心，借助这个抽象概念，许多问题会变得更容易理解。你可以引用历史类比，比如无理数（如 $\sqrt{2}$）——从古希腊的“有理几何构造”视角来看，当时的人也认为无理数不存在。但至少，你必须向他们展示欧拉公式的优美与实用：

$e^{i\theta }=\cos \theta +i\sin \theta$

(connecting them to trigonometry) and of the complex plane. Perhaps DeMoivre’s formula
（该公式将复数与三角学联系起来），以及复平面的概念。或许还可以介绍棣莫弗公式：

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

provides a nice example of a considerable simplification realized through complex numbers. Then, you should mention some of the multifarious connections that could come to mind, both applications (such as to alternating current & fluid dynamics) and mathematical extensions and related areas: vectors, quaternions, octonions, complex, Fourier & Harmonic analysis, Lie Groups, cyclotomic polynomials, analytic number theory, etc.
该公式是“通过复数简化问题”的绝佳例子。之后，你可以列举一些能想到的广泛关联：既有应用领域（如交流电、流体力学），也有数学扩展及相关领域（如向量、四元数、八元数、复分析、傅里叶分析与调和分析、李群、分圆多项式、解析数论等）。

Above all, one should mention that they have deep connections to algebra (e.g., its fundamental theorem) and by extension, to linear algebra and differential equations, two of the most useful areas of mathematics. And don’t hesitate to use pretty pictures.
最重要的是，要提到复数与代数学（如代数基本定理）有着深刻联系，进而与线性代数和微分方程相关——这两个是数学中最实用的领域。而且不妨用直观的图示来辅助讲解。

There is also some debate over, and well should there be room for, different fundamental viewpoints on what mathematics is and how it should be conducted. And one should probably also include here some perspectives more historically associated with statistics, such as pragmatism. With this in mind, one should have some tolerance for, or at least understanding of, non-mathematicians’ continued skepticism regarding mathematical concepts, since the proof of a concept’s utility is contingent on sufficient experience with it and, even within the field of mathematics, there are revisions and simplifications and room for alternative approaches.
关于“数学是什么”以及“数学应如何研究”，存在不同的基础观点，这其中也有争论——而这种争论理应存在空间。或许还应纳入一些与统计学历史相关的视角，如实用主义。考虑到这一点，我们应当对非数学人士对数学概念的持续质疑持包容态度（至少是理解态度），因为一个概念的实用性需要足够的实践来证明；即便在数学领域内部，也存在理论修正、简化以及不同研究路径的可能性。

Erase from your mind that ‘complex’ numbers have anything to do with the square root of $-1$. Instead ask yourself if this is feasible:
先忘掉“复数”与 $\sqrt{-1}$ 有任何关系，不妨问问自己：以下情况是否可行？

Intelligent life in a distant part of the universe developed in a completely different way than we did. They often say
在宇宙遥远角落的智慧生命，其发展路径与我们完全不同。他们常说：

God made the polygons, all else is the work of ___.
上帝创造了多边形，其余一切都是____的杰作。

This species made great strides in algebra at a very early stage. They built up considerable ‘mathematical potential energy’ in their ancient history.
这个物种在早期就在代数学上取得了巨大进步，在其远古历史中积累了大量“数学势能”。

In a short span of time, they went from the polygons to the circle, and then realized that there was a natural covering map of the 1-dim number line onto the circle group. For them, the circle group of rotations was just as useful as the number line.
在很短的时间内，他们从多边形研究过渡到圆形研究，随后发现：一维数轴到圆群存在自然的覆盖映射。对他们而言，旋转圆群和数轴同样有用。

As they worked with 2-dim space, one of their greatest minds realized that the most natural way to extend multiplication from the circle and number lines to all the points in the plane was to multiply number lengths and add their angles (dilation/rotation). It was a flash of intuition and insight, but the theory details gave then a tremendous technological advancement; many math problems they worked on had a new feel to them (for example, they found that every polynomial could now be completely factored).
在研究二维空间时，他们中最伟大的智者意识到：将乘法从圆群和数轴扩展到平面所有点的最自然方式，是“长度相乘、角度相加”（即伸缩与旋转结合）。这是一次直觉与洞察力的闪光，而理论细节则为他们带来了巨大的技术进步——许多数学问题在他们眼中有了新的解法（例如，他们发现所有多项式现在都能完全因式分解）。

As they advanced into quantum mechanics, they developed, what we humans call, the Schrödinger equation,
当他们深入研究量子力学时，提出了一个方程——也就是我们人类所说的薛定谔方程：

$i\hslash \frac{\partial }{\partial t}\Psi (\mathbf{r},t)=\hat{H}\Psi (\mathbf{r},t)$

Of course they did not express it as we do, and they were really surprised when they learned that $i$ meant “imaginary number” to so many people on earth. They were even more surprised to learn what percentage of people even knew what it stood for.
当然，他们的表达方式与我们不同。当他们得知，在地球上，许多人将 $i$ 称为“虚数单位”时，感到十分惊讶；而当他们了解到，知道 $i$ 代表什么的人比例有多低时，更是震惊不已。

– Neil Mayhew
Commented Jul 8, 2017 at 17:43
Nice idea. Could you flesh it out a bit? The connection between complex numbers and circle groups isn’t obvious.
想法很棒，能再详细阐述一下吗？复数和圆群之间的联系并不直观。

I usually say: “Believe it or not, electricity and radio waves actually do behave like complex numbers. You don’t see that in high school, but electric and electronics engineers do.”
我通常会说：“信不信由你，电和无线电波的行为真的和复数一样。你在高中阶段看不到这一点，但电气和电子工程师每天都在接触。”

I have just discovered this web page and your question. This is probably a late response.
我刚发现这个网页和你的问题，或许这是一个迟到的回答。

A lot of people think that at some point someone came along and simply asserted “yes there is a square root of -1 after all” and called it ‘i’. That’s a common misconception, and, unfortunately, seems to be what is often “taught” in “high schools” in the US.
很多人认为，某个时候突然有人站出来说“负一其实有平方根”，并把它叫做“i”。这是一个常见的误解，不幸的是，这似乎正是美国高中常“教”的内容。

The set of Real Numbers that you are used to is an example of what mathematicians call a “field”. That means you can add, subtract, multiply, and divide according to familiar properties. The field $\mathbb{C}$ contains the Reals: it is just the two-dimensional plane endowed with a very natural addition (vectors) and a really cool multiplication, which takes a little effort to understand.
你熟悉的实数集，是数学家所说的“域”（field）的一个例子。这意味着你可以按照熟悉的运算性质进行加、减、乘、除。复数域 $\mathbb{C}$ 包含实数域：它本质上是赋予了“自然加法（向量加法）”和“精妙乘法”的二维平面——理解这种乘法需要一点努力。

A complex number is just a point on the plane, an ordered pair. The absolute value of a complex number is its distance to the origin, and let’s call its “angle” the angle it forms with the positive x-axis. The Real numbers are just the x-axis, and “i” is just (0,1). So the real number 1 is (1,0) and -1 is (-1,0). Then multiplying complex numbers multiplies their absolute values and adds the angles. That’s why (0,1) times itself is -1 = (-1,0). 90 degrees plus 90 degrees = 180 degrees.
复数就是平面上的一个点，一个有序对。复数的绝对值是它到原点的距离，我们把它与 x 轴正方向形成的角称为“辐角”。实数就是 x 轴，“i”就是 (0,1)。因此，实数 1 对应 (1,0)，-1 对应 (-1,0)。复数相乘的规则是“模长相乘，辐角相加”——这就是为什么 (0,1) 乘以自身等于 -1（即 (-1,0)）：90 度加 90 度等于 180 度。

There is essentially no other way to imbed $\mathbb{R}$ (the reals) in a field in which all numbers have square roots. If you want to be technical, any other such field is isomorphic to $\mathbb{C}$. Furthermore, there is no way to make any higher ${\mathbb{R}}^n$ (${\mathbb{R}}^3$ = space for example) into a field in a natural way. The quaternions are a way to make ${\mathbb{R}}^4$ into a division ring, close but not a field.
本质上，不存在其他方法能将实数域 $\mathbb{R}$ 嵌入到一个“所有数都有平方根”的域中。从技术角度说，任何这样的域都与 $\mathbb{C}$ 同构。此外，无法以“自然方式”将更高维的 ${\mathbb{R}}^n$（如 ${\mathbb{R}}^3$，即三维空间）构造成域。四元数（quaternions）能将 ${\mathbb{R}}^4$ 构造成“除环”（division ring），与域接近，但并非域。

So, in short, the field of complex numbers is extremely real and concrete.
因此，简而言之，复数域是极其真实且具体的。

Last March Steven Strogatz wrote a column about $\mathbb{C}$ = the field of complex numbers in the New York Times. You might enjoy this exchange of emails I had with someone a few months later, after Strogatz’s article appeared in the New York Times.
去年三月，史蒂文·斯特罗加茨（Steven Strogatz）在《纽约时报》上发表了一篇关于复数域 $\mathbb{C}$ 的专栏文章。几个月后，在他的文章发表后，我与某人有过一段邮件交流，你或许会感兴趣。

http://home.bway.net/lewis/Complex.htm

Ordered pairs exist. If you define an operation on something that exists, the thing ‘with the operation’ exists. Certain operations can be defined on ordered pairs Complex numbers are ordered pairs with those operations defined on them. ∴ Complex numbers exist.
有序对是存在的。若你对某个存在的事物定义了运算，那么“带有该运算的事物”也是存在的。我们可以对有序对定义特定的运算，而复数就是定义了这些运算的有序对。因此，复数存在。

