The General Complex Number | 一般复数

📚 The General Complex Number | 一般复数

Complex numbers extend our familiar number system by introducing a new number i whose square is −1. They arise naturally when solving quadratic equations with negative discriminants and form the foundation for many areas of advanced mathematics, physics and engineering.

复数通过在原有数系中引入一个平方等于 −1 的新数 i，将我们熟悉的数系加以扩展。在求解判别式为负的二次方程时，复数自然出现，并成为高等数学、物理和工程学中众多领域的基础。

In AQA A-Level Mathematics, you need to master the general form of a complex number, its arithmetic, its geometric interpretation on an Argand diagram, and its modulus–argument (polar) form. This article guides you through every essential idea, carefully paired with Chinese explanations for deeper understanding.

在 AQA 进阶数学考试中，你需要掌握复数的一般形式、四则运算、其在阿尔冈图上的几何意义，以及“模—辐角”（极坐标）形式。本文将逐一讲解每一个核心概念，并配以中文解释，帮助你深入理解。

1. Definition: The General Form | 定义：一般形式

A complex number is written in the general form

复数的一般形式写作

z = x + yi

where x and y are real numbers, and i satisfies i² = −1. The symbol z is conventionally used to denote a complex number. The set of all complex numbers is denoted ℂ.

其中 x 和 y 是实数，i 满足 i² = −1。通常用符号 z 表示一个复数。全体复数的集合记作 ℂ。

If y = 0, then z is a real number; if x = 0, then z is purely imaginary. Thus the real numbers are a subset of the complex numbers.

如果 y = 0，则 z 是实数；如果 x = 0，则 z 是纯虚数。因此，实数集是复数集的子集。

2. Real and Imaginary Parts; Equality | 实部与虚部；相等条件

In the general form z = x + yi, we call x the real part and y the imaginary part. We write:

在一般形式 z = x + yi 中，x 称为实部，y 称为虚部。我们记：

Re(z) = x, Im(z) = y

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. That is:

两个复数相等当且仅当它们的实部相等且虚部相等。即：

a + bi = c + di ⇔ a = c and b = d

This simple fact allows us to solve equations involving unknown complex numbers by equating real and imaginary parts separately.

这一简单性质使我们能够通过分别令实部和虚部相等，来求解含有未知复数的方程。

3. The Complex Conjugate | 共轭复数

For a complex number z = x + yi, its complex conjugate is defined as z* = x − yi. The conjugate reflects z across the real axis in the Argand diagram.

对于复数 z = x + yi，其共轭复数定义为 z* = x − yi。在阿尔冈图中，共轭复数相当于把 z 关于实轴反射。

Key properties of the conjugate include:

共轭复数的重要性质包括：

z + z* = 2x (always real)
z − z* = 2yi (always purely imaginary)
z z* = x² + y² (a non-negative real number)
(z*)* = z
(w + z)* = w* + z*
(wz)* = w* z*

z + z* = 2x（恒为实数）
z − z* = 2yi（恒为纯虚数）
z z* = x² + y²（非负实数）
(z*)* = z
(w + z)* = w* + z*
(wz)* = w* z*

The product z z* is especially important because it gives the squared modulus of z.

乘积 z z* 特别重要，因为它给出 z 的模长的平方。

4. Addition and Subtraction of Complex Numbers | 复数的加法与减法

Complex numbers are added and subtracted by combining the real parts together and the imaginary parts together:

复数的加法和减法通过将实部与实部、虚部与虚部分别合并来实现：

(a + bi) + (c + di) = (a + c) + (b + d)i

(a + bi) − (c + di) = (a − c) + (b − d)i

For example, (3 + 4i) + (2 − 7i) = 5 − 3i. These operations satisfy the usual commutative and associative laws.

例如，(3 + 4i) + (2 − 7i) = 5 − 3i。这些运算满足通常的交换律和结合律。

Geometrically, addition of complex numbers corresponds to vector addition on the Argand diagram.

在几何上，复数的加法对应阿尔冈图上的向量加法。

5. Multiplication of Complex Numbers | 复数的乘法

Multiplication uses the distributive law and the key identity i² = −1:

复数乘法使用分配律以及关键恒等式 i² = −1：

(a + bi)(c + di) = (ac − bd) + (ad + bc)i

Notice the sign change in the real part because (bi)(di) = bd i² = −bd.

注意实部中出现的符号变化，因为 (bi)(di) = bd i² = −bd。

Example: (2 + 3i)(1 − 4i) = 2 − 8i + 3i − 12i² = 2 − 5i + 12 = 14 − 5i.

例：(2 + 3i)(1 − 4i) = 2 − 8i + 3i − 12i² = 2 − 5i + 12 = 14 − 5i。

Multiplication by i rotates a complex number by 90° anticlockwise about the origin.

乘以 i 相当于把复数绕原点逆时针旋转 90°。

6. Division and Realising the Denominator | 复数的除法与“有理化分母”

To divide one complex number by another, we multiply both numerator and denominator by the conjugate of the denominator. This process is called realising the denominator, because it produces a real denominator.

