📚 Complex Numbers: Basic Concepts and Operations | 复数基础概念与运算
Complex numbers form a crucial part of the IB mathematics curriculum, particularly for Higher Level students. They extend the real number system and provide a powerful toolkit for solving equations, modelling oscillations, and working with wave functions in physics and engineering. In this article, we cover the fundamental definitions, arithmetic operations, geometric representation, and the polar form of complex numbers, with worked examples to reinforce each idea.
复数是 IB 数学课程中的重要组成部分，尤其对高级水平学生而言。它将实数系统进行扩展，为求解方程、模拟振动以及处理物理和工程中的波动函数提供了强大工具。本文将讲解复数的基本定义、算术运算、几何表示和极坐标形式，并通过示例巩固每个概念。
1. The Imaginary Unit and Complex Numbers | 虚数单位与复数
We introduce the imaginary unit i defined by the property i² = -1. This ensures that every quadratic equation, such as x² + 1 = 0, has a solution. A complex number is written as z = a + bi, where a and b are real numbers. The set of complex numbers is denoted by ℂ, which contains all real numbers (when b = 0) and pure imaginary numbers (when a = 0).
我们引入由性质 i² = -1 定义的虚数单位 i。这保证了像 x² + 1 = 0 这样的二次方程都有解。复数写作 z = a + bi，其中 a 和 b 是实数。复数集用 ℂ 表示，它包含所有实数（当 b = 0 时）和纯虚数（当 a = 0 时）。
i² = -1, z = a + bi, a, b ∈ ℝ
2. Real and Imaginary Parts, Equality | 实部、虚部与复数相等
In z = a + bi, we call a the real part, written as Re(z), and b the imaginary part, written as Im(z). Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. This property allows us to set up equations to solve for unknown variables.
在 z = a + bi 中，a 称为实部，记作 Re(z)；b 称为虚部，记作 Im(z)。两个复数相等当且仅当它们的实部相等且虚部相等。这一性质使我们能够建立方程求解未知变量。
a + bi = c + di ⇔ a = c and b = d
For example, if x + yi = 3 − 2i, then x = 3 and y = −2.
例如，若 x + yi = 3 − 2i，则 x = 3，y = −2。
3. The Argand Diagram | 复平面
We can represent a complex number z = a + bi as a point (a, b) in a coordinate plane. The horizontal axis is called the real axis, and the vertical axis is called the imaginary axis. This visualisation is known as the Argand diagram, and it gives a geometric interpretation of complex numbers.
我们可以在坐标平面中将复数 z = a + bi 表示为点 (a, b)。横轴称为实轴，纵轴称为虚轴。这种可视化表示称为复平面，它为复数提供了几何解释。
The distance from the origin to the point (a, b) is the modulus |z| = √(a² + b²), while the angle made with the positive real axis is the argument. Both will be discussed in later sections.
从原点到点 (a, b) 的距离是模 |z| = √(a² + b²)，而与正实轴所成的角是辐角。这两者将在后面的章节中讨论。
4. Addition and Subtraction | 复数的加减法
To add or subtract complex numbers, we combine the real parts and the imaginary parts separately. This is similar to adding or subtracting two-dimensional vectors.
复数的加减法是将实部与实部、虚部与虚部分别相加减，这类似于二维向量的加减运算。
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi) − (c + di) = (a − c) + (b − d)i
Example: (3 + 2i) + (4 − i) = 7 + i. Subtraction: (5 − 3i) − (1 + 4i) = 4 − 7i.
示例：(3 + 2i) + (4 − i) = 7 + i。减法：(5 − 3i) − (1 + 4i) = 4 − 7i。
5. Multiplication of Complex Numbers | 复数的乘法
Multiplication of complex numbers follows the distributive law, remembering that i² = -1. The resulting real part is ac − bd, and the imaginary part is ad + bc.
复数的乘法遵循分配律，并牢记 i² = -1。结果的实部为 ac − bd，虚部为 ad + bc。
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
Example: (2 + 3i)(1 − i) = 2 − 2i + 3i − 3i² = 2 + i + 3 = 5 + i.
例如：(2 + 3i)(1 − i) = 2 − 2i + 3i − 3i² = 2 + i + 3 = 5 + i。
Another simple case: i(1 + 2i) = i + 2i² = i − 2 = −2 + i.
另一个简单情形：i(1 + 2i) = i + 2i² = i − 2 = −2 + i。
6. Complex Conjugate | 共轭复数
The conjugate of z = a + bi is z̄ = a − bi. Geometrically, it is the reflection of z across the real axis. Conjugates play a vital role in division and in finding the modulus.
复数 z = a + bi 的共轭为 z̄ = a − bi。几何上，它是 z 关于实轴的对称点。共轭在除法和求模中起着至关重要的作用。
Key properties of conjugates:
共轭的重要性质：
- z + z̄ = 2a (always real)
- z − z̄ = 2bi (pure imaginary)
- z · z̄ = a² + b² = |z|² (a non-negative real number)
- conjugate of a sum: (w + z)̄ = w̄ + z̄
- conjugate of a product: (w z)̄ = w̄ z̄
- z + z̄ = 2a（恒为实数）
- z − z̄ = 2bi（纯虚数）
- z · z̄ = a² + b² = |z|²（非负实数）
- 和的共轭：(w + z)̄ = w̄ + z̄
- 积的共轭：(w z)̄ = w̄ z̄
7. Division of Complex Numbers | 复数的除法
To divide one complex number by another, we multiply the numerator and denominator by the conjugate of the denominator. This eliminates the imaginary part in the denominator and produces a real denominator.
复数除法通过将分子分母同时乘以分母的共轭来消去分母中的虚部，从而得到实数分母。
(a + bi) / (c + di) = ((a + bi)(c − di)) / (c² + d²)
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