Since nobody has talked about Caspar Wessel I will, mainly because of this:
既然没人提到卡斯帕·韦塞尔（Caspar Wessel），我来补充一下，主要因为以下这点：

In 1799, Wessel was the first person to describe the geometrical interpretation of complex numbers as points in the complex plane.
1799 年，韦塞尔首次提出复数的几何解释，将其描述为复平面上的点。

As you can read from previous answers, all numbers are mathematical abstraction. 1 is not even defined in the real world, it is an abstraction to quantify what surrounds us. To every object we have, we define it’s “unity”: one potato is the whole elipsoidal tuber. But then if we cut it in half we’d say we have “half” a potato. We’re representing a concrete object with the abstract notion of unity, but the number 1 itself is our first numerical abstraction. What makes it so awfully mundane is that we are very used to unity. And, thus, we march steadily to 2, 3, …, to get what we name the “natural” numbers: $\mathbb{N}$. They are those who naturally arise in our day to day life, which we use to count and order, among other uses.
正如你从之前的回答中看到的，所有数都是数学抽象。数字“1”在现实世界中甚至没有定义——它是我们用来量化周围事物的抽象概念。对于每个物体，我们都定义了“单位”：一个土豆指的是整个椭圆形块茎。但如果我们把它切成两半，就会说有“半个”土豆。我们用“单位”这个抽象概念来表示具体物体，但数字“1”本身是我们第一个数值抽象。它之所以显得平淡无奇，只是因为我们太熟悉“单位”概念了。就这样，我们逐步得到了 2、3……即我们所说的“自然数”$\mathbb{N}$。它们自然地出现在日常生活中，用于计数、排序等。

Now, as you would probably imagine, we move onto integers. But as you are now doubtfull about the notion of complex numbers, many mathematicians were too about the negatives! They thought they were “impossible” and didn’t considered them numbers:
接下来，正如你可能想到的，我们扩展到了整数。但就像你现在对复数概念存疑一样，许多数学家也曾对负数存疑！他们认为负数是“不可能的”，不承认它们是数：

Prior to the concept of negative numbers, negative solutions to problems were considered “false” and equations requiring negative solutions were described as absurd.
Although the first set of rules for dealing with negative numbers was stated in the 7th century by the Indian mathematician Brahmagupta, it is surprising that in 1758 the British mathematician Francis Maseres was claiming that negative numbers “… darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple” .
在负数概念出现之前，问题的负解被视为“虚假的”，需要负解的方程被描述为“荒谬的”。
尽管印度数学家婆罗摩笈多（Brahmagupta）在 7 世纪就首次提出了处理负数的规则，但令人惊讶的是，1758 年英国数学家弗朗西斯·马塞雷斯（Francis Maseres）仍声称：负数“……使方程理论的整体变得晦涩，将本质上极其明显简单的事物变得模糊不清”。

So much for the “simple” integers, $\mathbb{Z}$. We can go on to the rational numbers $\mathbb{Q}$, the irrationals, and the real numbers $\mathbb{R}$, but let me focus on Wessels’ genius. He named his paper “On the Analytical Representation of Direction; An Attempt.” and began with
这就是所谓“简单”的整数集 $\mathbb{Z}$ 的历史。我们还可以继续讲有理数 $\mathbb{Q}$、无理数和实数 $\mathbb{R}$，但这里我想重点讲讲韦塞尔的天才之处。他将自己的论文命名为《论方向的解析表示；一次尝试》（On the Analytical Representation of Direction; An Attempt），并在开头写道：

“This present attempt deals with the question, how may be represet direction analytically; that is, how shall we express right lines so that in a single equation involving one unknow line and others known, both the length and the direction of the unknown line may be expressed.”
“本文试图解决的问题是：如何用解析方法表示方向？也就是说，如何表示直线，使得在包含一条未知直线和多条已知直线的单个方程中，既能表示未知直线的长度，又能表示其方向。”

He gives two propostition which seem to him “undeniable”:
他提出了两个在他看来“无可否认”的命题：

1. “…changes in direction which can be effected by algebraic operations shall be indicated by their signs.”
   “……代数运算所能引起的方向变化，应当用符号来表示。”
2. “…direction is not a subject for algebra except in so far as it can be changed by algebraic operations.”
   “……只有当方向能通过代数运算改变时，方向才是代数学的研究对象。”

He then explains how sum and substraction should be defined (he basically defines vector addition and substraction) and what he’s aiming in this exposition, among other remarks. But here’s the interesting part, after he defines multiplication:
随后，他解释了加法和减法应如何定义（本质上是定义了向量加法和减法），并阐述了本文的目标等内容。但有趣的部分在于他对“乘法”的定义之后：

“…so that the angle of the product (…) becomes equal to the sum of the direction angles of the factors.”
“……使得乘积的辐角（……）等于各因子方向角之和。”
 
“Let +1 designate the positive rectilinear unit and +$ϵ$ a certain other unit perpendicular to the positive unit and having the same origin; then the direction angle of +1 will be equal to 0º, that of -1 to 180º, that of +$ϵ$ to 90º and that of -$ϵ$ to -90º or 270º. By the rule that the direction angle of the product shal equal to sum of the angles of the factors, we have: (+1)(+1)=+1; (+1)(-1)=-1; (-1)(-1)=+1; (+1)(+$ϵ$)=+$ϵ$; (+1)(-$ϵ$)=-$ϵ$; (-1)(+$ϵ$)=-$ϵ$; (-1)(-$ϵ$)=+$ϵ$; (+$ϵ$)(+$ϵ$)=-1; (+$ϵ$)(-$ϵ$)=+1; (-$ϵ$)(-$ϵ$)=-1. From this it is seen that $ϵ$ is equal to $\sqrt{-1}$ (…)”
“设 +1 表示正的直线单位，+$ϵ$ 表示另一个与正单位垂直且同原点的单位；则 +1 的方向角为 0°，-1 的方向角为 180°，+$ϵ$ 的方向角为 90°，-$ϵ$ 的方向角为 -90° 或 270°。根据‘乘积的方向角等于各因子方向角之和’的规则，我们有：(+1)(+1)=+1；(+1)(-1)=-1；(-1)(-1)=+1；(+1)(+$ϵ$)=+$ϵ$；(+1)(-$ϵ$)=-$ϵ$；(-1)(+$ϵ$)=-$ϵ$；(-1)(-$ϵ$)=+$ϵ$；(+$ϵ$)(+$ϵ$)=-1；(+$ϵ$)(-$ϵ$)=+1；(-$ϵ$)(-$ϵ$)=-1。由此可见，$ϵ$ 等于 $\sqrt{-1}$（……）”

He then, after some reasoning and trignometrical though gives the know representation:
经过一些推理和三角学思考后，他给出了如今我们熟知的复数表示：

“In agreement with 1 and 6, the radius which begins at the center and diverges from the absolute or positive unit by angle $v$ is equal to $\cos v+ϵ\sin v$.”
“根据命题 1 和命题 6，从原点出发、与绝对单位（正单位）成角 $v$ 的射线（向量），可表示为 $\cos v+ϵ\sin v$。”

But what is most amazing is that he then gives a great ending to his paper!
但最令人惊叹的是，他在论文结尾还得出了一个重要结论：

Without knowing the angle which the indirect line $1+x$ makes with the absolute value, we may find, if the length of $x$ is less than 1, the power $(1+x{)}^m=1+\frac{mx}{1}+\frac{m\cdot m-1}{1\cdot 2}{x}^2+\text{etc.}$ If this series is arranged according to the powers of $m$, it has the same value and is changed into the form

不知道间接线 $1+x$ 与绝对值所成的角度，我们可以发现，如果 $x$ 的长度小于 1，则幂 $(1+x{)}^m=1+\frac{mx}{1}+\frac{m\cdot m-1}{1\cdot 2}{x}^2+\text{等等}$。如果这个级数按照 $m$ 的幂排列，它具有相同的值，并被改变为以下形式

$$1+\frac{ml}{1}+\frac{m^2{l}^2}{1\cdot 2}+\frac{m^3{l}^3}{1\cdot 2\cdot 3}+\text{etc.}$$

where

其中

$$l=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\text{etc.}$$

and is a sum of a direct (horizontal) and a perpendicular line. If we call the direct line $a$ and the perpendicular $b\sqrt{-1}$ then $b$ is the smallest measure of the angle which $1+x$ makes with $+1$. If we set

并且是一条直线（水平的）和一条垂直线的和。如果我们称直线为 $a$，垂直线为 $b\sqrt{-1}$，那么 $b$ 是 $1+x$ 与 $+1$ 所成角度的最小度量。如果我们设

$$1+\frac{1}{1}+\frac{1}{1\cdot 2}+\frac{1}{1\cdot 2\cdot 3}+\text{etc.}=e$$

then

那么

$$(1+x{)}^m$$

(…) may be represented by $e^{ma+mb\sqrt{-1}}$ (…)

(…) 可以表示为 $e^{ma+mb\sqrt{-1}}$ (…)

