📚 Complex Numbers for IGCSE Edexcel Mathematics | IGCSE Edexcel 数学：复数 考点精讲
Complex numbers appear in the IGCSE Edexcel Further Pure Mathematics course and are a vital tool for solving equations that have no real solutions. This guide covers all the key concepts—from the imaginary unit i to de Moivre's theorem—to help you revise efficiently and tackle exam questions with confidence.
复数属于 IGCSE Edexcel 进阶纯数学（Further Pure Mathematics）的核心内容，是求解无实数解方程的关键工具。本文梳理虚数单位 i、复数运算、共轭复数、极坐标形式及棣莫弗定理等重要考点，助你高效复习、从容应考。
1. The Imaginary Unit i | 虚数单位 i
fundamental property is i² = -1. From this definition, i³ = i²·i = -i and i⁴ = (i²)² = 1.
虚数单位 i 定义为 -1 的主平方根，核心性质是 i² = -1。由此可推得 i³ = -i，i⁴ = 1。
i = √-1, i² = -1
Any real number multiple of i is called a pure imaginary number, e.g. 5i, -√2 i.
任何实数与 i 的乘积称为纯虚数，例如 5i、-√2 i。
2. Complex Numbers: Standard Form a + bi | 复数的代数形式 a + bi
is the imaginary part Im(z). If b = 0 the number is real; if a = 0 it is purely imaginary.
复数写作 z = a + bi，其中 a, b 是实数。a 称为实部 Re(z)，b 称为虚部 Im(z)。若 b = 0，该数为实数；若 a = 0，则为纯虚数。
equal if and only if both their real parts and imaginary parts are equal: a + bi = c + di ⇒ a = c and b = d.
两个复数相等当且仅当它们的实部与虚部分别相等：a + bi = c + di ⇒ a = c 且 b = d。
3. Addition and Subtraction | 加法和减法
e real parts and the imaginary parts separately: (a + bi) ± (c + di) = (a ± c) + (b ± d)i.
复数的加减，只需分别合并实部和虚部：(a + bi) ± (c + di) = (a ± c) + (b ± d)i。
For example, (3 + 2i) + (4 - 7i) = 7 - 5i, and (6 - i) - (2 + 3i) = 4 - 4i.
例如，(3 + 2i) + (4 - 7i) = 7 - 5i；(6 - i) - (2 + 3i) = 4 - 4i。
4. Multiplication | 乘法
numbers by expanding the brackets as in ordinary algebra and then simplify using i² = -1.
复数相乘时，像普通代数一样展开括号，再利用 i² = -1 进行化简。
(a + bi)(c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i
For instance, (2 + 3i)(1 - 4i) = 2 - 8i + 3i - 12i² = 2 - 5i + 12 = 14 - 5i.
例如，(2 + 3i)(1 - 4i) = 2 - 8i + 3i - 12i² = 2 - 5i + 12 = 14 - 5i。
5. Complex Conjugate | 共轭复数
The complex conjugate of z = a + bi is denoted by z̄ or z* and is defined as z̄ = a - bi. The product of a complex number and its conjugate is always a non‑negative real number: z·z̄ = a² + b².
复数 z = a + bi 的共轭记作 z̄ 或 z*，定义为 z̄ = a - bi。一个复数与其共轭的乘积总是非负实数：z·z̄ = a² + b²。
Conjugates are essential for division and for finding real solutions from complex expressions.
共轭复数在除法运算和从复式中提取实数结果时至关重要。
6. Division | 除法
To divide one complex number by another, multiply both numerator and denominator by the conjugate of the denominator. This turns the denominator into a real number.
复数相除时，将分子分母同乘以分母的共轭复数，即可将分母转化为实数。
(a + bi) / (c + di) = (a + bi)(c - di) / (c² + d²)
i) / (2 - i) = (3 + i)(2 + i) / (2² + 1²) = (6 + 3i + 2i + i²) / 5 = (5 + 5i) / 5 = 1 + i.
例如：(3 + i) / (2 - i) = (3 + i)(2 + i) / (4 + 1) = (6 + 5i + i²) / 5 = (5 + 5i) / 5 = 1 + i。
7. Solving Quadratic Equations with Real Coefficients | 实系数二次方程的复数解
ac < 0), the roots are a conjugate pair of complex numbers given by the quadratic formula:
实系数二次方程 ax² + bx + c = 0，当判别式 Δ = b² - 4ac < 0 时，其根为一对共轭复数，由求根公式给出：
x = [-b ± i√4ac - b²] / (2a)
For example, solve x² - 4x + 13 = 0: Δ = 16 - 52 = -36. The roots are x = (4 ± i√36) / 2 = 2 ± 3i.
