Complex Numbers in IB Mathematics | IB数学：复数

📚 Complex Numbers in IB Mathematics | IB数学：复数

Complex numbers form a cornerstone of the IB Mathematics Analysis and Approaches (AA) and Applications and Interpretation (AI) syllabi. They provide a powerful extension of the real number system, enabling solutions to equations that were previously unsolvable and offering profound insights into geometry and trigonometry.

复数是IB数学分析与方法（AA）以及应用与解释（AI）课程体系中的核心内容。它作为实数系统的重要扩展，不仅能解决以往无法求解的方程，还为我们深入理解几何和三角学提供了强有力的工具。

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1. Introduction: What is a Complex Number? | 引言：什么是复数？

In IB Mathematics, a complex number is typically introduced through the imaginary unit i, defined by the property i² = −1. A general complex number is expressed in standard form as z = a + bi, where a and b are real numbers. Here, a is the real part (Re(z)) and b is the imaginary part (Im(z)).

在IB数学中，复数通常通过虚数单位 i 引入，其定义为 i² = −1。一般的复数以标准形式 z = a + bi 表示，其中 a 和 b 为实数。这里的 a 称为实部（Re(z)），b 称为虚部（Im(z)）。

For example, z = 3 + 4i is a complex number with Re(z) = 3 and Im(z) = 4. Notice that real numbers are simply complex numbers with b = 0.

例如，z = 3 + 4i 是一个复数，其实部 Re(z) = 3，虚部 Im(z) = 4。注意到实数其实就是虚部 b = 0 的复数。

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2. Equality of Complex Numbers | 复数的相等条件

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. In other words, if z₁ = a + bi and z₂ = c + di, then z₁ = z₂ implies a = c and b = d.

两个复数相等，当且仅当它们的实部和虚部分别相等。换句话说，若 z₁ = a + bi 且 z₂ = c + di，则 z₁ = z₂ 意味着 a = c 且 b = d。

This property is frequently used when solving equations involving complex numbers. Equating real and imaginary parts transforms a single complex equation into two simultaneous real equations.

这一性质在解复数方程时经常使用。通过令实部和虚部分别相等，可以将单个复数方程转化为两个联立的实数方程。

Solve: (x + y) + (x − y)i = 5 − i

Equating real parts: x + y = 5. Equating imaginary parts: x − y = −1. Solving simultaneously gives x = 2 and y = 3.

令实部相等：x + y = 5；令虚部相等：x − y = −1。联立解得 x = 2，y = 3。

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

Addition and subtraction of complex numbers are performed by adding or subtracting the real parts and the imaginary parts separately. Multiplication follows the distributive law, with i² replaced by −1. Division involves multiplying the numerator and denominator by the conjugate of the denominator.

复数的加减法分别对实部和虚部进行加减。乘法遵循分配律，并将 i² 替换为 −1。除法通过将分子分母同时乘以分母的共轭来实现。

For example, (2 + 3i) + (4 − i) = 6 + 2i, and (2 + 3i)(4 − i) = 8 − 2i + 12i − 3i² = 8 + 10i + 3 = 11 + 10i.

例如，(2 + 3i) + (4 − i) = 6 + 2i；(2 + 3i)(4 − i) = 8 − 2i + 12i − 3i² = 8 + 10i + 3 = 11 + 10i。

For division, consider (1 + i) ÷ (2 − i):

对于除法，例如 (1 + i) ÷ (2 − i)：

(1 + i)/(2 − i) = (1 + i)(2 + i) / [(2 − i)(2 + i)] = (2 + i + 2i + i²) / (4 + 1) = (1 + 3i)/5 = 1/5 + (3/5)i

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4. The Complex Conjugate | 共轭复数

The conjugate of a complex number z = a + bi, denoted as z̄, is defined as z̄ = a − bi. The conjugate is obtained by changing the sign of the imaginary part while keeping the real part unchanged.

复数 z = a + bi 的共轭复数记为 z̄，定义为 z̄ = a − bi。共轭复数的实部保持不变，虚部符号取反。

Key properties of the conjugate include:

共轭复数的重要性质包括：

• z + z̄ = 2Re(z), which is always real
• z − z̄ = 2i·Im(z), which is purely imaginary
• z·z̄ = a² + b², a non-negative real number
• Conjugation commutes with addition, subtraction, multiplication, and division

• z + z̄ = 2Re(z)，结果为实数
• z − z̄ = 2i·Im(z)，结果为纯虚数
• z·z̄ = a² + b²，为非负实数
• 共轭运算与加减乘除运算可交换顺序

These properties are essential for simplifying expressions and for finding real and imaginary parts of rational expressions.

这些性质对于化简表达式以及求有理表达式的实部和虚部至关重要。

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5. Modulus of a Complex Number | 复数的模

The modulus of z = a + bi, denoted |z|, is defined as the non-negative real number |z| = √(a² + b²). Geometrically, it represents the distance from the origin to the point (a, b) in the complex plane.

复数 z = a + bi 的模记为 |z|，定义为非负实数 |z| = √(a² + b²)。在几何上，它表示复平面中从原点到点 (a, b) 的距离。

Important properties of the modulus include:

模的重要性质包括：

• |z| ≥ 0, with equality if and only if z = 0
• |z₁·z₂| = |z₁|·|z₂|
• |z₁/z₂| = |z₁|/|z₂|, provided z₂ ≠ 0
• |z|² = z·z̄
• |z̄| = |z|

• |z| ≥ 0，当且仅当 z = 0 时取等号
• |z₁·z₂| = |z₁|·|z₂|
• |z₁/z₂| = |z₁|/|z₂|，其

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