So in trying to give direction an analytical representation Caspar Wessel has given the most important rules to represent complex numbers as lines in the plane, and made them be meaningfull for us.
因此，卡斯帕·韦塞尔在试图用解析方法表示方向的过程中，给出了将复数表示为平面直线的核心规则，使复数对我们而言具有了实际意义。

I hope you enjoyed this as much as I did when I read it. If you want, I can give you the whole article, which is amazingly interesting.
希望你和我第一次读到这段内容时一样感兴趣。如果你愿意，我可以给你整篇论文——内容非常精彩。

– Sniper Clown
Commented Jun 17, 2012 at 8:59
Do you have the link to the article?
你有这篇论文的链接吗？

There is a lovely way of motivating the “existence” of the complex numbers just by using a little calculus on the real numbers. I found this in Visual Complex Analysis, and it tickled me, so I thought I’d share it here, despite the lateness of the answer.
仅通过实数域上的一点微积分，就能巧妙地说明复数“存在”的必要性。我在《可视化复分析》（Visual Complex Analysis）中看到过这种方法，觉得很有趣，所以尽管回答迟到了，还是想在这里分享。

If $r_1,...,r_n$ are real numbers, define:
若 $r_1,...,r_n$ 为实数，定义：

$f(x)=\frac{1}{(x-r_1)(x-r_2)...(x-r_n)}$

When a ∉ { r 1 , . . . , r n } a \notin \{r_1, ..., r_n\} a∈/{r1​,...,rn​} we can find the Taylor series around $a$:
当 a ∉ { r 1 , . . . , r n } a \notin \{r₁, ..., rₙ\} a∈/{r1​,...,rn​} 时，我们可以求出 $f(x)$ 在 $a$ 点的泰勒级数：

${\sum }_{k=0}^{\infty }\frac{f^{(k)}(a)}{k!}(x-a{)}^k$

The question is, for what (real) $x$ does this series converge to $f(x)$?
问题是：对于哪些实数 $x$，该级数会收敛到 $f(x)$？

As it turns out, if we let $R={\min }_k|a-r_k|$, then if $|x-a|<R$ this series converges to $f(x)$ and if $|x-a|>R$, then it doesn’t converge.
结果表明，若令 $R={\min }_k|a-r_k|$（即 $a$ 到最近的极点 $r_k$ 的距离），则当 $|x-a|<R$ 时，级数收敛到 $f(x)$；当 $|x-a|>R$ 时，级数发散。

So, in a sense, the $r_i$ “block” the ability of the Taylor series to converge around them.
因此，从某种意义上说，极点 $r_i$ “阻碍”了泰勒级数在其周围的收敛。

Now, what about the Taylor series for $g(x)=\frac{1}{x^2+1}$?
那么，对于 $g(x)=\frac{1}{x^2+1}$ 的泰勒级数，情况又如何呢？

Given an $a$ , this function has no “real” blockages - it is defined on all of $\mathbb{R}$ - but the Taylor series for $g(x)$ around $a$ has a similar $R$ value, and that $R$ value is $\sqrt{1+a^2}$, a value that can be computed entirely with real number calculations.
对于任意实数 $a$，$g(x)$ 没有“实数”极点——它在整个 $\mathbb{R}$ 上都有定义——但 $g(x)$ 在 $a$ 点的泰勒级数仍有一个类似的收敛半径 $R$，且 $R=\sqrt{1+a^2}$——这个值完全可以通过实数计算得到。

That then looks like there is some geometric obstruction to the Taylor series, an obstruction not on the real line, but a unit distance away from the real line in a perpendicular direction away from 0. It “looks like” an “imaginary” root of $x^2+1=0$.
这似乎表明，泰勒级数的收敛存在某种几何障碍——这种障碍不在实轴上，而在与实轴垂直、距离原点 1 单位的“虚方向”上。它“看起来”就像是方程 $x^2+1=0$ 的“虚根”。

answered Mar 21, 2012 at 5:30
– Thomas Andrews

The fact that the Taylor series of $(x^2+1{)}^{-1}$ has convergence radius 1 is one of the best example for how complex-valued reasoning can be forced on you even though both the problem and the solution are real-valued. I find this more convincing than computing roots of cubics because for those one might object that Cardano maybe hasn’t found the “right” equation for the roots.
$(x^2+1{)}^{-1}$ 的泰勒级数收敛半径为 1，这是“即便问题和答案都是实值的，也不得不引入复值推理”的最佳例子之一。我认为这比三次方程求根更有说服力，因为对于三次方程，有人可能会反驳说卡尔达诺或许没有找到求根的“正确”方程。

– Thomas Andrews
Commented Sep 17 at 16:08

Yeah, and the general Taylor series around real $a$ of that function gives a radius of convergence $\sqrt{a^2+1}$, which really hints at the geometric nature of what is “obstructing” the power series… @M.Winter
是的，而且该函数在任意实数 $a$ 点的泰勒级数收敛半径为 $\sqrt{a^2+1}$，这确实暗示了“阻碍幂级数收敛”的障碍具有几何性质……@M.Winter

We will will first consider the most common definition of $i$, as the square root of $-1$. When you first hear this, it sounds crazy. 0 squared is 0; a positive times a positive is positive and a negative times a negative is positive too. So there doesn’t actually appear to be any number that we can square to get $-1$.
我们首先来看 $i$ 最常见的定义：$-1$ 的平方根。初次听到这个定义时，你可能会觉得很荒谬。0 的平方是 0，正数乘正数是正数，负数乘负数也是正数——似乎不存在一个数，其平方等于 $-1$。

A mathematician would collectively term 0, negative numbers and positive numbers as the real numbers. They would also define the term complex numbers as a group of numbers that includes these real numbers. So while we have shown that no real number can square to get $-1$, we haven’t even defined complex numbers at this point, so we can’t rule out that one might have this property.
数学家将 0、负数和正数统称为实数，并将复数定义为包含这些实数的数集。因此，尽管我们证明了“没有实数的平方等于 $-1$”，但此时我们尚未定义复数——所以不能排除“存在某种数（复数），其平方等于 $-1$”的可能性。

At this point, it makes sense to ask what does a mathematician mean by a number? It certainly isn’t what most people associate it with - as an abstract representation some kind of real world quantity. We need to understand that it isn’t uncommon for one word to have different meanings for different groups of people - after all words mean whatever we make them mean. Most people only need to real world quantities, so they find it convenient to call those numbers. On the other hand, mathematicians explore a variety of different number systems. Indeed some, such as complex numbers, are useful for solving problems that are actually about real numbers.
此时，有必要问：数学家所说的“数”到底是什么？它肯定不是大多数人理解的“现实世界数量的抽象表示”。我们需要明白，一个词对不同群体有不同含义是很常见的——毕竟词语的含义是由我们赋予的。大多数人只需要用“数”表示现实数量，因此他们将这类量称为“数”；而数学家则探索各种不同的数系，其中有些数系（如复数）即便用于解决与实数相关的问题，也十分有用。

So mathematicians define $i$ as a number that obeys most of the normal algebraic laws. They also defined $i\ast i$ to equal $-1$. From this we can derive all of the standard results about complex numbers.
因此，数学家将 $i$ 定义为一个“遵循大部分常规代数法则”的数，并规定 $i\ast i=-1$。由此，我们可以推导出关于复数的所有标准结论。

As for whether they are real - it depends on what you want to know. Obviously, they don’t correspond to quantities of physical objects. On the other hand, complex numbers can useful for representing resistance in an electric circuit. Ultimately, they are an idea and while ideas don’t exist physically, saying they don’t exist at all is inaccurate.
至于复数是否“真实”，这取决于你的定义。显然，它们不对应物理对象的数量，但复数可用于表示电路中的阻抗。归根结底，复数是一种思想——虽然思想没有物理实体，但说它们“完全不存在”是不准确的。

– stevenvh
Commented Sep 11, 2010 at 12:44

“When you first hear this, it sounds crazy.” I must say, when I first saw it being drawn in the Argand plane, it made perfectly sense to me: you introduce an imaginary axis because your imaginary number obviously can fit nowhere on the real axis. If you say i = 1 rotated CW over 90°, then ii = 1 rotated over 290°, and you’ve arrived at -1, back on the real axis! The Argand plane made it all clear to me. Unfortunately I have no such representation for quaternions.
“初次听到这个定义时，你可能会觉得很荒谬。”但我必须说，当我第一次在阿冈平面（复平面）上看到复数的几何表示时，一切都变得清晰了：之所以要引入虚轴，是因为虚数显然无法放在实轴上。如果你将 $i$ 理解为“1 顺时针旋转 90°”，那么 $i\ast i$ 就是“1 旋转 2×90°”，结果恰好是 -1——又回到了实轴上！阿冈平面让我彻底理解了复数，可惜我找不到四元数的类似直观表示。

– Simon
Commented Dec 4, 2011 at 4:55

@stevenvh: Exactly the same representation works, it’s just harder to visualize a 4D space!
@stevenvh：四元数其实有完全相同的表示方式，只是很难可视化四维空间而已！

Even natural numbers are abstract notions. Through abstract constructions the complex numbers can be demonstrated. So asking whether complex numbers is something like asking whether the natural numbers exist, whether the real numbers exist, or not. They are all things in the abstract domain of thought.
即便是自然数，也是抽象概念。通过抽象构造，我们可以证明复数的存在。因此，问“复数是否存在”，就像问“自然数是否存在”“实数是否存在”一样——它们都存在于抽象的思想领域中。

