虚数 | 存在性 / 精确定义 / 0 是否为虚数的争议

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注：本文为 “虚数 | 存在性 / 精确定义” 相关讨论合辑。
英文引文，机翻未校。
如有内容异常，请看原文。

Imaginary Numbers: How To Show They Exist

虚数：如何证明虚数确实存在？

One does it in exactly the same way one would show that fractions exist.
证明虚数存在的方法，与证明分数存在的方法完全一致。

Let’s look at a way to show that fractions exist. Of course, that’s something you know already; you don’t need a mathematical exposition to prove it to you. But the point of going through it is that exactly the same argument can be used to show that imaginary numbers exist. Having become convinced that the argument is a legitimate one by seeing it work in a familiar context, you should be more willing to accept it in the somewhat mysterious context of “imaginary” numbers.
让我们来看一种证明分数存在的方法。当然，你早就知道分数是存在的；你不需要一篇数学论述来向你证明这一点。但探讨它的意义在于，完全相同的论证可用于证明虚数存在。通过在熟悉的情境中看到该论证的合理性，你应该更愿意在“虚”数这种略带神秘的情境中接受它。

Argument that Fractions Exist

分数存在性的论证

Suppose the only things you knew about were the natural numbers (1, 2, 3, etc.), and you had to show that “three halves” exists. In other words, you need to show that there exists some number which, when doubled, gives you 3. You could argue as follows:
假设你只了解自然数（1、2、3 等），而你需要证明“二分之三”存在。换句话说，你需要证明存在某个数，当它乘以 2 时结果为 3。你可以按如下方式论证：

Granted, no such thing exists within the Natural Number System.
诚然，在自然数系中不存在这样的数。

However, there is a different number system in which such a thing does exist: the Rational Number System. The “numbers” in this different number system will be fractions: totally different objects from the natural numbers (they won’t represent sizes of sets; instead, they’ll represent ratios of sizes), but that doesn’t make them any less real.
然而，存在另一种数系，其中这样的数是存在的：有理数系。这个不同数系中的“数”是分数——与自然数完全不同的对象（它们不表示集合的大小，而是表示大小的比率），但这并不影响其存在的真实性。

Do fractions really exist? Yes. Do they really form a number system? Yes. Within this number system, is there a number which, when doubled, gives 3? Yes. Therefore, “three halves” exists.
分数真的存在吗？存在。它们真的构成一个数系吗？构成。在这个数系中，存在某个数乘以 2 得 3 吗？存在。因此，“ $\frac{3}{2}$ ”存在。

Validity of the Argument

论证的有效性

To see that the three key answers (in the last paragraph of the argument) really are “yes”, let’s look at the questions one by one.
为了确认这三个关键回答（论证最后一段中的）确实为“是”，我们逐一分析这些问题。

Do fractions really exist? Yes; they’re just pairs of natural numbers. (Let’s just talk about positive fractions here, to make the discussion as simple as possible and avoid having to worry about things like the denominator being zero). Pairs of natural numbers certainly exist, so fractions exist. We write such a pair by writing the first number over the second number, e.g. $\frac{a}{b}$ .
分数真的存在吗？ 存在；它们只是自然数的数对。（这里我们只讨论正分数，以尽可能简化讨论，避免分母为零等问题）。自然数的数对当然存在，所以分数存在。我们将这样的数对表示为“分子在上、分母在下”的形式，例如 $\frac{a}{b}$ 。

Do fractions really form a number system? Yes. A number system is just a collection of objects for which
分数真的构成一个数系吗？ 是的。一个数系只是满足以下条件的对象集合：

there’s a definition of what it means for two objects to be equal,
有关于“两个对象何时相等”的定义；
there is a rule for how to add two objects together and a rule for how to multiply two objects together (subtraction and division can be deduced from these, provided that all objects have corresponding negatives and some objects have corresponding reciprocals), and
有“两个对象相加”和“两个对象相乘”的规则（减法和除法可由这些规则推导得出，前提是所有对象都有对应的负数，且部分对象有对应的倒数）；
these rules for addition and multiplication satisfy the familiar properties of arithmetic, such as commutativity (order doesn’t matter), associativity (in a sum of three or more terms, it doesn’t matter which two you add first, and likewise for products), and distributivity ( $a (b + c) = ab + a c$ ).
这些加法和乘法规则满足算术的常见性质，例如交换律（运算顺序不影响结果）、结合律（三个或更多项相加时，先加哪两个不影响最终结果，乘法同理）和分配律（ $a (b + c) = ab + a c$ ）。

