📚 IGCSE WJEC Maths: Complex Numbers Key Points Explained | IGCSE WJEC 数学：复数考点精讲
Complex numbers extend the real number system by introducing the imaginary unit i, enabling solutions to equations that have no real answers, such as x² + 1 = 0. For IGCSE WJEC Mathematics, you are expected to understand the basic operations with complex numbers, their representation on an Argand diagram, and how to solve quadratic equations with negative discriminants. This guide covers every key point in a clear, bilingual format.
复数通过引入虚数单位 i 扩展了实数系统，使得像 x² + 1 = 0 这样没有实数解的方程得以求解。在 IGCSE WJEC 数学中，你需要掌握复数的基本运算、它们在 Argand 图上的表示，以及如何处理具有负判别式的二次方程。本指南将以清晰的中英双语形式覆盖每一个考点。
1. What are Complex Numbers? | 什么是复数？
Complex numbers are numbers that can be written 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 often denoted by ℂ. They allow us to find square roots of negative numbers and solve polynomial equations completely.
复数是可以写成 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。复数集常记作 ℂ。有了复数，我们就能够求负数的平方根，并彻底地求解多项式方程。
2. The Imaginary Unit i | 虚数单位 i
The imaginary unit i is defined by the property i² = −1. From this definition, we can derive further powers of i: i³ = i² × i = −i, i⁴ = (i²)² = 1, and then the pattern repeats every four powers. For any integer n, iⁿ can be simplified using the remainder when n is divided by 4.
虚数单位 i 由性质 i² = −1 定义。由此我们可以推导 i 的更高次幂：i³ = i² × i = −i，i⁴ = (i²)² = 1，然后这个模式每四次幂重复一次。对于任意整数 n，iⁿ 都可以通过 n 除以 4 的余数来化简。
3. Standard Form of a Complex Number | 复数的标准形式
The standard form of a complex number is z = a + bi, where a is called the real part and b is the imaginary part. Both a and b are real numbers. For example, 3 + 4i has real part 3 and imaginary part 4. A real number is just a complex number with b = 0, and a pure imaginary number has a = 0.
复数的标准形式是 z = a + bi，其中 a 称为实部，b 称为虚部。a 和 b 都是实数。例如 3 + 4i 的实部为 3，虚部为 4。实数就是虚部为 0 的复数，纯虚数则是实部为 0。
4. Real and Imaginary Parts | 实部与虚部
We often write Re(z) = a and Im(z) = b for a complex number z = a + bi. Note that the imaginary part is b itself, not bi. Identifying these parts correctly is essential when adding, subtracting, or comparing complex numbers.
对于复数 z = a + bi，我们通常记作 Re(z) = a，Im(z) = b。注意虚部是 b 本身，而不是 bi。正确识别实部和虚部对复数的加减和比较至关重要。
5. Equality of Complex Numbers | 复数的相等
Two complex numbers a + bi and c + di are equal if and only if their real parts are equal (a = c) and their imaginary parts are equal (b = d). This rule is used when solving equations involving complex unknowns, where we equate the real and imaginary parts separately to form a system of real equations.
两个复数 a + bi 和 c + di 相等当且仅当它们的实部相等 (a = c) 并且虚部相等 (b = d)。当求解含有复数未知数的方程时，我们利用这一规则，分别令实部和虚部相等，构成实方程组。
6. Addition and Subtraction | 复数的加法与减法
To add or subtract complex numbers, simply add or subtract the corresponding real parts and imaginary parts separately: (a + bi) ± (c + di) = (a ± c) + (b ± d)i. This works exactly like collecting like terms in algebra, where i is treated as a letter but with the knowledge that i² = −1 does not affect the linear combination.
对复数进行加减，只需分别对实部和虚部进行加减：(a + bi) ± (c + di) = (a ± c) + (b ± d)i。其操作与代数中的合并同类项完全一致，只是把 i 当作字母，同时知道 i² = −1 不会影响这种线性组合。
| Operation | Example | Result |
|---|---|---|
| Addition | (2 + 3i) + (5 − 2i) | 7 + i |
| Subtraction | (4 − i) − (1 + 2i) | 3 − 3i |
操作 | 示例 | 结果
加法 | (2 + 3i) + (5 − 2i) | 7 + i
减法 | (4 − i) − (1 + 2i) | 3 − 3i
7. Multiplication of Complex Numbers | 复数的乘法
Multiplication uses the distributive law (FOIL) together with i² = −1. When multiplying (a + bi)(c + di), we expand: ac + adi + bci + bdi². Replace i² by −1 to get (ac − bd) + (ad + bc)i. Always simplify the product into standard form a + bi.