If you want to reconcile it with your physical intuition, try the following. I suppose that your mental picture of the real numbers is by associating a line to it. Similarly, you can associate a plane to the set of complex numbers. Points are complex numbers. The real numbers will be the x-axis and imaginary numbers would be the y-axis. Addition is straightforward to visualize. For visualizing multiplication, write a complex number in the form $r{e}^{i\theta }$. Then multiplication is a combination of stretching by a magnitude of $r$ and rotation by an angle of $\theta$. I hope this helps you in mental visualization.
如果你想将复数与物理直觉联系起来，可以试试以下方法。我猜你对实数的直观印象是“与数轴关联”，类似地，你可以将复数集与“平面”关联：平面上的点就是复数，实轴（x 轴）对应实数，虚轴（y 轴）对应纯虚数。加法的可视化很直观；要可视化乘法，可将复数表示为 $r{e}^{i\theta }$ 的形式——此时乘法就是“按模长 $r$ 伸缩”与“按角度 $\theta$ 旋转”的结合。希望这能帮助你建立直观印象。

In electrical networks, complex numbers are very good for analysis of alternating current networks. If some current has an imaginary component, it means that the current is leading or lagging the voltage by some “phase difference”, or “angle”. A resistance having imaginary components mean that it has capacitance or inductance. Only the “real” parts contribute to energy loss. But for imepdance calculations you have to consider capacitance and inductance too. A similar analysis can be made for every kind of control systems. So complex number are all very much there in nature and all sorts of engineering problems. I cannot stress their importance enough.
在电路中，复数非常适合分析交流网络。若电流有虚部，说明电流与电压存在“相位差”（即角度差）——电流要么超前电压，要么滞后电压。若阻抗有虚部，说明电路中存在电容或电感。只有阻抗的“实部”会导致能量损耗，但计算总阻抗时，必须同时考虑电容和电感（即虚部）。各类控制系统的分析也类似。因此，复数在自然界和各种工程问题中都有切实应用，其重要性怎么强调都不为过。

– phv3773
Commented Oct 13, 2010 at 20:06

I agree that the complex plane is the easiest way to visualize the complex numbers. Part of the problem is that we use the term “imaginary.” We should have used a different word.
我同意复平面是可视化复数最简便的方式。问题之一在于我们使用了“虚数”（imaginary）这个术语——我们本该用一个更合适的词。

Here is a possible line of reasoning.
以下是一种可能的推理思路。

Say, we start with natural numbers, $\mathbb{N}$, then add 0, making them the whole numbers ${\mathbb{N}}_0$, then add negative numbers, making them integers $\mathbb{Z}$, then expand them to rationals $\mathbb{Q}$, and finally to reals $\mathbb{R}$. In some respect, we keep filling up the line, till there are no gaps left.
假设我们从自然数 $\mathbb{N}$ 开始，加入 0 得到非负整数 ${\mathbb{N}}_0$，再加入负数得到整数 $\mathbb{Z}$，扩展到有理数 $\mathbb{Q}$，最终得到实数 $\mathbb{R}$。从某种意义上说，我们一直在“填满”数轴，直到没有空隙为止。

Now the question: why do we call them “numbers”? What are the properties we expect from some object so we can call it a number. In fact, just a few: addition and multiplication must be defined for all of them, and we want these operations to be commutative and associative. The distributive law of multiplication over addition must hold. It’s nice to have a notion of ordering, so we can say $a<b$, and multiplication by some positive number must be compatible with that notion: if $a<b$ and $x$ is positive, then $ax<bx$. Those are the minimal requirements, met by $\mathbb{N}$. As we expand our numbering system, we get more and more useful properties, and the structure (both algebraic and topological) gets more complicated: from semigroup in case of $\mathbb{N}$ to complete ordered field in case of $\mathbb{R}$, but those basic ones always hold.
现在的问题是：我们为什么称它们为“数”？一个对象需要具备哪些性质，我们才会称之为“数”？其实只需要几点：必须定义全体元素的加法和乘法，且运算需满足交换律和结合律，乘法对加法需满足分配律。若能定义序关系（即可以说 $a<b$）就更好了，且正数乘法需与序关系兼容：若 $a<b$ 且 $x$ 为正数，则 $ax<bx$。这些是自然数 $\mathbb{N}$ 也能满足的最低要求。随着数系扩展，我们获得了越来越多的有用性质，代数和拓扑结构也越来越复杂（从 $\mathbb{N}$ 的半群结构，到 $\mathbb{R}$ 的完备有序域结构），但上述基本性质始终保持。

Now, the next step in our “quest for expanding our system” would be to look at ${\mathbb{R}}^2$, ${\mathbb{R}}^3$ etc. In general, the objects (“points”) populating the space ${\mathbb{R}}^n$ are called “vectors” and they differ from numbers in one important respect: we can’t define vector multiplication so it meets our requirements for numbers (commutativity, associativity and having multiplication distributed over addition). So we don’t treat vectors in ${\mathbb{R}}^n$ as the numbers, in particular, we don’t think of them as atomic objects, but rather in terms of the components or coordinates.
接下来，在“扩展数系”的探索中，我们会考虑 ${\mathbb{R}}^2$、${\mathbb{R}}^3$ 等空间。通常，${\mathbb{R}}^n$ 中的元素（“点”）被称为“向量”，它们与“数”的关键区别在于：我们无法定义满足“数”的要求（交换律、结合律、乘法对加法分配律）的向量乘法。因此，我们不将 ${\mathbb{R}}^n$ 中的向量视为“数”——尤其不会将它们视为不可分割的“原子对象”，而是会从分量或坐标的角度去看待它们。

There’s one important exception, however: i is ${\mathbb{R}}^2$. In fact we can define multiplication of two objects (points, vectors) in ${\mathbb{R}}^2$ so it meets all our requirements and we can call these objects “numbers”. To be precise, all except one: the ordering. Multiplication we define is no longer compatible with the ordering. So we agree that probably not having that feature isn’t such a big deal. Once we introduced multiplication, we end up with complete field (but not ordered field) of entities which we have a right to call the numbers. So we call them the complex numbers and designate them as $\mathbb{C}$. There are some similarities between ${\mathbb{R}}^2$ and $\mathbb{C}$, but there are also some difference, the most important one being the fact that multiplication is defined so $\mathbb{C}$ is a field rather than vector space. In fact we can treat the elements of $\mathbb{C}$ as the scalars and use them to build a vector space, a complex matrices, etc. Well, this is roughly, why we can call complex numbers “numbers”. They exhibit the same behavior (except for ordering) as the numbers we’re familiar with do. One of my favorite examples: when we define the derivative in vector analysis, the formulas look quite different from those we got used in Calculus. But when we define the derivative for complex-valued functions, the formula is exactly the same, because multiplication and hence division) is defined.
但有一个重要例外：${\mathbb{R}}^2$。事实上，我们可以在 ${\mathbb{R}}^2$ 中定义两个元素（点、向量）的乘法，使其满足“数”的所有要求——因此我们可以将这些元素称为“数”。更准确地说，除了一点：序关系。我们定义的乘法不再与序关系兼容，但我们认为这个缺陷无关紧要。引入乘法后，我们得到了一个“完备域”（但非有序域），其元素完全有资格被称为“数”——这就是复数域 $\mathbb{C}$。${\mathbb{R}}^2$ 与 $\mathbb{C}$ 有相似之处，但也有关键区别：正是因为乘法的定义，$\mathbb{C}$ 成为了“域”，而非“向量空间”。事实上，我们可以将 $\mathbb{C}$ 的元素视为“标量”，用它们构建向量空间、复矩阵等。这大致就是我们将复数称为“数”的原因——它们（除了序关系外）表现出与我们熟悉的“数”相同的性质。我最喜欢的例子之一是：向量分析中的导数公式与微积分中的导数公式差异很大，但复值函数的导数公式与微积分中的完全相同——因为复数定义了乘法（进而定义了除法）。

Now, why complex numbers are important is a different question. What I wanted to focus is the fact that we have a right to call them “numbers”. But, just like we can think of rational numbers as a special case of reals, and when move from reals to rationals, we loose the important property of completeness, we can think of real numbers as a special case of complex numbers, and when we move from complex to reals we loose some nice properties as well. In many respects, studying complex numbers helps us to better understand real numbers.
复数为何重要是另一个问题，我想强调的是：我们完全有理由称它们为“数”。就像我们可以将有理数视为实数的特例（从实数回到有理数会失去“完备性”这一重要性质）一样，我们也可以将实数视为复数的特例（从复数回到实数会失去一些优良性质）。在很多方面，研究复数能帮助我们更好地理解实数。