Roughly speaking, any collection of objects that satisfies these properties is, by definition, a number system. (Strictly speaking, some of these properties need to be stated a little more precisely, but the rough statement is quite enough for our purposes!)
大致而言，根据定义，任何满足这些性质的对象集合都是数系。（严格来说，部分性质需要更精确的表述，但大致说明对我们的目的来说已经足够！）

These properties are all satisfied by fractions. We have a definition of when two fractions are to be considered equal:
分数满足所有这些性质。我们定义了“两个分数何时被视为相等”：

$\frac{a}{b} = \frac{c}{d}$ if and only if $a d = b c$ .
$\frac{a}{b} = \frac{c}{d}$ 当且仅当 $a d = b c$ 。

We have a rule for adding two fractions:
我们有两个分数相加的规则：

$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$

and a rule for multiplying two fractions:
以及两个分数相乘的规则：

$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$ .

One can check that these rules do indeed satisfy the familiar properties of arithmetic.
可以验证，这些规则确实满足算术的常见性质。

Therefore, fractions form a number system.
因此，分数构成一个数系。

Within this number system, is there an object which, when doubled, gives 3? Yes. It is the fraction $\frac{3}{2}$ . When you double it, you get the fraction $\frac{3}{1}$ .
在这个数系中，是否存在某个对象，乘以 2 后结果为 3？ 存在。这个对象就是分数 $\frac{3}{2}$ 。将其乘以 2，得到分数 $\frac{3}{1}$ 。

Strictly speaking, $\frac{3}{1}$ is something different from the natural number 3. After all, it’s a pair of natural numbers, 3 and 1 (representing the ratio “3 to 1”), not a single natural number.
严格来说， $\frac{3}{1}$ 与自然数 3 并不相同。毕竟，它是自然数的数对（3 和 1，代表“3 比 1”的比率），而非单个自然数。

However, fractions of the form $\frac{a}{1}$ behave identically to the way ordinary natural numbers $a$ behave. They add and multiply in exactly the same way that ordinary natural numbers do:
然而，形如 $\frac{a}{1}$ 的分数，其运算行为与普通自然数 $a$ 完全一致。它们的加法和乘法运算与普通自然数的运算方式完全相同：

$\frac{a}{1} + \frac{b}{1} = \frac{a + b}{1}$
$\frac{a}{1} \cdot \frac{b}{1} = \frac{ab}{1}$ .

The “/1” just “comes along for the ride”.
这里的“/1”只是“附带的形式”，不影响运算本质。

Since numbers are just abstract concepts anyway, and since natural numbers $a$ and fractions of the form $\frac{a}{1}$ are completely identical as far as their arithmetic behaviour is concerned, it is perfectly legitimate to view them as just two different representations of the same underlying concept.
既然数本质上只是抽象概念，且自然数 $a$ 与形如 $\frac{a}{1}$ 的分数在算术行为上完全一致，那么将它们视为“同一底层概念的不同表示形式”是完全合理的。

With this in mind, we can consider the fraction $\frac{3}{1}$ (the ratio “3 to 1”) and the natural number “3” to be the same thing. This enables us to say that $\frac{3}{2}$ , when doubled, gives 3.
基于此，我们可以认为分数 $\frac{3}{1}$ （“3 比 1”的比率）与自然数“3”是同一事物。这使我们能够说： $\frac{3}{2}$ 乘以 2 的结果为 3。

This completes the argument that “three halves” exists. Of course, that’s something you knew already; it’s obvious that fractions exist. But even though you already knew that fractions exist, and didn’t need this long argument proving it, the point of going through the details of the argument is that exactly the same argument can be used to show that imaginary numbers exist.
这就完成了“二分之三存在”的论证。当然，这是你早就知道的——分数的存在性显而易见。但即便你已知道分数存在，且无需这一长串论证来证明，探讨该论证细节的意义仍在于：完全相同的论证可用于证明虚数存在。

The argument that “imaginary” numbers exist is almost word-for-word identical to the above argument. So, being convinced that the above argument is a valid one, you should be better able to accept the argument that imaginary numbers exist.
“虚”数存在性的论证与上述论证几乎逐字相同。因此，既然你确信上述论证有效，就应该更易接受“虚数存在”的论证。

Argument that Imaginary Numbers Exist

虚数存在性的论证

This argument is patterned after the above argument that fractions exist; you’ll probably find it helpful to open another window on your web browser and view the two of them side by side.
该论证模仿了上述“分数存在性”的论证；你或许会发现，在浏览器中打开另一个窗口，将两者并排放置查看会很有帮助。

The issue is the existence of the mysterious quantity “ $i$ ”, since imaginary numbers are just multiples of $i$ . In other words, we want to see that there exists some number which, when squared, gives you $- 1$ . Here is such an argument:
问题的核心是神秘量“ $i$ ”的存在性——因为虚数只是 $i$ 的倍数。换句话说，我们要证明“存在某个数，其平方结果为 $- 1$ ”。论证如下：