复数乘法运用分配律（FOIL），并结合 i² = −1。将 (a + bi)(c + di) 展开得到：ac + adi + bci + bdi²。用 −1 替换 i²，得到 (ac − bd) + (ad + bc)i。最后结果一定要化成标准形式 a + bi。
Multiplication formula: (a + bi)(c + di) = (ac − bd) + (ad + bc)i
乘法公式：(a + bi)(c + di) = (ac − bd) + (ad + bc)i
8. Complex Conjugate | 复共轭
The complex conjugate of z = a + bi is denoted by z* (or sometimes z̅) and is defined as z* = a − bi. Conjugates reflect the number across the real axis on an Argand diagram. A key property is that multiplying a complex number by its conjugate gives a real number: z × z* = a² + b², which is the square of the modulus of z.
复数 z = a + bi 的共轭记作 z*（有时也记作 z̅），定义为 z* = a − bi。在 Argand 图上，共轭是关于实轴反射的点。一个重要性质是：一个复数乘上它的共轭得到实数：z × z* = a² + b²，这正是 z 的模的平方。
9. Division of 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 turns the denominator into a real number. For example, to calculate (3 + 2i) / (1 − i), multiply top and bottom by the conjugate (1 + i): (3 + 2i)(1 + i) / ((1 − i)(1 + i)) = (3 + 3i + 2i + 2i²) / (1 + 1) = (1 + 5i) / 2 = 0.5 + 2.5i.
要将两个复数相除，将分子和分母同时乘以分母的共轭复数，使得分母变为实数。例如，计算 (3 + 2i) / (1 − i)，用共轭 (1 + i) 同乘分子分母：分子展开得 3 + 3i + 2i + 2i² = 1 + 5i，分母得 (1)² + (1)² = 2，结果为 0.5 + 2.5i。
(a + bi) / (c + di) = ((a + bi)(c − di)) / (c² + d²)
除法公式：(a + bi) / (c + di) = ((a + bi)(c − di)) / (c² + d²)
10. 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 complex plane. It is defined as |z| = √(a² + b²). The modulus is always a non-negative real number. For example, |3 + 4i| = √(3² + 4²) = 5. The modulus satisfies |z₁z₂| = |z₁||z₂| and |z₁ / z₂| = |z₁| / |z₂|.
复数 z = a + bi 的模（绝对值）是复平面上原点到点 (a, b) 的距离，定义为 |z| = √(a² + b²)。模永远是非负实数。例如 |3 + 4i| = 5。模满足性质 |z₁z₂| = |z₁||z₂| 以及 |z₁ / z₂| = |z₁| / |z₂|。
11. Argand Diagram | Argand 图
An Argand diagram is a plot of complex numbers on a plane with horizontal real axis Re and vertical imaginary axis Im. The complex number a + bi is represented by the point (a, b) or by a position vector from the origin. This visual tool helps in understanding addition as vector addition, and multiplication as a combination of scaling and rotation (beyond IGCSE scope but useful for recognition). The conjugate is the reflection across the real axis.
Argand 图是一个在平面上绘制复数的坐标图，横轴为实轴 Re，纵轴为虚轴 Im。复数 a + bi 用点 (a, b) 或从原点出发的位置向量表示。这种可视化工具有助于理解加法即向量加法，乘法则是缩放与旋转的组合（虽然超出 IGCSE 范围，但识别它很有用）。共轭就是关于实轴的镜像反射。
12. Solving Quadratic Equations with Complex Roots | 解二次方程得到复数根
When solving a quadratic equation ax² + bx + c = 0 using the quadratic formula x = (−b ± √(b² − 4ac)) / (2a), if the discriminant Δ = b² − 4ac is negative, the roots are complex conjugates. For example, solve x² + 4x + 13 = 0. Here a = 1, b = 4, c = 13. Discriminant Δ = 16 − 52 = −36. Using i, √(−36) = 6i. So x = (−4 ± 6i) / 2 = −2 ± 3i. The two roots are −2 + 3i and −2 − 3i, a conjugate pair.
当用求根公式 x = (−b ± √(b² − 4ac)) / (2a) 解二次方程 ax² + bx + c = 0 时，如果判别式 Δ = b² − 4ac 为负，则根为一对共轭复数。例如解 x² + 4x + 13 = 0。其中 a=1, b=4, c=13，判别式 Δ = 16 − 52 = −36。引入 i，√(−36) = 6i，于是 x = (−4 ± 6i) / 2 = −2 ± 3i。两根分别为 −2 + 3i 和 −2 − 3i，是共轭对。
Whenever a quadratic with real coefficients has a complex root, its conjugate is also a root. This is a very common exam question: ‘One root of the equation is 1 + 2i, find the equation.’ The sum of roots gives the linear coefficient, and the product gives the constant term.
只要实系数的二次方程有一个复数根，它的共轭也必定是根。这是考试中常见的题目：“已知方程的一个根是 1 + 2i，求该方程。”利用根的和可得到一次项系数，根的积得到常数项。
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