As a sidenote, again, when we move from $\mathbb{R}$ to $\mathbb{C}$, we loose the ordering. Complex numbers form a complete field, but this is not an ordered field. (There is only one complete ordered field, up to isomorphism.) Now suppose we want to explore ${\mathbb{R}}^n$ further. That’s the next best? ${\mathbb{R}}^4$. We can define multiplication of objects (vectors) in ${\mathbb{R}}^4$ which is associative, but - alas - not commutative. It’s kind of hard to think of numbers whose multiplication is not commutative. So we no longer call them (quaternions, $\mathbb{H}$) numbers. Finally, the next interesting space is ${\mathbb{R}}^8$, where we still can define multiplication, but it is no longer associative (in addition to being non-commutative). It’s really hard to deal with something which is not associative, and certainly it’s hard to think of those entities as numbers.
再补充一点：从 $\mathbb{R}$ 扩展到 $\mathbb{C}$ 时，我们失去了序关系。复数域是完备域，但不是有序域（在同构意义下，只有一个完备有序域，即实数域）。若我们想进一步探索 ${\mathbb{R}}^n$，下一个有意义的空间是 ${\mathbb{R}}^4$：我们可以在 ${\mathbb{R}}^4$ 中定义向量乘法，满足结合律，但遗憾的是不满足交换律。很难将“乘法不交换”的对象视为“数”，因此我们不将四元数（$\mathbb{H}$）称为“数”。再下一个有趣的空间是 ${\mathbb{R}}^8$（八元数）：这里的乘法既不满足交换律，也不满足结合律。处理非结合的对象非常困难，更不用说将它们视为“数”了。

Re: “What’s the best way to explain to a non-mathematician that complex numbers are necessary and meaningful, in the same way that real numbers are?”
回复：“向非数学专业人士解释复数与实数同样必要且有意义的最佳方式是什么？”

I am a non-mathematician…so fully qualified to answer this question…
我是非数学专业人士……因此完全有资格回答这个问题……

• first explain imaginary numbers
  首先解释虚数
• then explain that complex numbers are simply numbers that use imaginary numbers
  然后解释复数就是包含虚数的数
• then tell 'em about how they are applied in weather forecasting
  接着告诉他们复数在天气预报中的应用
• bang!..I geddit
  砰！……我懂了

I got all this from an amazing radio show :
这些都是我从一个很棒的广播节目里学到的：

“In Our Time - Imaginary Numbers”
《我们的时代——虚数》

Melvyn Bragg and his guests discuss imaginary numbers - important mathematical phenomena which provide us with useful tools for understanding the world.
梅尔文·布拉格（Melvyn Bragg）和嘉宾们探讨了虚数——这一重要的数学现象为我们提供了理解世界的有用工具。

A number of answerers have answered by asking the question of whether the real numbers exist.
很多回答者通过反问“实数是否存在”来回应原问题。

There are lots of physical examples that a person can give to a lay-person of real numbers, including fractions, zero and negative numbers. Vector addition all by itself provides plenty of examples. For instance, most anyone can develop an intuition for the idea that if a vector pointing north has a positive component, then it can be combined with a vector pointing south, and that the south-pointing vector intuitively and usefully can be assigned a negative component.
向普通人解释实数时，有很多物理例子可用，包括分数、零和负数。向量加法本身就提供了大量例子：例如，大多数人都能理解“若向北的向量分量为正，则向南的向量分量可赋值为负”——这种直观的赋值方式很有用。

So the question is, are there intuitive physical examples of imaginary numbers. I’ve never seen one! For example, I’d like anyone to show me something even resembling or approximating a stick of length 2i, that when lined up perpendicularly with another stick of length 2i outlines a region with area -4. Given this, lay people have a point when they deny complex numbers exist.
因此问题在于：是否有直观的物理例子能解释虚数？我从未见过！例如，谁能给我看一根“长度为 2i”的棍子，将它与另一根同样长度的棍子垂直放置，围成一个“面积为 -4”的区域？正因如此，普通人质疑复数的存在，并非没有道理。

Two advanced comments on this:
对此有两点深入看法：

(1) A lovely and non-obvious fact about $\sqrt{-1}$ is that once you have introduced it and called it “i” you don’t have to introduce a pile of other numbers. For example, the $\sqrt{i}=1/\sqrt{2}+i/\sqrt{2}$. Imaginary numbers would be a lot less compelling to mathematicians if introducing them immediately led to a bunch of other quantities having to be hypothesized and named.
（1）关于 $\sqrt{-1}$，有一个美妙且不直观的事实：一旦引入它并称之为“i”，就无需再引入一堆其他新数。例如，$\sqrt{i}=1/\sqrt{2}+i/\sqrt{2}$（仍是复数）。若引入虚数后，还需不断假设和命名新的量，那么复数对数学家的吸引力会大打折扣。

(2) Quantum mechanics makes heavy use of imaginary numbers. Unlike in electrical engineering, it isn’t just a convenience. At the end of the day, in electrical engineering, after having solved a difficult problem, you’ll never actually go measure an imaginary voltage or current. The solution to electrical engineering problems always involves “throwing away” the imaginary part of the answer, even though it makes the legwork a lot simpler when coming to an answer. In quantum mechanics, you don’t throw the complex part of the wave function away. You work with it at every step of the way, until at the end, you use it’s norm-squared. This turns a complex “probability amplitude” into a “probability density.”
（2）量子力学大量使用复数，且与电气工程不同，这并非仅仅为了方便。在电气工程中，即便用复数解决了难题，最终也不会去测量“虚电压”或“虚电流”——总是要“舍弃”答案的虚部，尽管虚部能简化计算过程。而在量子力学中，波函数的复部不会被舍弃：每一步计算都要用到它，直到最后取其模的平方，将复“概率幅”转化为“概率密度”。

So you might say that quantum mechanics provides an example of the “existence” of complex numbers. However, the problem with this example is that quantum mechanics remains difficult to intuit even for the most experienced physicists. “I think I can safely say that nobody understands quantum mechanics.” https://en.wikiquote.org/wiki/Richard_Feynman
因此，你或许会说量子力学为复数的“存在”提供了例子。但这个例子的问题在于：即便对最有经验的物理学家来说，量子力学也难以直观理解。正如理查德·费曼（Richard Feynman）所说：“我想我可以有把握地说，没人真正理解量子力学。”

– bobobobo
Commented Apr 6, 2012 at 0:39

no, “we” (I’m EE bkg) don’t throw away the imaginary component. We sometimes want to know the reactive power, we study ccts that have “purely reactive load” (inductor/capacitor) etc
不对，“我们”（我是电气工程背景）不会舍弃虚部。我们有时需要知道无功功率，会研究“纯感性/容性负载”（电感/电容）的电路等。

– Cheerful Parsnip
Commented Mar 20, 2013 at 1:33

Penrose has made the point that the fundamental nature of reality, as measured by quantum wave functions take values in complex Hilbert spaces, is essentially ruled by the complex numbers. In a way the complex numbers are more fundamental than the real numbers.
彭罗斯（Penrose）指出，量子波函数取值于复希尔伯特空间，这表明现实的基本性质本质上由复数支配——从某种意义上说，复数比实数更基础。

---

“Where” exactly are complex numbers used “in the real world”?

“现实世界”中复数究竟用于何处？

I’ve always enjoyed solving problems in the complex numbers during my undergrad. However, I’ve always wondered where are they used and for what? In my domain (computer science) I’ve rarely seen it be used/applied and hence am curious.
本科期间，我一直很喜欢解决复数相关的问题，但始终好奇它们在实际中用于何处、有何用途。在我所在的计算机科学领域，很少见到复数的应用，因此我对此十分好奇。

So what practical applications of complex numbers exist and what are the ways in which complex transformation helps address the problem that wasn’t immediately addressable?
那么，复数有哪些实际应用？复数变换在解决“无法直接解决的问题”时，具体起到了什么作用？

Way back in undergrad when I asked my professor this he mentioned that “the folks in mechanical and aerospace engineering use it a lot” but for what? (Don’t other domains use it too?). I’m well aware of its use in Fourier analysis but that’s the farthest I got to a ‘real world application’. I’m sure that’s not it.
本科时我曾问过教授这个问题，他说“机械和航空航天工程领域的人经常用”，但具体用在什么地方呢？（其他领域难道不用吗？）我知道复数在傅里叶分析中有用，但这是我对“现实应用”的唯一了解，显然这并非全部。

PS: I’m not looking for the ability to make one problem easier to solve, but a bigger picture where the result of the complex analysis is used for something meaningful in the real world. A naive analogy is deciding the height of tower based on trigonometry. That’s going from paper to the real world. Similarly, what is it that is analyzed in the complex world and the result is used in the real world without imaginaries clouding the problem?
附言：我并非想知道复数“如何简化某一问题”，而是想了解更宏观的图景——复分析的结果如何用于现实世界中有意义的事物。一个简单的类比是：通过三角学计算确定塔的高度，这是从理论（纸面上）到现实应用的过程。类似地，在复数领域分析的对象是什么？其结果如何在现实中应用，且不会因虚数而让问题变得复杂？

The question: Interesting results easily achieved using complex numbers is nice but covers a more mathematical perspective on interim results that make solving a problem easier. It covers different ground IMHO.
问题《借助复数可轻松获得的有趣结果》很有价值，但它更多从数学角度讨论“复数如何简化中间计算”，与我的问题侧重点不同。

– Mariano Suárez-Álvarez
Commented Jan 24, 2013 at 2:45

I am pretty sure this has been asked…
我很确定这个问题之前有人问过……

– Trevor Wilson
Commented Jan 24, 2013 at 2:48

See Electrical_impedance#Complex_impedance
参见《电气阻抗#复阻抗》

– user856
Commented Jan 24, 2013 at 2:52

I can still duplicate my comment: have you checked Complex_number#Applications ?
我再重复一下我的评论：你看过《复数#应用》吗？