Granted, no such thing exists within any of the four familiar number systems (the Natural Number System, the Integers, the Rational Number System, or the Real Number System).
诚然，在四种熟悉的数系（自然数系、整数系、有理数系、实数系）中，均不存在这样的数。

However, there is a different number system in which such a thing does exist: the Complex Number System. The “numbers” in this different number system will be totally different objects from the familiar real numbers (they will in fact be pairs of real numbers), but that doesn’t make them any less real.
然而，存在另一种数系，其中这样的数是存在的：复数系。这个数系中的“数”是与熟悉的实数完全不同的对象（实际上是实数对），但这并不影响其存在的真实性。

Do complex numbers really exist? Yes. Do they really form a number system? Yes. Within this number system, is there a number which, when squared, gives $- 1$ ? Yes. Therefore, $i$ exists.
复数真的存在吗？存在。它们真的构成一个数系吗？构成。在这个数系中，存在某个数的平方为 $- 1$ 吗？存在。因此， $i$ 存在。

Validity of the Argument

论证的有效性

Do complex numbers really exist? Yes; we just define a complex number to be a pair of real numbers. Real numbers certainly exist, so pairs of them exist.
复数真的存在吗？ 存在；我们只需将复数定义为“实数对”即可。实数当然存在，因此它们的数对也存在。

Do complex numbers really form a number system? Yes. Remember that any collection of objects for which
复数真的构成一个数系吗？ 构成。请记住，任何满足以下条件的对象集合，根据定义就是一个数系：

there is a definition of what the objects are and when two objects are equal,
有“对象的定义”和“两个对象何时相等”的规则；
there is a rule for how to add two objects,
有“两个对象相加”的规则；
there is a rule for how to multiply two objects, and
有“两个对象相乘”的规则；
these rules obey familiar arithmetic laws like commutativity, associativity, and distributivity,
这些规则遵循交换律、结合律、分配律等常见算术定律。

These properties are all satisfied by complex numbers.
复数满足所有这些性质。

We have a definition of when two complex numbers are to be considered equal: they are equal if and only if they are the same pair of real numbers.
我们定义了“两个复数何时被视为相等”：当且仅当它们是相同的实数对时，这两个复数相等。

We have a rule for adding two complex numbers (which, remember, are nothing more than pairs of real numbers):
我们有两个复数相加的规则（请记住，复数无非是实数对）：

$(a, b) + (c, d) = (a + c, b + d)$

and a rule for multiplying two complex numbers:
两个复数相乘的规则为：

$\cdot (c, d) = (ac - bd, ad + bc)$

The rule for multiplication may look very strange, but there’s nothing wrong with that; one can still verify that these rules do indeed satisfy the familiar properties of arithmetic.
乘法规则可能看起来很奇怪，但这并无问题；我们仍可验证，这些规则确实满足算术的常见性质。

Therefore, complex numbers form a number system.
因此，复数构成一个数系。

Within this number system, is there an object which, when squared, gives $- 1$ ? Yes. It is the pair $(0, 1)$ . When you square it using the above rule of multiplication, you get
在这个数系中，是否存在某个对象，其平方结果为 $- 1$ ？ 存在。这个对象就是数对 $(0, 1)$ 。使用上述乘法规则对其平方，得到：

$\cdot (0, 1) = (0 \cdot 0 - 1 \cdot 1, 0 \cdot 1 + 1 \cdot 0) = (-1, 0)$ .

Strictly speaking, the complex number $(- 1, 0)$ is something different from the real number $- 1$ . After all, it’s a pair of real numbers, $- 1$ and $0$ , not a single real number.
严格来说，复数 $(- 1, 0)$ 与实数 $- 1$ 并不相同。毕竟，它是实数的数对（ $- 1$ 和 $0$ ），而非单个实数。

However, complex numbers of the form $(a, 0)$ behave identically to the way ordinary real numbers $a$ behave. They add and multiply in exactly the same way that ordinary real numbers do:
然而，形如 $(a, 0)$ 的复数，其运算行为与普通实数 $a$ 完全一致。它们的加法和乘法运算与普通实数的运算方式完全相同：

$(a, 0) + (b, 0) = (a + b, 0)$
$\cdot (b, 0) = (ab, 0)$ .

