📚 Complex Numbers as 2D Vectors | 复数与二维向量的联系
At first glance, complex numbers and two-dimensional vectors seem to belong to different parts of mathematics. Complex numbers are written with an imaginary unit i, while vectors are drawn as arrows with magnitude and direction. Yet the two ideas are deeply connected. In the Argand diagram, every complex number can be viewed as a point in a plane, and every such point can be joined to the origin by a position vector. This article explores that link and shows how vector ideas such as length, direction, dot product and rotation can all be expressed through complex arithmetic.
乍看之下，复数与二维向量似乎分属数学的不同部分：复数用虚数单位 i 书写，向量则被画成具有大小和方向的箭头。然而两者之间存在深刻联系。在阿甘图中，每个复数都可以看成平面中的一个点，而每个这样的点都可以与原点相连，成为一个位置向量。本文探讨这种联系，并说明长度、方向、点积、旋转等向量概念如何用复数运算表达。
1. The Argand Diagram and Position Vector | 阿甘图与位置向量
A complex number z = a + bi can be plotted on an Argand diagram. The horizontal axis contains the real part a, and the vertical axis contains the imaginary part b. This is exactly the same as plotting the Cartesian point (a, b).
复数 z = a + bi 可以画在阿甘图上：水平轴对应实部 a，竖直轴对应虚部 b。这与在笛卡尔平面中描出点 (a, b) 完全相同。
From the origin to this point we can draw a vector. This is called the position vector of the complex number. Thus every complex number is equivalent to a 2D vector, and every 2D vector can be labelled by a unique complex number. The two objects may look different, but they represent the same geometric information: a horizontal component and a vertical component.
从原点指向该点可画出一个向量，称为这个复数的位置向量。因此，每个复数都等价于一个二维向量；反过来，每个二维向量也都可以用唯一一个复数来标记。两者看起来不同，但表达的几何信息相同：一个水平分量和一个竖直分量。
2. Components and Equality | 分量与相等
The real part of a complex number is the x-component of the vector, and the imaginary part is the y-component. If z = a + bi, then its vector form is (a, b). The table below summarises the correspondence.
复数的实部就是向量的 x 分量，虚部就是向量的 y 分量。若 z = a + bi，则它的向量形式是 (a, b)。下表总结对应关系。
| Complex number | Vector meaning |
|---|---|
| a = Re(z) | x-component |
| b = Im(z) | y-component |
| |z| = √(a² + b²) | magnitude or length |
| arg(z) = θ | direction angle from the positive x-axis |
Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. In vector language, two vectors are equal if and only if their components are equal. Both statements express the same idea: the geometric object is completely determined by its two components.
两个复数相等，当且仅当它们的实部相等且虚部相等。用向量语言来说，两个向量相等当且仅当它们的分量相等。这两种表述表达的是同一个思想：几何对象完全由其两个分量决定。
3. Addition and Subtraction | 加法与减法
Complex addition is componentwise. If z₁ = a + bi and z₂ = c + di, then their sum is z₁ + z₂ = (a + c) + (b + d)i. This is the same as adding two vectors component by component.
复数加法按分量进行。若 z₁ = a + bi，z₂ = c + di，则和为 z₁ + z₂ = (a + c) + (b + d)i。这与向量逐分量相加完全相同。
(a + bi) + (c + di) = (a + c) + (b + d)i
Geometrically, addition follows the parallelogram law. Place the tail of z₂ at the head of z₁; the vector from the origin to the final head is z₁ + z₂. It is also the diagonal of the parallelogram formed by the two vectors.
从几何上看，加法遵循平行四边形法则。将 z₂ 的起点放在 z₁ 的终点，从原点到最终终点的向量就是 z₁ + z₂，它也是由两个向量构成的平行四边形的对角线。
Subtraction is therefore the reverse operation. The difference z₁ − z₂ = (a − c) + (b − d)i is the displacement vector from z₂ to z₁. This is extremely useful in geometry: the vector from point A to point B is simply B − A.
减法因此是反向操作。差 z₁ − z₂ = (a − c) + (b − d)i 就是从 z₂ 指向 z₁ 的位移向量。这在几何中非常有用：从点 A 到点 B 的向量就是 B − A。
4. Real Scalar Multiplication | 实数标量乘法
Multiplying a complex number by a real number k changes its modulus by a factor of |k|. If k > 0, the direction is unchanged; if k < 0, the direction is reversed. This is exactly scalar multiplication of a vector.
用实数 k 乘以复数，会使其模长变为原来的 |k| 倍。若 k > 0，方向不变；若 k < 0，方向反转。这正是向量的标量乘法。
k(a + bi) = (ka) + (kb)i
If |k| > 1, the vector is stretched; if 0 < |k| < 1, the vector is compressed. Real scalar multiplication therefore represents dilation or contraction along the same line.