– Gerry Myerson
Commented Aug 4, 2018 at 9:41

Mathematicians use complex numbers because we are too cool for regular vectors. xkcd.com/2028
数学家使用复数，是因为我们觉得普通向量不够“酷”。

complex numbers aren’t just vectors.they’re a profound extension of realnumbers, laying the foundation for the fundamental theorem of algebra and the entire field of complex analysis.
复数不仅仅是向量。它们是实数的深刻扩展，为代数基本定理和整个复分析领域奠定了基础。

– Alex Amato
Commented Jun 5, 2021 at 20:13

There are plenty of applications of complex numbers, but from what I have seen they are typically used to simplify solving a math equation, and the end result is still a real number. Or in some cases (lile quantum) a 2d vectors are represented with complex numbers, but could be represented with 2d vectors. The point is these equations could be solved without introducing complex numbers, but sometimes the math feels easier to do with conplex numbers. Note: introducing a complex number by taking the squareroot of -1 is an invertable operation. This is the key that allows the math to work out.
复数有很多应用，但据我观察，它们通常用于简化数学方程求解，最终结果仍是实数。在某些情况下（如量子力学），二维向量可用复数表示，但也可用普通二维向量表示。关键在于：即便不引入复数，这些方程也能求解，但用复数会让计算更轻松。需注意：通过对 -1 开平方引入复数，是一种可逆操作——这是确保计算有效的核心。

Answers

Complex numbers are used in electrical engineering all the time, because Fourier transforms are used in understanding oscillations that occur both in alternating current and in signals modulated by electromagnetic waves.
电气工程中经常用到复数，因为理解“交流电中的振荡”和“电磁波调制信号中的振荡”时，需要用到傅里叶变换——而傅里叶变换依赖复数。

– PhD
Commented Jan 24, 2013 at 2:42

An example would go a long way for those who don’t know this :
举个例子会让不了解的人更容易理解：

– EuYu
Commented Jan 24, 2013 at 2:46

Complex numbers in a sense embody certain aspects of trigonometry. Therefore it is not unexpected for them to arise in situations involving trigonometric functions, such as waves and oscillations mentioned by Michael Hardy. A concrete example of their use is in phasors for example.
复数在某种意义上蕴含了三角学的某些性质，因此在涉及三角函数的场景（如迈克尔·哈迪提到的波动和振荡）中出现复数，并不意外。例如，“相量”（phasors）就是复数的一个具体应用。

– Michael Hardy
Commented Jan 24, 2013 at 2:52

“one begin the current” . . . I presume “one being the current” was meant.
“one begin the current”……我猜你想写的是“one being the current”（其中一个是电流）。

– Michael Hardy
Commented Jan 24, 2013 at 2:55

The letter $j$ rather than $i$ is used in electrical engineering for the imaginary unit. The expression $t\mapsto e^{j\omega t}$ occurs frequently. The real and imaginary parts of that are $\cos (\omega t)$ and $\sin (\omega t)$. Frequency is $\omega$ and $t$ is time.
电气工程中常用字母 $j$ 表示虚数单位，而非 $i$（避免与电流符号 $i$ 混淆）。表达式 $t\mapsto e^{j\omega t}$ 十分常见，其虚部和实部分别为 $\sin (\omega t)$ 和 $\cos (\omega t)$，其中 $\omega$ 是频率，$t$ 是时间。

– Michael Hardy
Commented Jan 24, 2013 at 2:56

Fourier transforms are also used in studying time series in statistics, including the stock market and biorhythms.
傅里叶变换也用于统计学中的时间序列分析，包括股票市场和生物节律研究。

I was asked this exact question by my wife last night. She was looking for an everyday example of the use of complex numbers to explain to her 8th grade math class (whose knowledge of complex numbers consists of $i=\sqrt{-1}$).
昨晚我妻子正好问了我这个问题。她需要一个复数在日常生活中的应用例子，用来给她教的八年级学生讲解——这些学生对复数的认知仅限于 $i=\sqrt{-1}$。

My response was this:
我的回答是这样的：

Imagine an electronic piano. Each key produces a different tone—think of each tone as a “signal” with its own “frequency” (how high or low the note is). A volume control changes the loudness of all keys by the same amount: turn it up, every note gets louder; turn it down, every note gets softer. That’s how real numbers work with signals—they only adjust one thing (like loudness) across the board.
想象一台电子钢琴，每个琴键都能发出不同的音调——可以把每个音调看作一个“信号”，每个信号都有自己的“频率”（音调的高低）。音量旋钮会同时改变所有琴键的音量：调大旋钮，所有音符都变响；调小旋钮，所有音符都变轻。这就是实数在信号处理中的作用——它们只能统一调整一个维度（比如音量）。
 
Now, imagine a filter—like the “bass boost” or “treble control” on a speaker. If you turn up the bass, the low-pitched keys (like the left side of the piano) get louder, but the high-pitched keys (right side) stay the same (or get quieter). If you tweak the treble, the opposite happens. This filter doesn’t just “turn everything up or down”—it adjusts different notes based on their frequency. That’s what complex numbers do: they add an “extra dimension” to calculations, letting us tweak specific parts of a signal (like different frequencies) instead of just one general setting.
现在再想象一个“滤波器”——就像音箱上的“低音增强”或“高音调节”功能。调高低音时，低音区的琴键（比如钢琴左侧）会变响，而高音区的琴键（右侧）音量不变（甚至变轻）；调高高音时则相反。这种滤波器不会“统一调整所有声音”，而是会根据“频率”调整不同音符的音量。这就是复数的作用：它们给计算增加了一个“额外维度”，让我们能针对性调整信号的特定部分（比如不同频率），而不是只能进行统一设置。

(Yes, I know about phase shifts and Fourier transforms, but these are 8th graders, and for their comprehensive testing, they’re only required to grasp a real-world link to complex numbers—not the math behind it. I don’t love the simplicity, but it’s what works for their level.)
（我当然知道相位偏移和傅里叶变换，但面对的是八年级学生，他们的综合测试只要求掌握复数与现实世界的“关联”，而非背后的数学原理。虽然这个例子过于简化，但对他们的认知水平来说刚刚好。）

– user85798
Commented Apr 16, 2014 at 12:21

8th grade is young for complex numbers.
八年级接触复数确实太早了。

– sfmiller940
Commented Nov 11, 2019 at 18:38

This seems more like an example of a 2D vector space. The defining feature of complex numbers is their algebra (like multiplication), but I get that explaining that to 8th graders is way harder—your analogy works for showing “why we need something beyond real numbers,” which is the first step.
这更像二维向量空间的例子，但复数的核心是其代数运算（比如乘法）。不过我理解，给八年级学生解释乘法太复杂了——你的类比能说明“为什么我们需要实数之外的东西”，这已经是第一步了。

Since you mentioned “real world,” let’s ground this in the tiny stuff that makes up everything.
既然你提到了“现实世界”，那我们就从构成万物的微观粒子说起。

The “real world” is built from particles—protons, electrons, photons—but quantum mechanics tells us these aren’t just tiny balls. They act like waves, and every wave has an “amplitude” (a measure of its “strength”). Here’s the key: that amplitude isn’t a real number—it’s a complex number.
“现实世界”由粒子构成——质子、电子、光子等——但量子力学告诉我们，这些粒子不只是微小的“球”，它们还表现出“波”的特性。每一种波都有“振幅”（衡量波“强度”的物理量），而关键在于：这个振幅不是实数，而是复数。

Think of it this way: if a real number is like saying “this wave is 5 units tall,” a complex number says “this wave is 5 units tall and shifted by a certain phase” (like how two ocean waves might crash at slightly different times). That phase shift matters—when two quantum waves interact, their complex amplitudes “add up” in a way that real numbers can’t capture (this is called “quantum interference,” and it’s why things like lasers or semiconductors work).
可以这样理解：如果用实数描述波是“这个波高5个单位”，那么用复数描述就是“这个波高5个单位，且存在一定的相位偏移”（就像两个海浪可能在略微不同的时间撞击海岸）。这种相位偏移至关重要——当两个量子波相互作用时，它们的复振幅会以实数无法描述的方式“叠加”（这一现象称为“量子干涉”，也是激光、半导体等技术能实现的核心原因）。

So complex numbers aren’t just a “math trick”—they’re how we describe the fundamental behavior of the particles that make up your phone, your laptop, even the air you breathe. Every time you use a GPS, a solar panel, or a digital camera, you’re relying on technology that depends on quantum waves—waves described by complex numbers.
因此，复数不只是“数学技巧”——它们是描述微观粒子基本行为的工具，而这些粒子构成了你的手机、笔记本电脑，甚至你呼吸的空气。每次你使用GPS、太阳能板或数码相机时，所依赖的技术都离不开量子波——而这些量子波正是用复数描述的。

– orion
Commented Dec 24, 2014 at 17:49

Quantum mechanics is the best answer here because unlike classical waves (like sound or water), where complex numbers are just a shortcut, in quantum mechanics, they’re unavoidable. The complex amplitude isn’t just “extra info”—it’s part of the wave’s identity.
量子力学是最好的例子，因为和经典波（如声波、水波）中“复数只是捷径”不同，量子力学里的复数是必需的。复振幅不是“额外信息”——它是波的本质属性之一。