The “,0” just “comes along for the ride”.
这里的“,0”只是“附带的形式”，不影响运算本质。

Since numbers are just abstract concepts anyway, and since real numbers $a$ and complex numbers of the form $(a, 0)$ are completely identical as far as their arithmetic behaviour is concerned, it is perfectly legitimate to view them as just two different representations of the same underlying concept.
既然数本质上只是抽象概念，且实数 $a$ 与形如 $(a, 0)$ 的复数在算术行为上完全一致，那么将它们视为“同一底层概念的不同表示形式”是完全合理的。

With this in mind, we can consider the complex number $(- 1, 0)$ and the real number $- 1$ to be the same thing (this may seem a little hard to swallow, but remember it is no different from saying that the fraction $\frac{3}{1}$ and the natural number 3 are the same thing, something that we do all the time; it may be helpful to re-read the corresponding paragraph for fractions to see just how similar the two cases are).
基于此，我们可以认为复数 $(- 1, 0)$ 与实数 $- 1$ 是同一事物（这可能有点难以接受，但请记住，这与我们常说“分数 $\frac{3}{1}$ 和自然数 3 是同一事物”并无不同——我们一直都是这么做的；重新阅读分数对应的段落，看看这两种情况有多相似，或许会有所帮助）。

This enables us to say that $(0, 1)$ , when squared, gives $- 1$ . Therefore, $i$ exists; it is merely the pair of numbers $(0, 1)$ under the above rules for adding and multiplying.
这使我们能够说： $(0, 1)$ 的平方结果为 $- 1$ 。因此， $i$ 存在——在上述加法和乘法规则下，它无非就是数对 $(0, 1)$ 。

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What’s the precise meaning of imaginary number?

虚数的精确含义是什么？

The same to the title, what’s the precise meaning of imaginary number? And on the other hand, how can the imaginary number be reflected in Physics?
与标题一致，虚数的精确含义是什么？另一方面，虚数在物理学中如何体现？

edited Jul 13, 2012 at 14:22
– J. M. ain’t a mathematician

asked Jul 13, 2012 at 13:35
– mathon

What do you mean by “meaning?”
你说的“含义”具体指什么？

– Thomas Andrews
Jul 13, 2012 at 13:37

The mass of a particle with velocity $\mathbf{v}$ is given by $m_0 \sqrt{1 - \frac{v^2}{c^2}}$ (Theory of special relativity). From this follows, that tachyons (particles travelling faster than $c$ ) must have an imaginary mass, if they exist…
速度为 $\mathbf{v}$ 的粒子，其质量由公式 $m_0 \sqrt{1 - \frac{v^2}{c^2}}$ （狭义相对论）给出。由此可推出，若快子（速度超过 $c$ 的粒子）存在，则其质量必定为虚质量…

– draks …
Jul 13, 2012 at 13:56

@draks +1 but not if they cancel that by complex-valued velocities which, if you take Wick-rotations more serious than is probably healthy, indicate a particle hopping through space-time…
@draks +1（赞同），但如果快子通过复数值速度抵消了虚质量——若过于较真地看待维克旋转，这种复速度意味着粒子在时空中跳跃——情况就不同了…

– Tobias Kienzler
Commented Jul 13, 2012 at 20:41

Schrödinger equation uses complex numbers. It seems to me its use of complex numbers is essential. This is different from the use of complex numbers in electrical circuits, fluid mechanics, etc.
薛定谔方程用到了复数。在我看来，复数在该方程中的使用是必不可少的，这与复数在电路、流体力学等领域的应用不同。

– Makoto Kato
Jul 18, 2012 at 10:31

Answers

I’m unsure what you mean by the “precise” meaning of an imaginary number, but to me it seems best to talk about the prototypical example of an imaginary number, the imaginary unit $i$ , and then work from there.
我不确定你所说的虚数“精确”含义具体指什么，但对我而言，最好从虚数的典型例子——虚数单位 $i$ 入手，再逐步展开。

You can think of the imaginary unit as being a solution of the equation $X^2 + 1 = 0$ i.e. you define $i$ to be a number such that $i^2 = -1$ (note that I cannot say “the” number since it is not unique, as $- i$ also has this property). To me, this is the precise meaning of the imaginary unit. One then defines the complex numbers $\mathbb{C}$ in terms of this imaginary unit as
你可以将虚数单位视为方程 $X^2 + 1 = 0$ 的解，即定义 $i$ 为满足 $i^2 = -1$ 的数（需注意，不能称其为“唯一”的解，因为 $- i$ 也满足该性质）。对我来说，这就是虚数单位的精确含义。随后，基于该虚数单位，复数 $\mathbb{C}$ 可定义为

