Edexcel Maths: Complex Numbers – Key Points Explained | Edexcel 数学：复变函数 考点精讲

📚 Edexcel Maths: Complex Numbers – Key Points Explained | Edexcel 数学：复变函数 考点精讲

Complex numbers form a cornerstone of the Edexcel Further Pure Mathematics syllabus, appearing in both FP1 and FP2. They extend the real number system by introducing the imaginary unit i, enabling us to solve equations that have no real solutions and to model a wide range of geometric and algebraic problems. This article distils the essential concepts, from basic arithmetic and conjugates to polar forms, de Moivre’s theorem, roots of unity, loci, and elementary complex functions – all presented as key revision points for exam success.

复数在 Edexcel 进阶纯数学（FP1 和 FP2）中处于核心地位。通过引入虚数单位 i，复数将实数系进行了扩展，使我们能够求解没有实数解的方程，并用以描述丰富的几何与代数问题。本文提炼了考试中的必考要点，包括基本运算、共轭、极坐标形式、德莫弗定理、单位根、轨迹以及复变函数初步，帮助同学们高效备考。

1. Complex Numbers Basics | 复数基础

A complex number is written in Cartesian form as z = a + ib, where a, b ∈ ℝ and i is the imaginary unit satisfying i² = −1. The real part is Re(z) = a, and the imaginary part is Im(z) = b (note that Im(z) is a real number, it does not include the i).

复数通常用代数形式 z = a + ib 表示，其中 a 与 b 为实数，i 是虚数单位，满足 i² = −1。实部记为 Re(z) = a，虚部记为 Im(z) = b（注意 Im(z) 本身是实数，不含 i）。

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. This principle is used repeatedly when solving equations that involve real and imaginary terms.

两个复数相等当且仅当它们的实部相等且虚部相等。在解含实部和虚部的方程时，这一基本原理会被反复使用。

2. Complex Conjugate | 共轭复数

The complex conjugate of z = a + ib is denoted by z̄ (or z*), and is defined as z̄ = a − ib. Key properties include: z + z̄ = 2a = 2 Re(z) and z z̄ = a² + b² = |z|², which is always a real, non‑negative number.

复数 z = a + ib 的共轭复数记为 z̄（或 z*），定义为 z̄ = a − ib。重要性质有：z + z̄ = 2a = 2 Re(z)；z z̄ = a² + b² = |z|²，恒为非负实数。

Conjugates are essential for simplifying quotients. To divide by a complex number, multiply the numerator and denominator by the conjugate of the denominator. For example, (3+2i)/(1−i) = ((3+2i) (1+i))/((1−i)(1+i)) = (1+5i)/2.

共轭复数在简化分式时不可或缺。要除以一个复数，可将分子分母同时乘以分母的共轭。例如，(3+2i)/(1−i) = ((3+2i)(1+i))/((1−i)(1+i)) = (1+5i)/2。

3. Arithmetic of Complex Numbers | 复数的四则运算

Addition and subtraction are carried out by handling real and imaginary parts separately: (a+ib) ± (c+id) = (a±c) + i(b±d). Multiplication uses the distributive law together with i² = −1: (a+ib)(c+id) = (ac−bd) + i(ad+bc).

加减法分别合并实部与虚部：(a+ib) ± (c+id) = (a±c) + i(b±d)。乘法利用分配律并结合 i² = −1：(a+ib)(c+id) = (ac−bd) + i(ad+bc)。

For division, the trick is always to make the denominator real by multiplying top and bottom by the conjugate of the denominator, as shown in the previous section. All operations keep the result in the form x + iy.

对于除法，关键是将分母实数化：分子分母同乘分母的共轭，如上节所示。所有运算的结果都可以写成 x + iy 的形式。

4. Modulus and Argument | 模与辐角

The modulus of z = a + ib is |z| = √(a² + b²). Geometrically, it represents the distance from the origin to the point (a, b) in the complex plane. The argument, arg(z), is the angle θ measured from the positive

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