– Mark Viola
Commented Jun 5, 2018 at 23:04

Protons and electrons are particles, not waves. Are you familiar with Quantum Electrodynamics (QED)? You’re referring to the fields around charged particles, not the particles themselves.
质子和电子是粒子，不是波。你了解量子电动力学（QED）吗？你说的其实是带电粒子周围的场，而非粒子本身。

– lesnik
Commented Jun 6, 2018 at 7:16
@MarkViola If they’re “just particles,” why can’t we measure both their position and speed exactly (Heisenberg’s uncertainty principle)? QED says fields and particles are two sides of the same coin—but even so, the field’s amplitude is still complex.
@MarkViola 如果它们“只是粒子”，为什么我们无法同时精确测量它们的位置和速度（海森堡不确定性原理）？量子电动力学认为场和粒子是同一事物的两面——但无论如何，场的振幅仍是复数。

– Mark Viola
Commented Jun 6, 2018 at 14:06
@lesnik The uncertainty principle is about measurement, not the particle’s nature. Electrons are particles—they have mass, charge, and spin—but their behavior is described by a wavefunction (which uses complex numbers). The wavefunction isn’t the particle itself; it’s a tool to predict its behavior.
@lesnik 不确定性原理和测量有关，和粒子的本质无关。电子是粒子——它们有质量、电荷和自旋——但它们的行为由波函数（用复数描述）预测。波函数不是粒子本身，而是预测其行为的工具。

– lesnik
Commented Jun 6, 2018 at 15:06

@MarkViola But if the wavefunction (with complex numbers) is the only way to predict the particle’s behavior, doesn’t that make complex numbers “real” in the sense that they describe how the universe works?
@MarkViola 但如果波函数（含复数）是预测粒子行为的唯一方式，那复数在“描述宇宙运行规律”的意义上，不就是“真实”的吗？

The other answers cover AC circuits and quantum mechanics, but let’s zoom out: complex numbers shine anywhere there’s oscillation or rotation—two things that show up in almost every branch of engineering and science.
其他回答提到了交流电路和量子力学，不过我们可以更宏观地看：只要存在“振动”或“旋转”，复数就能发挥作用——这两种现象几乎出现在每一个工程和科学领域。

Take mechanical engineering, for example. If you’re designing a car’s suspension, you need to model how the springs and shock absorbers react to bumps. A spring doesn’t just “bounce”—it oscillates (moves up and down) with a specific frequency. To calculate how that oscillation dies down (damping), engineers use complex numbers. Here’s why: the math for damped oscillation involves a differential equation, and its solutions are complex numbers. The “real part” tells you how far the spring moves; the “imaginary part” tells you how fast the oscillation fades.
以机械工程为例，设计汽车悬架时，需要模拟弹簧和减震器在遇到颠簸时的反应。弹簧不只是“弹跳”——它会以特定频率振动（上下移动）。要计算这种振动如何减弱（阻尼效应），工程师就会用到复数。原因如下：阻尼振动的数学模型涉及微分方程，而该方程的解是复数——实部表示弹簧的位移，虚部表示振动减弱的速度。

Or consider aerospace engineering: when designing a plane’s wings, engineers need to model how air flows around them (fluid dynamics). Airflow has “vortices” (tiny whirlpools) that rotate— and rotation is exactly what complex numbers describe well (remember: multiplying a complex number by $i$ is a 90-degree rotation). Using complex analysis, engineers can calculate how these vortices affect lift and drag—critical for making sure the plane stays in the air.
再看航空航天工程：设计飞机机翼时，工程师需要模拟空气绕机翼的流动（流体力学）。气流中会形成旋转的“涡流”（微小的漩涡）——而旋转正是复数擅长描述的现象（还记得吗？一个复数乘以 $i$ 相当于旋转90度）。通过复分析，工程师能计算这些涡流对升力和阻力的影响——这对确保飞机能在空中飞行至关重要。

The key pattern here: complex numbers don’t just “simplify math”—they let us model two related things at once (like oscillation amplitude + damping, or rotation speed + direction) that real numbers can’t handle in a single value. That’s why they’re non-negotiable for designing things that move, vibrate, or flow—from cars to planes to bridges.
这里的核心规律是：复数不只是“简化数学计算”——它们能让我们同时描述两个相关的物理量（比如振动振幅与阻尼、旋转速度与方向），而实数无法用单一数值同时表示这两个量。因此，在设计汽车、飞机、桥梁等涉及运动、振动或流动的物体时，复数是必不可少的工具。

– cowboydan
Commented Aug 30, 2016 at 16:25

I know this is an old post, but I use complex numbers in cubic spline interpolation for oscilloscope displays. If you have an oscilloscope showing a signal with few data points, you need to “fill in the gaps” to make the wave look smooth. Cubic splines use complex numbers to calculate how the curve bends between points—especially important for high-frequency signals where jagged edges would hide details.
我知道这是个老帖子，但我在示波器显示的三次样条插值中用到了复数。如果示波器显示的信号数据点很少，就需要“填补空白”让波形看起来平滑。三次样条用复数计算曲线在点之间的弯曲方式——对于高频信号尤其重要，因为锯齿状的边缘会掩盖细节。

– KenWSmith
Commented Feb 6, 2018 at 19:28

I want to back up orion’s point: complex numbers reveal that exponential and trigonometric functions are part of the same family. Euler’s formula ($e^{i\theta }=\cos \theta +i\sin \theta$) isn’t just a trick—it’s a bridge. That’s why Fourier transforms (which turn signals into frequency components) are so much easier with complex numbers: you’re not switching between two separate math systems (exponentials vs. trigonometry)—you’re using one.
我想支持orion的观点：复数揭示了指数函数和三角函数其实属于“同一家族”。欧拉公式（$e^{i\theta }=\cos \theta +i\sin \theta$）不只是技巧——它是连接两种函数的桥梁。这也是为什么用复数进行傅里叶变换（将信号分解为频率分量）会简单得多：你无需在两种独立的数学体系（指数与三角）之间切换，只用一种体系即可。

You said you’re not looking for “making problems easier,” but let’s be clear: “easier” often means “possible.” For example, designing a Wi-Fi router’s antenna requires calculating how radio waves travel through walls and around objects (electromagnetism). Without complex numbers, this calculation would be so messy it’s practically impossible—you’d have to track hundreds of separate trigonometric terms instead of a few complex ones.
你说不想了解复数“如何简化问题”，但需要明确的是：“简化”往往意味着“可行”。比如设计Wi-Fi路由器天线时，需要计算无线电波如何穿过墙壁、绕过物体传播（电磁学问题）。如果没有复数，这个计算会极其繁琐，几乎无法完成——你需要跟踪数百个独立的三角函数项，而不是几个复数。

But let’s get to the “end result” you care about: when you use Wi-Fi, the router’s antenna is positioned based on complex analysis. The signal’s strength, the direction it’s beamed, even how it avoids interference from other devices—all of that depends on calculating the complex “propagation” of radio waves. You don’t see the complex numbers, but the router’s placement (and whether your phone gets a signal) is their real-world output.
但我们来看看你关心的“最终结果”：当你使用Wi-Fi时，路由器天线的位置是基于复分析确定的。信号强度、传播方向，甚至如何避免其他设备的干扰——这些都依赖于对无线电波复“传播特性”的计算。你看不到复数，但路由器的摆放位置（以及你的手机能否接收到信号）就是复数在现实中的应用成果。

Another example: medical imaging, like MRI machines. MRIs use radio waves to interact with hydrogen atoms in your body. The atoms emit signals that the machine detects—and those signals are analyzed using complex numbers. The “image” you see of your brain or bones is created by converting complex signal data into real-world pixels. Without complex numbers, MRIs would be blurry or unable to distinguish between healthy and damaged tissue.
再举一个例子：医学成像，比如核磁共振（MRI）。MRI利用无线电波与人体内的氢原子相互作用，氢原子会发出机器可检测的信号——这些信号需用复数分析。你看到的脑部或骨骼图像，是通过将复信号数据转换为现实中的像素生成的。如果没有复数，MRI图像会模糊不清，无法区分健康组织和受损组织。

In short: complex numbers are the “middleman” between raw physical data (waves, signals) and the useful things we build from that data (Wi-Fi, MRIs, cars). You don’t see them, but the technology you use every day can’t exist without them.
简而言之：复数是连接原始物理数据（波、信号）与我们基于这些数据制造的实用设备（Wi-Fi、MRI、汽车）的“中间人”。你看不到它们，但你每天使用的技术都离不开它们。

– rrogers
Commented Aug 7, 2018 at 19:43

You should mention linear filter design—this is where complex numbers are non-negotiable. If you’re designing a filter for a phone’s microphone (to cancel out background noise), you need to “block” specific frequencies (like traffic or loud talking) and “pass” others (the user’s voice). This is done using a “transfer function”—which is a complex number. The transfer function tells you how much each frequency is amplified or reduced, and without complex numbers, you can’t design a filter that works accurately.
你应该提一下线性滤波器设计——这是复数必不可少的领域。设计手机麦克风的滤波器（用于消除背景噪音）时，需要“阻断”特定频率（如交通噪音、大声交谈），“放行”其他频率（用户的声音）。这一过程通过“传递函数”实现，而传递函数是复数。传递函数能告诉我们每个频率被放大或减弱的程度，没有复数，就无法设计出精确工作的滤波器。