$\mathbb{C} = \{ a + bi \,|\, a, b \in \mathbb{R} \}$

More formally, the complex numbers are obtained by adjoining a root of $X^2 + 1$ to $\mathbb{R}$ by taking the quotient of $\mathbb{R}[X]$ by the maximal ideal $X^2 + 1)$ . Let $\mathbb{C} = \mathbb{R}[X] / (X^2 + 1)$ . Note that ${1, X\}$ is a basis for $\mathbb{C}$ and $X$ has the property that $X^2 = -1$ . From this it is seen that the map $\phi: \mathbb{C} \rightarrow \mathbb{C}$ defined by $\phi(a + bX) = a + bi$ is an isomorphism of fields.
更正式地说，复数是通过在实数集 $\mathbb{R}$ 中添加多项式 $X^2 + 1$ 的一个根得到的，具体方式是取多项式环 $\mathbb{R}[X]$ 对极大理想 $X^2 + 1)$ 的商环。设 $\mathbb{C} = \mathbb{R}[X] / (X^2 + 1)$ ，可知 ${1, X\}$ 是 $\mathbb{C}$ 的一组基，且 $X$ 满足 $X^2 = -1$ 。由此可看出，由 $\phi(a + bX) = a + bi$ 定义的映射 $\phi: \mathbb{C} \rightarrow \mathbb{C}$ 是一个域同构。

Another way of viewing the complex numbers is as follows:
看待复数的另一种方式如下：

The set $\mathbb{R}$ of matrices of the form
形如以下形式的矩阵构成的集合 $\mathbb{R}$ ：

$\begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix}$

where $\in \mathbb{R}$ , behaves exactly like the real numbers with respect to the operations of matrix addition and multiplication i.e. they are isomorphic as fields $(\mathbb{R} \cong \mathbb{R})$ . Consider the matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ . Notice that this matrix has the property that
其中 $\in \mathbb{R}$ ，该集合在矩阵加法和乘法运算下的表现与实数集完全一致，即它们作为域是同构的（ $\mathbb{R} \cong \mathbb{R}$ ）。考虑矩阵 $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ ，可发现该矩阵满足以下性质：

$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$

Setting $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ and using our identification above, we see that $i$ is a solution of the equation $X^2 + 1 = 0$ . So with this one could reasonably believe that the precise meaning of the imaginary unit $i$ is $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ . One then notices that the set of matrices $\mathbb{C}$ of the form
设 $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ ，结合上述等价性可知， $i$ 是方程 $X^2 + 1 = 0$ 的解。因此，据此可合理认为虚数单位 $i$ 的精确含义为 $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ 。随后可发现，形如以下形式的矩阵构成的集合 $\mathbb{C}$ ：

$\begin{pmatrix} a & -b \\ b & a \end{pmatrix}$

where $\in \mathbb{R}$ behave precisely like the complex number $\mathbb{C}$ i.e. they are isomorphic as fields $(\mathbb{C} \cong \mathbb{C})$ , and takes this to be the precise meaning of the complex numbers.
其中 $\in \mathbb{R}$ ，其表现与复数集 $\mathbb{C}$ 完全一致，即它们作为域是同构的（ $\mathbb{C} \cong \mathbb{C}$ ），因此可将此视为复数的精确含义。

Additionally, one could formally develop the complex numbers by defining $\mathbb{C}$ to be the set of pairs $\in \mathbb{R}^2$ such that additions and multiplication are defined by
此外，还可通过以下方式形式化构造复数：定义 $\mathbb{C}$ 为有序对 $\in \mathbb{R}^2$ 构成的集合，其中加法和乘法运算定义为

$(a, b) + (c, d) = (a + c, b + d)$

and
以及

$(a, b) (c, d) = (a c - b d, a d + b c)$

First note that subset $D$ of $\mathbb{C}$ consisting of elements of the form $(a, 0)$ behaves just like elements of $\mathbb{R}$ , so, in particular, $(1, 0)$ is the multiplicative identity of $\mathbb{C}$ . One then notices that the element $\in \mathbb{C}$ has the property that $(0, 1) (0, 1) = (- 1, 0)$ and so it is a solutions to the equation $X^2 + 1 = 0$ .
首先注意到， $\mathbb{C}$ 中形如 $(a, 0)$ 的元素构成的子集 $D$ ，其表现与实数集 $\mathbb{R}$ 的元素完全一致，因此 $(1, 0)$ 是 $\mathbb{C}$ 的乘法单位元。随后可发现，元素 $\in \mathbb{C}$ 满足 $(0, 1) (0, 1) = (- 1, 0)$ ，因此它是方程 $X^2 + 1 = 0$ 的解。

edited Sep 20, 2012 at 16:11
answered Jul 13, 2012 at 14:05
– Holdsworth88

+1: Good explanation. Your multiplicative identity is wrong though…
+1：解释得很好，但你的乘法单位元写错了…

– donroby
Commented Jul 13, 2012 at 23:28

@donroby thank you for pointing that out. I have fixed the typo.
@donroby 感谢指出这个问题，我已经修正了拼写错误。