要用一个复数除以另一个复数，我们将分子和分母同时乘以分母的共轭复数。这个过程称为“实数化分母”，因为它使分母变为实数。

z / w = (z × w*) / (w × w*)

Since w w* is real, the division is then straightforward. For example:

因为 w w* 是实数，之后的除法便很直接。例如：

(3 + 2i) / (1 − i) = (3 + 2i)(1 + i) / ((1 − i)(1 + i)) = (1 + 5i) / 2 = 1/2 + (5/2)i

Always write the result in the general form x + yi.

始终把结果写成一般形式 x + yi。

7. The Argand Diagram | 阿尔冈图

A complex number z = x + yi can be represented as a point (x, y) on a plane, called the Argand diagram. The horizontal axis is the real axis and the vertical axis is the imaginary axis.

复数 z = x + yi 可以表示为平面上的一个点 (x, y)，这个平面称为阿尔冈图。横轴是实轴，纵轴是虚轴。

Several complex numbers can be plotted together, and we can visualise addition as a parallelogram rule and multiplication as a combination of scaling and rotation.

我们可以在同一图上绘制多个复数，并利用平行四边形法则来可视化加法，乘法则可视为缩放与旋转的组合。

Complex number	Point / vector
z = 3 + 4i	(3, 4)
z* = 3 − 4i	(3, −4)
−z = −3 − 4i	(−3, −4)

The conjugate z* is the reflection of z in the real axis.

共轭复数 z* 是 z 关于实轴的对称点。

8. Modulus and Argument | 模与辐角

The modulus of a complex number z = x + yi, denoted |z|, is its distance from the origin:

复数 z = x + yi 的模，记作 |z|，是它到原点的距离：

|z| = √(x² + y²)

The argument, denoted arg(z), is the angle θ measured anticlockwise from the positive real axis to the line joining the origin to z. It is usually given in radians, with the principal value in the interval (−π, π].

辐角记作 arg(z)，是从正实轴逆时针旋转到原点与 z 的连线所成的角度 θ。通常用弧度表示，主值位于区间 (−π, π] 内。

We can relate these to x and y:

它们与 x、y 的关系为：

x = |z| cos θ, y = |z| sin θ, tan θ = y / x

To find the correct argument, you must consider the quadrant in which the point lies, not simply take the inverse tangent.

要得到正确的辐角，必须考虑点所在的象限，而不能简单使用反正切函数。

9. Modulus–Argument Form (Polar Form) | 模—辐角形式（极坐标形式）

Using the relationships above, any non-zero complex number can be written in modulus–argument form:

利用上述关系，任何非零复数都可以写成模—辐角形式：

z = r(cos θ + i sin θ)

where r = |z| and θ = arg(z). This is sometimes written as z = r cis θ for brevity.

其中 r = |z|，θ = arg(z)。为简洁起见，有时也写成 z = r cis θ。

For example, z = 1 + i has r = √2 and θ = π/4, so z = √2 cis(π/4).

例如，z = 1 + i 的 r = √2，θ = π/4，所以 z = √2 cis(π/4)。

This form is extremely useful for multiplication and division because moduli multiply and arguments add.

这种形式对乘法和除法非常有用，因为模相乘、辐角相加。

10. Solving Quadratic Equations with Complex Roots | 解具有复数根的二次方程

Consider a quadratic equation ax² + bx + c = 0 with real coefficients. Its discriminant Δ = b² − 4ac determines the nature of the roots. If Δ < 0, the roots are complex conjugates.

考虑实系数二次方程 ax² + bx + c = 0。其判别式 Δ = b² − 4ac 决定根的性质。若 Δ < 0，则根为一对共轭复数。

Using the quadratic formula and the fact that √(−1) = i, we can write the roots in the form p ± qi.

利用二次求根公式以及 √(−1) = i，我们可以将根写成 p ± qi 的形式。

x = [−b ± √(b² − 4ac)] / (2a)

Example: x² + 2x + 5 = 0. Here Δ = 4 − 20 = −16, so x = (−2 ± 4i) / 2 = −1 ± 2i.

例：x² + 2x + 5 = 0。此处 Δ = 4 − 20 = −16，所以 x = (−2 ± 4i) / 2 = −1 ± 2i。

Notice that the two complex roots are conjugates of each other. This is always true when the coefficients of the quadratic are real.

注意这两个复数根互为共轭。当二次方程系数为实数时，这一点总是成立。

By mastering the general form z = x + yi, its conjugate, arithmetic, and geometric representation, you lay a solid foundation for later topics such as complex roots of polynomials, Euler’s formula, and transformations in the complex plane. Practice drawing Argand diagrams and converting between Cartesian and modulus–argument forms until each step becomes automatic.

掌握一般形式 z = x + yi、共轭、四则运算及其几何表示，你便为后续主题奠定了坚实基础，例如多项式的复根、欧拉公式以及复平面上的变换。请多练习绘制阿尔冈图，并在直角坐标形式与模—辐角形式之间反复转换，直到每一步都驾轻就熟。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

The General Complex Number | 一般复数