📚 Complex Numbers | 复数考点精讲
Complex numbers are a natural extension of the real number system and form an essential part of the CIE IGCSE Additional Mathematics syllabus. They arise when we try to solve equations like x² + 1 = 0, which have no real solutions. In this guide, we will walk through the core concepts: the imaginary unit i, the standard form a + bi, arithmetic operations, the complex conjugate, modulus, Argand diagram, and solving quadratic equations with complex roots. Each section presents key ideas in both English and Chinese, followed by worked‑style explanations so you can master this topic for your exam.
复数是实数系的自然延伸，也是 CIE IGCSE 附加数学考纲的重要组成部分。当我们试图求解 x² + 1 = 0 这种在实数范围内无解的方程时，复数便应运而生。本文将逐一梳理核心考点：虚数单位 i、标准形式 a + bi、四则运算、共轭复数、模、阿尔冈图以及含复根的二次方程。每个要点均配有中英文对照讲解，帮助你扎实掌握这一章节，自信应对考试。
1. Introduction to Complex Numbers | 复数简介
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit satisfying i² = –1. The set of complex numbers is denoted by ℂ. The real part is a, and the imaginary part is b (not bi). Complex numbers allow us to solve all polynomial equations, providing a complete algebraic system.
复数是可以表示为 a + bi 形式的数，其中 a 和 b 是实数，i 是满足 i² = –1 的虚数单位。复数集通常记作 ℂ。a 称为实部，b 称为虚部（注意虚部是 b，不是 bi）。复数使得所有多项式方程都有解，构成了一个完备的代数系统。
2. The Imaginary Unit i | 虚数单位 i
The imaginary unit i is defined as i = √(–1). Therefore i² = –1. Higher powers of i follow a cyclic pattern: i¹ = i, i² = –1, i³ = –i, i⁴ = 1, and then the cycle repeats every four powers. For example, i⁵ = i, i⁶ = –1, i⁷ = –i, i⁸ = 1. When simplifying expressions involving powers of i, divide the exponent by 4 and use the remainder.
虚数单位 i 定义为 i = √(–1)。因此 i² = –1。i 的高次幂遵循循环规律：i¹ = i，i² = –1，i³ = –i，i⁴ = 1，之后每四次一个循环。例如 i⁵ = i，i⁶ = –1，i⁷ = –i，i⁸ = 1。化简含 i 的高次幂时，可将指数除以 4，根据余数得到结果。
| Power of i | Simplified value |
|---|---|
| i¹ | i |
| i² | –1 |
| i³ | –i |
| i⁴ | 1 |
| iⁿ (general) | i^(n mod 4) |
3. Standard Form a + bi | 标准形式 a + bi
Any complex number can be written uniquely as z = a + bi, where a, b ∈ ℝ. The real part is Re(z) = a, the imaginary part is Im(z) = b. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal: a + bi = c + di ⇒ a = c and b = d. Zero is represented as 0 + 0i. Writing complex numbers in standard form makes addition, subtraction, and multiplication straightforward.
任何复数都可以唯一地写成 z = a + bi 的形式，其中 a、b ∈ ℝ。实部记作 Re(z) = a，虚部记作 Im(z) = b。两个复数相等当且仅当它们的实部相等且虚部相等：a + bi = c + di ⇒ a = c 且 b = d。零表示为 0 + 0i。将复数写成标准形式后，加减法和乘法都一目了然。
4. Adding and Subtracting Complex Numbers | 复数的加法和减法
To add or subtract complex numbers, simply combine the real parts and combine the imaginary parts separately. For z₁ = a + bi and z₂ = c + di: z₁ + z₂ = (a + c) + (b + d)i; z₁ – z₂ = (a – c) + (b – d)i. This operation is commutative and associative, just like real number addition. No special rules are needed – treat i as a constant while collecting like terms.
复数的加减法只需分别合并实部和虚部。对于 z₁ = a + bi 和 z₂ = c + di：z₁ + z₂ = (a + c) + (b + d)i；z₁ – z₂ = (a – c) + (b – d)i。该运算满足交换律和结合律，与实数加法一致。无需特殊规则——把 i 当作一个常数，合并同类项即可。
5. Multiplying Complex Numbers | 复数的乘法
Multiply complex numbers as you would expand binomials, using the distributive property and remembering i² = –1. For z₁ = a + bi and z₂ = c + di: z₁ z₂ = (a + bi)(c + di) = ac + adi + bci + bdi² = (ac – bd) + (ad + bc)i. This is simply applying FOIL and then simplifying the i² term. Always give the final answer in standard form a + bi.
复数乘法如同展开二项式一样，利用分配律并牢记 i² = –1。对于 z₁ = a + bi 和 z₂ = c + di：z₁ z₂ = (a + bi)(c + di) = ac + adi + bci + bdi² = (ac – bd) + (ad + bc)i。实质就是先进行 FOIL 展开，再化简 i² 项。最终答案务必写成标准形式 a + bi。
6. The Complex Conjugate | 共轭复数
The complex conjugate of z = a + bi is z* = a – bi (sometimes written as z̅). Geometrically, it is a reflection of z across the real axis on the Argand diagram. The conjugate has useful properties: z + z* = 2a (purely real), z – z* = 2bi (purely imaginary), and z z* = a² + b² (a non‑negative real number). The conjugate is especially helpful when dividing complex numbers.