– Holdsworth88
Commented Jul 14, 2012 at 4:58

This is a mathematician’s view.
以下是数学家视角的解读。

At heart, anything involving symmetry involves group theory. We like to represent our groups as matrices. One of the most powerful ways to study matrices is to study the eigenvalues and eigenvectors of the matrices.
本质上，任何涉及对称性的问题都与群论相关。我们倾向于将群表示为矩阵形式，而研究矩阵最有效的方法之一就是分析其特征值和特征向量。

The eigenvalues of real matrices can still be complex, however. So if we are studying matrices of symmetries in real vector spaces, we often end up “breaking up” the elements of that space into components that are complex, which has the paradoxical affect of simplifying the study of the underlying matrix.
然而，实矩阵的特征值仍可能是复数。因此，当我们研究实向量空间中的对称矩阵时，往往会将空间中的元素“分解”为复分量——这看似矛盾，却能简化对底层矩阵的研究。

The further you get into physics, the more you become interested in various types of symmetries, and there is really no way to study those symmetries without looking at complex numbers. After a while, you start to think of these complex components as parts of the underlying universe that we just don’t see because they “cancel out.”
在物理学领域研究得越深入，你就会对各类对称性越感兴趣，而研究这些对称性离不开复数。久而久之，你会逐渐将这些复分量视为宇宙底层结构的一部分——只是它们因“相互抵消”而无法被我们直接观测到。

edited Aug 7, 2012 at 14:00
answered Jul 13, 2012 at 14:01
– Thomas Andrews

Exactly，in Physics, we can view complex functions as vector fields.
确实，在物理学中，我们可以将复函数视为向量场。

– mathon
Commented Jul 14, 2012 at 7:00

Studying some LRC circuits might be a good way to get an idea of how complex numbers can be used in physics. As opposed to Quantum Mechanics, LRC circuits can be studied by the average first or second year undergraduate. Yale has a couple of pretty decent lectures on them.
研究一些 LRC 电路，或许能帮助理解复数在物理学中的应用。与量子力学不同，LRC 电路的知识普通本科大一或大二学生即可掌握，耶鲁大学有几节关于该主题的优质课程。

answered Jul 13, 2012 at 13:59
– dayar

LRC circuits don’t really explain what $i$ is. In reality, they’re just stealing the notation to represent phase as a vector in imaginary space. But the phase is very much real, and all LRC circuits can be solved using only real math. It’s just tedious to do so.
LRC 电路并不能真正解释 $i$ 是什么。实际上，它只是借用复数符号，将相位表示为虚空间中的向量——但相位本身是实数，而且所有 LRC 电路问题都能仅用实数数学解决，只是过程较为繁琐。

– Chris Cudmore
Commented Jul 13, 2012 at 21:00

In fairness, any problem that has real solutions can be solved using only real math. In fact, I claim that any problem involving complex numbers at all can be phrased in a way that only uses real numbers, and then can be solved using only real numbers.
客观来说，任何有实数解的问题都能仅用实数数学解决。事实上，我认为所有涉及复数的问题都能转化为仅含实数的表述，进而用实数数学求解。

– Chris Taylor
Commented Jul 18, 2012 at 10:18

The precise meaning of a complex number is that it is a particular complex number. It is what it is, no more, no less.
复数的精确含义就是“它是某个特定的复数”，仅此而已，不多也不少。

When one uses numbers (or other sorts of objects) to quantify things, one shouldn’t confuse the number with the thing being quantified. For example, the picture
当我们用数字（或其他对象）对事物进行量化时，不应将数字与被量化的事物混淆。例如，图形

* * * * *

is not the number “5”. It is simply a collection of asterisks. By using “5” to quantify the number of asterisks, we can better understand and work with the picture. Sometimes we can use such a picture to help us understand and work with “5”. But “5” is “5” and the picture is the picture, and you shouldn’t confuse the two.
不是数字“5”，它只是一组星号。用“5”来量化星号的数量，能帮助我们更好地理解和分析这个图形；有时也能通过这个图形辅助理解数字“5”。但“5”就是“5”，图形就是图形，二者不可混淆。

Complex numbers are often useful for quantifying a variety of things, such as various properties of waves, the impedance of a circuit, or position on a planar surface. They often can be used to simplify formulas and calculations. These are all uses of a complex number.
复数常被用于量化各类事物，如波的各种属性、电路的阻抗、平面上的位置等，还能简化公式和计算——这些都是复数的应用场景。

But what is a number like “1+3 $i$ ”? “1+3 $i$ ” is simply “1+3 $i$ ”, just like “5” is just “5”.
但像“1+3 $i$ ”这样的数究竟是什么？它就是“1+3 $i$ ”，就像“5”就是“5”一样。

answered Jul 13, 2012 at 23:10
user14972

Is 0 an imaginary number?