例如，解方程 x² - 4x + 13 = 0：Δ = -36，根为 x = (4 ± 6i) / 2 = 2 ± 3i。
8. Argand Diagram | Argand 图
axis for the imaginary part. The complex number z = a + bi is plotted at the point (a, b).
Argand 图将复数表示为平面上的点或向量，其中水平轴表示实部，垂直轴表示虚部。复数 z = a + bi 对应点 (a, b)。
This visual representation makes it easy to understand addition (parallelogram law) and the geometric meaning of modulus and argument.
这种直观表示有助于理解复数的加法（平行四边形法则）以及模和辐角的几何意义。
9. Modulus and Argument | 模与辐角
The modulus of a complex number z = a + bi, written |z| he distance from the origin to the point (a, b) on the Argand diagram. It is calculated as |z| = √a² + b².
复数 z = a + bi 的模，记作 |z|，是 Argand 图中原点到点 (a, b) 的距离，计算公式为 |z| = √a² + b²。
e range (-π, π] or [0, 2π). tan θ = b/a, but careful adjustment of the quadrant is needed.
辐角 arg(z) 是该向量与正实轴之间的夹角 θ，通常以弧度为单位，主值范围为 (-π, π] 或 [0, 2π)。tan θ = b/a，但需根据象限进行调整。
For example, if z = 1 - i, then |z| = √1+1 = √2, and arg(z) = -π/4 (or 7π/4).
例如，z = 1 - i，则 |z| = √2，arg(z) = -π/4（或 7π/4）。
10. Modulus-Argument Form (Polar Form) | 极坐标形式
A complex number can be expressed in modulus-argument form using its modulus r and argument θ:
复数可以用模 r 和辐角 θ 表示为极坐标形式：
z = r(cos θ + i sin θ), where r = |z|, θ = arg(z)
This form is extremely useful for multiplication, division and raising to powers. Always ensure the argument is correct for the quadrant.
这种形式对于乘法、除法及幂运算极为便利。务必根据象限确定正确的辐角。
Example: Express 1 - √3 i in polar form. r = √1+3 = 2; tan θ = -√3, fourth quadrant ⇒ θ = -π/3, so z = 2(cos(-π/3) + i sin(-π/3)).
例如：将 1 - √3 i 表示为极坐标形式。r = 2，θ = -π/3，故 z = 2(cos(-π/3) + i sin(-π/3))。
11. Multiplication and Division in Polar Form | 极坐标形式下的乘除运算
When multiplying two complex numbers in polar form, multiply their moduli and add their arguments. When dividing, divide the moduli and subtract the arguments.
极坐标形式下，两复数相乘时，模长相乘、辐角相加；相除时，模长相除、辐角相减。
z₁z₂ = r₁r₂ [cos(θ₁+θ₂) + i sin(θ₁+θ₂)]
z₁ / z₂ = r₁/r₂ [cos(θ₁-θ₂) + i sin(θ₁-θ₂)]
π/3 + i sin π/3) and z₂ = 2(cos π/6 + i sin π/6), then z₁z₂ = 6(cos π/2 + i sin π/2) = 6i.
例如，z₁ = 3(cos π/3 + i sin π/3)，z₂ = 2(cos π/6 + i sin π/6)，则 z₁z₂ = 6(cos π/2 + i sin π/2) = 6i。
12. de Moivre’s Theorem | 棣莫弗定理
= rⁿ (cos nθ + i sin nθ). This provides a quick way to compute powers of complex numbers.
对于任意整数 n，棣莫弗定理指出：若 z = r(cos θ + i sin θ)，则 zⁿ = rⁿ (cos nθ + i sin nθ)。这为计算复数的高次幂提供了捷径。
Example: Find (1 + i)⁴. First write 1 + i in polar form: r = √2, θ = π/4. Then (1 + i)⁴ = (√2)⁴ (cos π + i sin π) = 4(-1 + 0i) = -4.
例如：计算 (1 + i)⁴。先将 1 + i 化为极坐标形式：r = √2，θ = π/4。则 (1 + i)⁴ = (√2)⁴ (cos π + i sin π) = 4(-1) = -4。
The theorem is also the basis for finding roots of complex numbers when n is a rational exponent.
该定理也是求复数方根（当 n 为有理指数时）的基础。
Published by TutorHao | Further Pure Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图