若 |k| > 1，向量被拉伸；若 0 < |k| < 1，向量被压缩。因此实数标量乘法表示同一直线上的伸长或收缩。
5. Modulus as Magnitude | 模作为长度
The modulus of a complex number z = a + bi is defined as |z| = √(a² + b²). This is the Euclidean length of the corresponding position vector (a, b).
复数 z = a + bi 的模定义为 |z| = √(a² + b²)，这就是对应位置向量 (a, b) 的欧氏长度。
In geometry problems involving complex numbers, the modulus is almost always the key. The distance from the origin to the point z is |z|. The distance between two points z₁ and z₂ is |z₁ − z₂|, since z₁ − z₂ is the displacement vector between them.
在涉及复数的几何题中，模常常是关键。点 z 到原点的距离是 |z|。点 z₁ 与 z₂ 之间的距离是 |z₁ − z₂|，因为 z₁ − z₂ 是它们之间的位移向量。
The triangle inequality also holds: |z₁ + z₂| ≤ |z₁| + |z₂|. This is the geometric statement that a direct path can never be longer than a path made of two intermediate vectors.
三角不等式也成立：|z₁ + z₂| ≤ |z₁| + |z₂|。这在几何上说明，直达路径不可能比经由两个中间向量的路径更长。
6. Argument as Direction | 辐角作为方向
The direction of a vector is measured by its angle from the positive real axis. For a complex number z = a + bi, this angle is called the argument and is written arg(z). It is usually measured anticlockwise.
向量的方向通过与正实轴之间的夹角来度量。对于复数 z = a + bi，这个角称为辐角，记为 arg(z)，通常按逆时针方向度量。
tan θ = b / a, with a quadrant check
The formula tan θ = b/a alone is not enough, because the same tangent value appears in two different quadrants. The quadrant must be chosen according to the signs of a and b. The principal argument is normally chosen in the interval (−π, π].
仅凭 tan θ = b/a 并不够，因为同一个正切值可能出现在两个不同象限。必须根据 a 和 b 的符号确定象限。辐角的主值通常取在区间 (−π, π] 内。
For example, z = 1 + i lies in the first quadrant, so its modulus is √2 and its argument is π/4. The vector has equal components, so it makes a 45° angle with the real axis.
例如，z = 1 + i 位于第一象限，因此它的模为 √2，辐角为 π/4。该向量的两个分量相等，所以它与实轴夹角为 45°。
7. Multiplication by i is a 90° Rotation | 乘以 i 是 90° 旋转
One of the most striking links between complex numbers and vectors appears when a complex number is multiplied by i. If z = a + bi, then iz = i(a + bi) = −b + ai. Thus the vector (a, b) is mapped to (−b, a).
复数与向量之间最引人注目的联系之一出现在乘以 i 的时候。若 z = a + bi，则 iz = i(a + bi) = −b + ai。于是向量 (a, b) 被映射到 (−b, a)。
(a, b) → (−b, a)
This is exactly an anticlockwise rotation through 90°. Repeating the operation explains the powers of i: multiplying by i² = −1 rotates by 180°, multiplying by i³ = −i rotates by 270°, and multiplying by i⁴ = 1 returns to the original direction.
这正是逆时针旋转 90°。重复该操作可以解释 i 的幂：乘以 i² = −1 就是旋转 180°，乘以 i³ = −i 就是旋转 270°，乘以 i⁴ = 1 则回到原来方向。
8. General Multiplication: Rotation and Scaling | 一般乘法：旋转与伸缩
If a complex number is written in modulus-argument form, its geometric effect on another complex number becomes clear. Let z₁ have modulus r₁ and argument θ₁, and let z₂ have modulus r₂ and argument θ₂. Write cis θ = cos θ + i sin θ. Then:
若采用模—辐角形式，一个复数对另一个复数的几何作用就变得很清楚。设 z₁ 的模为 r₁，辐角为 θ₁；z₂ 的模为 r₂，辐角为 θ₂。记 cis θ = cos θ + i sin θ。则：
(r₁ cis θ₁)(r₂ cis θ₂) = r₁ r₂ cis(θ₁ + θ₂)
Therefore, when z₂ is multiplied by z₁, its length is multiplied by r₁ and its direction is rotated by θ₁. In vector language, complex multiplication by a non-real number combines a scaling with a rotation.
因此，用 z₁ 去乘 z₂ 时，z₂ 的长度被放大 r₁ 倍，方向被旋转 θ₁。用向量语言说，乘以一个非实数等价于“缩放 + 旋转”的复合变换。
This is why complex numbers are so useful for rotations in the plane. A rotation about the origin through angle α is simply multiplication by cis α. A rotation
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