0 是虚数吗？

My question is due to an edit to the Wikipedia article: Imaginary number.
我的问题源于对维基百科词条《虚数》（Imaginary number）的一次编辑。

The funny thing is, I couldn’t find (in three of my old textbooks) a clear definition of an “imaginary number”. (Though they were pretty good at defining “imaginary component”, etc.)
有趣的是，我在三本旧教材中都没能找到“虚数”的明确定义——尽管它们对“虚部”等概念的定义很清晰。

I understand that the number zero lies on both the real and imaginary axes.
我知道数字 0 既在实轴上，也在虚轴上。

But is 0 both a real number and an imaginary number?
但 0 既是实数也是虚数吗？

We know certainly, that there are complex numbers that are neither purely real, nor purely imaginary. But I’ve always previously considered, that a purely imaginary number had to have a square that is a real and negative number (not just non-positive).
我们当然知道，存在既非纯实数也非纯虚数的复数。但我之前一直认为，纯虚数的平方必须是实数且为负数（而非仅仅是非正数）。

Clearly we can (re)define a real number as a complex number with an imaginary component that is zero (meaning that $0$ is a real number), but if one were to define an imaginary number as a complex number with real component zero, then that would also include $0$ among the pure imaginaries.
显然，我们可以（重新）将实数定义为虚部为 $0$ 的复数（这意味着 $0$ 是实数），但如果将虚数定义为实部为 $0$ 的复数，那么 $0$ 也会被归入纯虚数的范畴。

What is the complete and formal definition of an “imaginary number” (outside of the Wikipedia reference or anything derived from it)?
那么，“虚数”的完整正式定义是什么（不参考维基百科及其衍生内容）？

edited Jan 19, 2018 at 15:11
– Elements In Space

asked Apr 6, 2017 at 15:49
– CandidFlakes

See Complex number : “A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number.”
参见《复数》词条：“实部为 $0$ 的复数称为纯虚数，而虚部为 $0$ 的复数是实数。”

– Mauro ALLEGRANZA
Commented Apr 6, 2017 at 15:51

Here’s what wolfram says
以下是沃尔夫勒姆（Wolfram）给出的定义：

在这里插入图片描述

– kingW3
Commented Apr 6, 2017 at 15:52

I do not think this question should be down voted. It is well edited and clearly there was decent thought put into it.
我认为这个问题不应该被投反对票。它的表述很规范，显然提问者投入了足够的思考。

– user416426
Commented Apr 6, 2017 at 15:58

The downvotes are sad. The premise might seem silly, but the question is well-written and clearly thought-out. I like it.
反对票的存在很可惜。这个问题的前提或许看似简单，但表述清晰、逻辑严谨，我很认可。

– The Count
Commented Apr 6, 2017 at 16:02

And why not? Mathematics is full of similar cases. For example, the zero function is the unique function that is both even and odd.
为什么不能呢？数学中满是类似的案例。例如，零函数是唯一既为偶函数又为奇函数的函数。

– MJD
Commented Apr 6, 2017 at 16:02

在数学中，

偶函数的定义：如果对于所有 $x$ ，都有 $f (- x) = f (x)$ ，那么函数 $f (x)$ 是偶函数。
奇函数的定义：如果对于所有 $x$ ，都有 $f (- x) = - f (x)$ ，那么函数 $f (x)$ 是奇函数。

对于零函数 $f (x) = 0$ ：

满足偶函数的条件： $f (- x) = 0 = f (x)$
满足奇函数的条件： $f (- x) = 0 = - f (x)$

零函数 $f (x) = 0$ 是唯一既为偶函数又为奇函数的函数。

Answers

The Wikipedia article cites a textbook that manages to confuse the issue further:
维基百科词条引用了一本教材，但该教材的定义反而让问题更易混淆：

11.2 Complex Numbers

复数

Purely imaginary (complex) number

纯虚数（复数）

A complex number $z = x + i y$ is called a purely imaginary number if and only if $x = 0$ , i.e., $\text{Re}(z) = 0$ .
复数 $z = x + i y$ 被称为纯虚数，当且仅当 $x = 0$ ，即 $\text{Re}(z) = 0$ 。

Imaginary number

虚数

A complex number $z = x + i y$ is said to be an imaginary number if and only if $\neq 0$ , i.e., $\text{Im}(z) \neq 0$ .
复数 $z = x + i y$ 被称为虚数，当且仅当 $\neq 0$ ，即 $\text{Im}(z) \neq 0$ 。

EXAMPLES:

示例：

(i) $z_1 = -3 - 4i$ . $\hspace{3em}$ Here $\text{Re}(z_1) = -3$ and $\text{Im}(z_1) = -4$

