IB Mathematics: Core Concepts of Complex Numbers | IB数学：复数核心概念梳理

📚 IB Mathematics: Core Concepts of Complex Numbers | IB数学：复数核心概念梳理

Complex numbers are a fundamental topic in IB Mathematics Analysis and Approaches (AA) HL and Applications and Interpretation (AI) HL. They extend the real number system and provide powerful tools for solving equations, analyzing periodic phenomena, and understanding two-dimensional geometry. This article consolidates the core concepts every IB student must master before sitting for exams.

复数是IB数学分析与方法（AA）高级水平和应用与解释（AI）高级水平中的基础课题。它扩展了实数系，为解方程、分析周期现象和理解二维几何提供了强大工具。本文系统梳理每位IB学生在考试前必须掌握的核心概念。

1. The Imaginary Unit i | 虚数单位 i

The imaginary unit is defined as i² = −1. It is the fundamental building block of all complex numbers. This definition allows us to take square roots of negative numbers and solve equations such as x² + 1 = 0, which has no real solutions.

虚数单位定义为 i² = −1。它是所有复数的基础构件。这一定义使我们能够对负数开平方，并求解 x² + 1 = 0 之类在实数范围内无解的方程。

Key properties of powers of i:

i 的幂次的关键性质：

i¹ = i, i² = −1, i³ = −i, i⁴ = 1
The powers of i cycle every four terms: i^n = i^(n mod 4)
For a > 0, √(−a) = i√a

For any integer n, i^n cycles through four values. To simplify i^n, divide n by 4 and take the remainder r; then i^n = i^r where r ∈ {0, 1, 2, 3}. For instance, i^27 = i^(24+3) = i³ = −i.

对任意整数 n，i^n 以四个值为周期循环。化简 i^n 时，将 n 除以 4 取余数 r，则 i^n = i^r，其中 r ∈ {0, 1, 2, 3}。例如，i^27 = i^(24+3) = i³ = −i。

2. Definition of Complex Numbers | 复数的定义

A complex number z is written as z = a + bi, where a and b are real numbers. Here, a is called the real part and b is called the imaginary part. We write Re(z) = a and Im(z) = b. Note that the imaginary part is the coefficient of i, not the product bi itself.

复数 z 写作 z = a + bi，其中 a 和 b 是实数。a 称为实部，b 称为虚部。我们记 Re(z) = a，Im(z) = b。注意虚部是 i 的系数，而不是 bi 这个整体。

The set of complex numbers is denoted ℂ. Real numbers are a subset of complex numbers (when b = 0), and purely imaginary numbers have a = 0. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal: if a + bi = c + di, then a = c and b = d.

复数集记作 ℂ。实数集是复数集的子集（当 b = 0 时），纯虚数则满足 a = 0。两个复数相等当且仅当它们的实部相等且虚部相等：若 a + bi = c + di，则 a = c，b = d。

3. The Complex Plane (Argand Diagram) | 复平面（阿甘图）

A complex number z = a + bi can be represented as a point (a, b) in the complex plane, also known as an Argand diagram. The horizontal axis represents the real part, and the vertical axis represents the imaginary part.

复数 z = a + bi 可表示为复平面上的点 (a, b)，复平面也称为阿甘图。横轴代表实部，纵轴代表虚部。

The real axis contains all points (a, 0), representing real numbers
The imaginary axis contains all points (0, b), representing purely imaginary numbers
The complex number z = a + bi corresponds to the position vector from the origin to (a, b)

In the complex plane, the complex number z corresponds to a position vector from the origin to (a, b). Addition of complex numbers corresponds exactly to vector addition: (a + bi) + (c + di) = (a + c) + (b + d)i. This geometric interpretation is central to many IB problems involving transformations.

在复平面中，复数 z 对应从原点到点 (a, b) 的位置向量。复数的加法与向量的加法完全对应：(a + bi) + (c + di) = (a + c) + (b + d)i。这一几何解释是许多涉及变换的IB题目的核心。

4. Modulus and Argument | 模与辐角

The modulus of z = a + bi is its distance from the origin, given by:

复数 z = a + bi 的模是它到原点的距离：

|z| = √(a² + b²)

The argument is the angle θ between the positive real axis and the vector representing z, measured counterclockwise. We write arg(z) = θ. The principal argument Arg(z) is restricted to the interval −π < θ ≤ π, which is the convention used in IB Mathematics.

辐角是正实轴与表示 z 的向量之间的夹角 θ，逆时针方向为正。记 arg(z) = θ。主辐角 Arg(z) 限定在区间 −π < θ ≤ π 内，这也是IB数学所采用的约定。

For z = a + bi, the following relations hold:

对于 z = a + bi，以下关系成立：

cos θ = a / |z|
sin θ = b / |z|
tan θ = b / a (when a ≠ 0), with the quadrant determined by the signs of a and b

When calculating the argument, always check which quadrant the point lies in. Relying on tan alone can produce a wrong angle: for z = −1 + i, tan θ = −1, but the correct principal argument is 3π/4, not −π/4.

计算辐角时，务必检查点所在的象限。仅凭 tan 可能得出错误角度：对于 z = −1 + i，tan θ = −1，但正确的主辐角是 3π/4，而非 −π/4。

5. Complex Conjugates | 共轭复数

The complex conjugate of z = a + bi is z̄ = a − bi. Geometrically, z̄ is the reflection of z across the real axis in the Argand diagram.

复数 z = a + bi 的共轭复数是 z̄ = a − bi。几何上，z̄ 是 z 在阿甘图中关于实轴的反射。

Key properties of conjugates:

共轭的关键性质：

z + z̄ = 2a = 2Re(z)
z − z̄ = 2bi
z · z̄ = a² + b² = |z|²
(z̄)̄ = z
z₁ + z₂ = z̄₁ + z̄₂
(z₁ · z₂)̄ = z̄₁ · z̄₂

The product z · z̄ is always a non-negative real number. This fact is essential for dividing complex numbers, because it allows us to eliminate the imaginary part from the denominator.

乘积 z · z̄ 始终是非负实数。这一事实对复数除法至关重要，因为它使我们能够消去分母中的虚部。

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

Given z₁ = a + bi and z₂ = c + di, the operations are defined as follows:

设 z₁ = a + bi，z₂ = c + di，运算定义如下：

Addition/Subtraction: z₁ ± z₂ = (a ± c) + (b ± d)i
Multiplication: z₁ · z₂ = (ac − bd) + (ad + bc)i
Multiplication by a real scalar: kz = ka + kbi

For division, multiply the numerator and denominator by the conjugate of the denominator to make the denominator real:

做除法时，将分子和分母同时乘以分母的共轭复数，使分母变成实数：

(a + bi) / (c + di) = [(a + bi)(c − di)] / (c² + d²)

This rationalization technique is essential in IB exams and appears in nearly every complex number problem. For example, (2 + 3i) / (1 − i) = (2 + 3i)(1 + i) / (1 + 1) = (−1 + 5i) / 2 = −1/2 + (5/2)i.

这一分母有理化技巧在IB考试中至关重要，几乎出现在每道复数题中。例如，(2 + 3i) / (1 − i) = (2 + 3i)(1 + i) / (1 + 1) = (−1 + 5i) / 2 = −1/2 + (5/2)i。

7. Polar Form | 极坐标形式

Using the modulus r = |z| and argument θ, any complex number z = a + bi can be written in polar form:

利用模 r = |z| 和辐角 θ，任意复数 z = a + bi

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IB Mathematics: Core Concepts of Complex Numbers | IB数学：复数核心概念梳理