📚 GCSE CIE Mathematics: Complex Numbers Key Points | GCSE CIE 数学：复数 考点精讲
While the core CIE IGCSE Mathematics (0580) syllabus does not explicitly cover complex numbers, this topic frequently appears in the IGCSE Additional Mathematics (0606) course and serves as an essential bridge to A-level Pure Mathematics. This article unpacks all fundamental concepts of complex numbers, from the imaginary unit to polar form, with clear bilingual explanations to support revision and deep understanding.
尽管 CIE IGCSE 数学 (0580) 大纲不直接考查复数，但在 IGCSE 附加数学 (0606) 中复数常作为重点登场，也是衔接 A-level 纯数的关键桥梁。本文从虚数单位到极坐标形式，用清晰的中英双语梳理复数全部基础概念，助力复习与深度理解。
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 a solution of the equation x² = −1. Complex numbers allow us to solve equations that have no real solutions, such as x² + 1 = 0.
复数是可以表示为 a + bi 形式的数，其中 a 和 b 是实数，i 是方程 x² = −1 的一个解。复数使我们能够求解无实数解的方程，例如 x² + 1 = 0。
The real part of a complex number is a, and the imaginary part is b. If b = 0, the number is purely real; if a = 0, the number is purely imaginary.
复数的实部是 a，虚部是 b。当 b = 0 时，该数为纯实数；当 a = 0 时，该数为纯虚数。
2. Imaginary Unit i | 虚数单位 i
The imaginary unit i is defined as i = √(−1). It follows that i² = −1, i³ = −i, i⁴ = 1, and the cycle repeats every four powers. This cyclic property is often tested in simplifying higher powers of i.
虚数单位 i 定义为 i = √(−1)。由此可得 i² = −1，i³ = −i，i⁴ = 1，每四次幂一循环。这一周期性质常用于简化 i 的高次幂。
To simplify iⁿ, divide n by 4 and use the remainder: if remainder is 0, result is 1; if 1, i; if 2, −1; if 3, −i.
化简 iⁿ 的方法是将 n 除以 4 取余数：余 0 得 1；余 1 得 i；余 2 得 −1；余 3 得 −i。
3. Complex Number Form a + bi | 复数形式 a + bi
Every complex number is written uniquely as z = a + bi, where a = Re(z) and b = Im(z). Both a and b are real numbers. This standard rectangular form makes arithmetic operations straightforward.
每个复数都可以唯一地写作 z = a + bi，其中 a = Re(z)，b = Im(z)，a 和 b 均为实数。这种标准的矩形形式使得算术运算变得直接。
For example, in z = 3 − 2i, the real part is 3 and the imaginary part is −2, not −2i. The imaginary part is always a real number.
例如，在 z = 3 − 2i 中，实部是 3，虚部是 −2，而不是 −2i。虚部总是一个实数。
4. Equality of Complex Numbers | 复数相等
Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. That is, a + bi = c + di implies a = c and b = d.
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 意味着 a = c 且 b = d。
This property is used to find unknown real numbers in equations involving complex numbers. For instance, solving (x + yi) + (2 − 3i) = 5 + i gives x + 2 = 5 and y − 3 = 1, so x = 3, y = 4.
这一性质用于求解含复数方程中的未知实数。例如，解 (x + yi) + (2 − 3i) = 5 + i 得到 x + 2 = 5 且 y − 3 = 1，因此 x = 3，y = 4。
5. Addition and Subtraction of Complex Numbers | 复数的加减法
To add or subtract complex numbers, simply combine the real parts and combine the imaginary parts separately. For (a + bi) ± (c + di) = (a ± c) + (b ± d)i.
复数的加减法只需分别合并实部和虚部。即 (a + bi) ± (c + di) = (a ± c) + (b ± d)i。
This operation follows the same commutative and associative laws as real numbers. For example, (4 + 2i) + (−1 + 5i) = 3 + 7i.
该运算遵循与实数相同的交换律和结合律。例如，(4 + 2i) + (−1 + 5i) = 3 + 7i。
6. Multiplication of Complex Numbers | 复数的乘法
Complex numbers are multiplied using the distributive law and the fact that i² = −1. For (a + bi)(c + di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i.
复数乘法利用分配律以及 i² = −1 展开。(a + bi)(c + di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i。
It is often easier to treat the multiplication as binomial expansion, then replace i² with −1 and simplify. e.g., (2 + i)(3 − 4i) = 6 − 8i + 3i − 4i² = 6 − 5i + 4 = 10 − 5i.
通常可以将乘法视为二项式展开，再将 i² 替换为 −1 并化简。例如 (2 + i)(3 − 4i) = 6 − 8i + 3i − 4i² = 6 − 5i + 4 = 10 − 5i。
7. Complex Conjugate | 共轭复数
The complex conjugate of z = a + bi is denoted as z̄ or z* and defined as a − bi. Geometrically, the conjugate is a reflection of z across the real axis.