– skyking
Commented Aug 8, 2018 at 5:36

@rrogers I agree filters are important, but let’s not overstate “non-negotiable.” You could do it with real numbers—you’d just use two separate real functions (one for amplitude, one for phase) instead of one complex function. But that’s like using two hands to catch a ball when one works—possible, but inefficient. The point is, complex numbers let you merge those two functions into one, which makes the design process feasible.
@rrogers 我同意滤波器很重要，但别夸大“必不可少”。用实数也能设计——只是需要两个独立的实函数（一个描述振幅，一个描述相位），而不是一个复函数。但这就像能用一只手接球，却要用两只手——可行，但低效。关键是，复数能将这两个函数合并为一个，让设计过程变得可行。

– rrogers
Commented Aug 8, 2018 at 14:14

@skyking For analog filters (the kind in old radios), you can’t separate amplitude and phase—they’re linked (this is called “minimum phase”). So you need complex numbers to model that link. Digital filters are different, but analog filters (still used in many devices) rely on complex numbers directly.
@skyking 对于模拟滤波器（老式收音机里的那种），你无法将振幅和相位分开——它们是关联的（这称为“最小相位”特性）。因此需要用复数建模这种关联。数字滤波器不同，但模拟滤波器（仍用于许多设备）直接依赖复数。

Two-dimensional problems involving Laplace’s equation—think heat flow, water flow, or static electricity—are solved using complex analysis, specifically “conformal mapping.”
涉及拉普拉斯方程的二维问题——比如热传导、水流或静电问题——会用复分析解决，具体来说是“保角映射”方法。

Let’s take heat flow as an example. Suppose you’re designing a computer chip and need to make sure a hot component (like the CPU) doesn’t overheat. The chip is a flat, 2D surface, and heat spreads across it in a pattern described by Laplace’s equation. Solving this equation for an irregular shape (like a chip with tiny circuits) is hard with real numbers—but conformal mapping lets you “bend” the complex plane to turn the irregular chip shape into a simple shape (like a rectangle) where the equation is easy to solve.
以热传导为例，假设你在设计计算机芯片，需要确保发热部件（如CPU）不会过热。芯片是扁平的二维表面，热量在芯片上的传播模式可用拉普拉斯方程描述。用实数求解不规则形状（比如带有微小电路的芯片）的拉普拉斯方程很困难，但保角映射能“弯曲”复平面，将不规则的芯片形状转换为简单形状（如矩形）——在矩形中，拉普拉斯方程很容易求解。

Here’s how it works: you use a complex function to map the chip’s weird shape onto a rectangle. Solve the heat flow equation for the rectangle (simple!), then map the solution back to the chip’s original shape using the same complex function. The result is a heat map that tells engineers where to place cooling fans or heat sinks—critical for preventing the chip from melting.
具体过程是：用一个复函数将芯片的不规则形状“映射”到矩形上，求解矩形的热传导方程（很简单！），再用同一个复函数将解“映射”回芯片的原始形状。最终得到的热分布图能告诉工程师哪里需要放置散热风扇或散热片——这对防止芯片熔化至关重要。

This isn’t just “math for math’s sake”: every computer chip you use (in your phone, laptop, gaming console) was designed with heat flow calculations that rely on complex numbers. The chip’s size, the placement of components, even the material it’s made of—all of these are influenced by the complex analysis of heat flow.
这不是“为了数学而数学”：你使用的每一块计算机芯片（手机、笔记本电脑、游戏主机中的芯片），其设计都离不开基于复数的热传导计算。芯片尺寸、元件布局，甚至制造材料的选择——这些都受热传导复分析的影响。

– Alexander Suraphel
Commented Feb 24, 2019 at 17:04

Why wouldn’t these be solved using vectors instead?
为什么不用向量来解决呢？

– Jam
Commented Jun 1, 2019 at 17:16

@Alexander Suraphel Complex numbers are like vectors with extra rules—specifically, multiplication. Vectors let you add, but not multiply in a way that preserves the “shape” of the problem (conformal mapping requires multiplication to keep angles the same). So vectors can describe heat flow, but they can’t “bend” the problem into a simpler shape—only complex numbers can do that.
@Alexander Suraphel 复数就像带有额外规则的向量——关键是乘法规则。向量可以相加，但无法通过乘法“保持问题的形状”（保角映射需要通过乘法保持角度不变）。因此，向量能描述热传导，但无法将问题“转换”为简单形状——只有复数能做到这一点。

– fshabashev
Commented Sep 19, 2022 at 20:57

数学上，一个共形变换（保角变换）是一个保持角度不变的映射。
共形映射是复变函数论的一个分支，它从几何的观点来研究复变函数，其通过一个解析函数把一个区域映射到另一个区域进行研究。这个性质可以将一些不规则或者不好用数学公式表达的区域边界映射成规则的或已成熟的区域边界。

Yep, complex numbers can be represented as 2x2 matrices (nagwa.com/en/explainers/152196980513). To be precise, the field of complex numbers is isomorphic to a set of 2x2 real matrices. But using matrices instead of complex numbers would make the math bulkier—you’d be writing out matrices every time instead of a single complex number. So complex numbers are a “shorthand” that’s also more intuitive for shape-based problems like conformal mapping.
没错，复数可以用2×2矩阵表示（参考nagwa.com/en/explainers/152196980513）。准确地说，复数域与一组2×2实矩阵同构。但用矩阵代替复数会让计算更繁琐——你每次都要写出完整矩阵，而不是一个简单的复数。因此，复数是一种“简写”，而且对于保角映射这类与形状相关的问题，复数也更直观。

Electrical engineering relies on complex numbers for almost every AC circuit—let’s break this down with a concrete example: a simple LED lightbulb connected to a wall outlet.
电气工程中，几乎所有交流电路都依赖复数——我们用一个具体例子拆解：一个连接到墙上插座的简单LED灯泡。

Wall outlets provide AC (alternating current), which means the current’s direction flips 60 times per second (in the US) or 50 times per second (in Europe). This flipping is an oscillation, and oscillations are easiest to model with complex numbers. Here’s how it works:
墙上的插座提供交流电（AC），这意味着电流方向每秒翻转60次（美国）或50次（欧洲）。这种翻转是一种振动，而振动用复数建模最简便，具体如下：

• The voltage from the outlet is a complex number: $V=V_0{e}^{i\omega t}$, where $V_0$ is the maximum voltage (like 120V in the US), $\omega$ is the frequency (60Hz), and $t$ is time.
  插座输出的电压是复数：$V=V_0{e}^{i\omega t}$，其中 $V_0$ 是最大电压（如美国的120伏），$\omega$ 是频率（60赫兹），$t$ 是时间。

• The LED has “impedance” (a combination of resistance and reactance, which resists current flow). Impedance is also a complex number: $Z=R+iX$, where $R$ is resistance (real part) and $X$ is reactance (imaginary part).
  LED具有“阻抗”（电阻和电抗的组合，用于阻碍电流流动），阻抗也是复数：$Z=R+iX$，其中 $R$ 是电阻（实部），$X$ 是电抗（虚部）。

• To find the current flowing through the LED, you use Ohm’s Law—for AC circuits, this is $I=V/Z$ (current = voltage / impedance). Since both $V$ and $Z$ are complex, the current $I$ is also complex.
  要计算流过LED的电流，需使用欧姆定律——交流电路的欧姆定律为 $I=V/Z$（电流=电压/阻抗）。由于 $V$ 和 $Z$ 都是复数，电流 $I$ 也是复数。

The end result? Engineers use this complex current to calculate two things:
最终结果是什么？工程师用这个复电流计算两件事：

1. The “real part” of $I$ tells them how much current actually powers the LED (this is the “useful” current that makes the bulb light up).
   $I$ 的实部表示实际为LED供电的电流（即让灯泡发光的“有效”电流）；

2. The “imaginary part” tells them how much current is “wasted” (this is the current that heats up the wires but doesn’t power the bulb).
   $I$ 的虚部表示“浪费”的电流（即让导线发热但不供电的电流）。

Without complex numbers, engineers would have to use two separate equations (one for the useful current, one for the wasted current) and constantly convert between trigonometric functions. With complex numbers, it’s one equation—simple enough to design the LED’s circuit so it doesn’t overheat or burn out.
如果没有复数，工程师需要用两个独立的方程（一个计算有效电流，一个计算浪费的电流），还需频繁转换三角函数。有了复数，只需一个方程——简单到能轻松设计LED电路，避免灯泡过热或烧毁。

Every AC device you use—refrigerators, TVs, phone chargers—relies on this exact calculation. The complex numbers stay behind the scenes, but the device’s safety and efficiency depend on them.
你使用的每一个交流设备——冰箱、电视、手机充电器——都依赖这种计算。复数虽然隐藏在幕后，但设备的安全性和效率都离不开它们。

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• 复数 | 存在意义 / 实际应用（篇 2 ）-优快云博客
  https://blog.youkuaiyun.com/u013669912/article/details/154322228

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via:

• Some older questions, e.g. here and here, have some brilliant answers.
  • soft question - Do complex numbers really exist?
    https://math.stackexchange.com/questions/154/do-complex-numbers-really-exist
  • soft question - “Where” exactly are complex numbers used “in the real world”?
    https://math.stackexchange.com/questions/285520/where-exactly-are-complex-numbers-used-in-the-real-world