复数 z = a + bi 的共轭复数记为 z* = a – bi（有时也写作 z̅）。从几何上看，它是 z 关于实轴在阿尔冈图上的镜像反射。共轭复数具有以下有用性质：z + z* = 2a（纯实数），z – z* = 2bi（纯虚数），以及 z z* = a² + b²（非负实数）。在复数除法中，共轭复数起着关键作用。
7. Dividing Complex Numbers | 复数的除法
To divide two complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator. This makes the denominator real. For example: (a + bi)/(c + di) = [(a + bi)(c – di)] / [(c + di)(c – di)] = [(ac + bd) + (bc – ad)i] / (c² + d²). Write the result as (ac + bd)/(c² + d²) + [(bc – ad)/(c² + d²)] i. Always write the final quotient in standard form.
作复数除法时，将分子分母同时乘以分母的共轭复数，使分母变为实数。例如：(a + bi)/(c + di) = [(a + bi)(c – di)] / [(c + di)(c – di)] = [(ac + bd) + (bc – ad)i] / (c² + d²)。最后将结果写成 (ac + bd)/(c² + d²) + [(bc – ad)/(c² + d²)] i 的标准形式。牢记最终答案要整理成 a + bi 格式。
8. Solving Quadratic Equations with Complex Roots | 解含复根的二次方程
When solving a quadratic equation ax² + bx + c = 0 with real coefficients, the discriminant Δ = b² – 4ac determines the nature of the roots. If Δ < 0, the roots are complex conjugates of the form p ± qi. Using the quadratic formula: x = [–b ± √(b² – 4ac)] / 2a, when Δ < 0, √Δ = √(–|Δ|) = i√|Δ|. Thus the roots are x = (–b ± i√|Δ|) / 2a. Always express the final roots as two separate complex numbers in standard form.
解实系数二次方程 ax² + bx + c = 0 时，判别式 Δ = b² – 4ac 决定根的性质。若 Δ < 0，则两根为共轭复数，形如 p ± qi。使用求根公式 x = [–b ± √(b² – 4ac)] / 2a，当 Δ < 0 时，√Δ = √(–|Δ|) = i√|Δ|。因此根为 x = (–b ± i√|Δ|) / 2a。最终要将两个根分别写成标准形式 a + bi。
x = (–b ± i√|b² – 4ac|) / 2a
9. The Modulus of a Complex Number | 复数的模
The modulus (or absolute value) of a complex number z = a + bi is the distance from the origin to the point (a, b) on the Argand diagram. It is denoted by |z| and calculated as |z| = √(a² + b²). Note that |z| is always a non‑negative real number. The modulus also equals √(z z*), connecting the concept to the complex conjugate. Modulus obeys the multiplicative property: |z₁ z₂| = |z₁|·|z₂|, and |z₁ / z₂| = |z₁| / |z₂| (for z₂ ≠ 0).
复数 z = a + bi 的模（或绝对值）是阿尔冈图上原点到点 (a, b) 的距离。记作 |z|，计算公式为 |z| = √(a² + b²)。模永远是非负实数。它同时也满足 |z| = √(z z*)，与共轭复数概念紧密相连。模运算满足乘法性质：|z₁ z₂| = |z₁|·|z₂|，以及 |z₁ / z₂| = |z₁| / |z₂|（z₂ ≠ 0）。
10. The Argand Diagram | 阿尔冈图
The Argand diagram is a plane where the horizontal axis represents the real part and the vertical axis represents the imaginary part of a complex number. The complex number a + bi is plotted as the point (a, b) or as a vector from the origin. This visualisation helps interpret addition (vector sum), subtraction, modulus, and the complex conjugate. A purely real number lies on the horizontal axis, a purely imaginary number on the vertical axis.
阿尔冈图是一个平面，其中横轴表示复数的实部，纵轴表示虚部。复数 a + bi 可绘制为点 (a, b) 或从原点出发的向量。这种图形表示有助于理解加法（向量和）、减法、模以及共轭复数。纯实数位于横轴上，纯虚数位于纵轴上。
11. Equality of Complex Numbers | 复数相等
Two complex numbers are equal exactly when their real parts are equal and their imaginary parts are equal. This simple but powerful fact is used to find unknown real values in an equation involving complex numbers. For example, if (x + 2) + (y – 3)i = 5 – 4i, then we equate Re and Im: x + 2 = 5 ⇒ x = 3, and y – 3 = –4 ⇒ y = –1. Always express both sides in standard form before comparing parts.
两个复数相等当且仅当它们的实部相等且虚部相等。这个简单而有力的事实常用来求解含有复数的方程中的未知实数。例如，若 (x + 2) + (y – 3)i = 5 – 4i，则分别令实部和虚部相等：x + 2 = 5 ⇒ x = 3，y – 3 = –4 ⇒ y = –1。比较前务必将等式两边都写成标准形式。
12. Real and Imaginary Parts – Identifying Components | 识别实部与虚部
When given an expression that is not in standard form, simplify it first. For instance, to find the real part of (3 + 2i)/(1 – i), perform the division (multiply by conjugate) to obtain a standard form, then read off Re and Im. Expressions like z z* or z + z* can directly give real numbers without explicit imaginary parts, which is a common examination trick. Always remember: Re(z) and Im(z) are real numbers.
若给出的表达式并非标准形式，应先进行化简。例如，要找出 (3 + 2i)/(1 – i) 的实部，应先做除法（乘以共轭）得到标准形式，再读取实部和虚部。像 z z* 或 z + z* 这类表达式可直接得出实数，不含显式虚部，这是考试中常见的技巧。牢记：Re(z) 和 Im(z) 均为实数。
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