复数 z = a + bi 的共轭记作 z̄ 或 z*，定义为 a − bi。几何上，共轭是 z 关于实轴的反射。
Key properties: z·z̄ = a² + b² (a real number), and (z₁/z₂) = (z₁·z̄₂)/(z₂·z̄₂) for division. The sum z + z̄ = 2a is purely real; the difference z − z̄ = 2bi is purely imaginary.
关键性质：z·z̄ = a² + b²（实数）；除法时 (z₁/z₂) = (z₁·z̄₂)/(z₂·z̄₂)。和 z + z̄ = 2a 为纯实数；差 z − z̄ = 2bi 为纯虚数。
8. Division of Complex Numbers | 复数的除法
To divide two complex numbers, multiply the numerator and denominator by the conjugate of the denominator. This eliminates the imaginary part from the denominator and expresses the result in the form a + bi.
两个复数相除时，将分子分母同乘分母的共轭复数。这样可消去分母中的虚部，并将结果表示为 a + bi 的形式。
For example, (3 + 2i)/(1 − i) = (3 + 2i)(1 + i)/(1 − i)(1 + i) = (3 + 3i + 2i + 2i²)/(1 − i²) = (3 + 5i − 2)/(1 + 1) = (1 + 5i)/2 = 0.5 + 2.5i.
例如，(3 + 2i)/(1 − i) = (3 + 2i)(1 + i)/(1 − i)(1 + i) = (3 + 3i + 2i + 2i²)/(1 − i²) = (3 + 5i − 2)/(1 + 1) = (1 + 5i)/2 = 0.5 + 2.5i。
9. Argand Diagram | 阿尔冈图
An Argand diagram represents complex numbers as points or vectors on a plane, with the horizontal axis as the real part and the vertical axis as the imaginary part. The number z = a + bi corresponds to the point (a, b).
阿尔冈图将复数表示为平面上的点或向量，横轴为实部，纵轴为虚部。复数 z = a + bi 对应点 (a, b)。
This visual tool helps to understand operations like addition (vector addition) and multiplication by i (rotation by 90° anticlockwise).
这一可视化工具有助于理解加法（向量加法）和乘以 i（逆时针旋转 90°）等运算。
10. Modulus and Argument | 模与辐角
The modulus of z, written |z|, is the distance from the origin to point (a, b) in the Argand diagram, given by |z| = √(a² + b²). For example, |3 − 4i| = √(9 + 16) = 5.
复数 z 的模记作 |z|，是阿尔冈图中原点到点 (a, b) 的距离，计算公式为 |z| = √(a² + b²)。例如 |3 − 4i| = √(9 + 16) = 5。
The argument of z, arg(z), is the angle θ (in radians or degrees) that the vector makes with the positive real axis, usually taken in (−π, π] or [0, 2π). tan θ = b/a, but the quadrant must be considered.
复数 z 的辐角 arg(z) 是向量与正实轴之间的夹角 θ（以弧度或度为单位），通常取 (−π, π] 或 [0, 2π)。tan θ = b/a，但需考虑象限。
11. Polar Form | 极坐标形式
A complex number can be expressed in polar form as z = r(cos θ + i sin θ), where r = |z| and θ = arg(z). This form is especially useful for multiplication, division, and finding powers and roots.
复数可用极坐标形式表示为 z = r(cos θ + i sin θ)，其中 r = |z|，θ = arg(z)。该形式尤其适用于乘除运算、求幂和开方。
Using Euler’s formula, cos θ + i sin θ = e^(iθ), so we can also write z = re^(iθ). Multiplication law: r₁e^(iθ₁)·r₂e^(iθ₂) = r₁r₂ e^(i(θ₁+θ₂)).
利用欧拉公式 cos θ + i sin θ = e^(iθ)，也可写作 z = re^(iθ)。乘法法则：r₁e^(iθ₁)·r₂e^(iθ₂) = r₁r₂ e^(i(θ₁+θ₂))。
12. Solving Quadratic Equations with Complex Roots | 解有复数根的二次方程
When the discriminant (b² − 4ac) of a quadratic equation ax² + bx + c = 0 is negative, the roots are complex conjugates. They can be found using the quadratic formula and simplifying √(negative number) in terms of i.
当二次方程 ax² + bx + c = 0 的判别式 (b² − 4ac) 为负时，其根为共轭复数。可利用求根公式，并将负数的平方根用 i 表示来求解。
For example, solve x² + 4x + 13 = 0: discriminant = 16 − 52 = −36, so √(−36) = 6i. Thus x = (−4 ± 6i)/2 = −2 ± 3i.
例如，解 x² + 4x + 13 = 0：判别式 = 16 − 52 = −36，故 √(−36) = 6i。因此 x = (−4 ± 6i)/2 = −2 ± 3i。
These skills are directly examinable and frequently appear in questions requiring both algebraic manipulation and graphical interpretation.
这些技能是直接可考的，常出现在要求代数操作和图形解释的问题中。
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