(ii) $z_2 = -4i = 0 - 4i$ , $\hspace{0.9em}$ Here $\text{Re}(z_2) = 0$ and $\text{Im}(z_2) = -4$

(iii) $z_3 = -3 = -3 + 0i$ . $\hspace{0.1em}$ Here $\text{Re}(z_3) = -3$ and $\text{Im}(z_3) = 0$

(iv) $z_4 = 0 = 0 + i0$ . $\hspace{1.6em}$ Here $\text{Re}(z_4) = 0$ and $\text{Im}(z_4) = 0$

Note:

注意：

Every real number is a complex number.
每个实数都是复数。

0 is both purely real and purely imaginary.
0 既是纯实数也是纯虚数。

For purely imaginary numbers, the real part must be zero, while the imaginary part may be zero or non-zero.
对于纯虚数，实部必须为零，虚部可以为零也可以不为零。

Every real number is purely real, and every purely real number is real. However, an imaginary number may or may not be purely imaginary.
每个实数都是纯实数，每个纯实数都是实数，但虚数可以是纯虚数也可以不是纯虚数。

$i y$ is also written as $y i$ .
$i y$ 也可以写作 $y i$ 。

All purely imaginary numbers except zero are imaginary numbers. However, there are infinitely many imaginary numbers that are not purely imaginary. For example, $2 + 3 i$ is an imaginary number but not a purely imaginary number.
除了零以外的所有纯虚数都是虚数，但有无限多的虚数不是纯虚数。例如， $2 + 3 i$ 是虚数但不是纯虚数。

Do not think that the word “imaginary” means that these numbers are mystical, fictitious, or unreal in the usual sense of the word, or that “complex” means complicated.
不要认为“虚数”这个词意味着这些数字是神秘的、虚构的或在通常意义上是不真实的，或者“复数”意味着复杂。

Imaginary numbers do exist and have very real applications in many branches of physical science. We call them numbers for the obvious reason that the familiar operations can be defined for them and that they can be shown to satisfy the familiar laws like the commutative, distributive, and associative laws, as in the case of real numbers.
虚数确实存在，并且在许多物理科学的分支中有非常实际的应用。我们称它们为数字，显然是因为可以为它们定义熟悉的运算，并且可以证明它们满足熟悉的法则，如交换律、分配律和结合律，就像实数一样。

This is a slightly different usage of the word “imaginary”, meaning “non-real”: among the complex numbers, those that aren’t real we call imaginary, and a further subset of those (with real part $0$ ) are purely imaginary. Except that by this definition, $0$ is clearly purely imaginary but not imaginary!
这里“虚数”的用法略有不同，指代“非实数”：在复数集中，非实数的数被称为虚数，而其中实部为 $0$ 的子集被称为纯虚数。但按此定义， $0$ 显然是纯虚数，却不属于虚数范畴！

Anyway, anybody can write a textbook, so I think that the real test is this: does $0$ have the properties we want a (purely) imaginary number to have?
不过，任何人都能编写教材，因此真正的判断标准应是： $0$ 是否具备我们期望（纯）虚数应有的性质？

I can’t (and MSE can’t) think of any useful properties of purely imaginary complex numbers $z$ apart from the characterization that $e^z| = 1$ . But $0$ clearly has this property, so we should consider it purely imaginary.
除了“ $e^z| = 1$ ”这一特征外，我想不出纯虚数 $z$ 还有其他有用性质。而 $0$ 显然满足该性质（ $e^0| = |1| = 1$ ），因此应将其归为纯虚数。

(On the other hand, $0$ has all of the properties a real number should have, being real; so it makes some amount of sense to also say that it’s purely imaginary but not imaginary at the same time.)
（另一方面， $0$ 具备实数应有的所有性质，因此是实数；所以从这个角度说，“ $0$ 是纯虚数但非虚数”，在一定程度上也是合理的。）

edited Apr 13, 2017 at 12:21
answered Apr 6, 2017 at 16:09
– Misha Lavrov

I don’t think there is a
我认为并不存在一个

complete and formal definition of “imaginary number”
“虚数”的完整正式定义

It’s a useful term sometimes. It’s an author’s responsibility to make clear what he or she means in any particular context where precision matters. If $0$ should count, or not, then the text must say so.
“虚数”有时是个有用的术语，但在需要精确性的特定语境下，作者有责任明确其含义——比如需说明 $0$ 是否应被归入虚数范畴。

Your question shows clearly that you understand the structure of the complex numbers, so you should be able to make sense of any passage you encounter.
从你的问题可以看出，你已理解复数的结构，因此面对任何相关表述，都能准确解读其含义。

answered Apr 6, 2017 at 16:10
– Ethan Bolker