📚 The Conjugate of a Complex Number and Division of Complex Numbers | 复数的共轭与复数除法
Complex conjugates are central to AQA A Level Mathematics, especially when simplifying quotients such as (3 + 4i)/(1 − 2i). Understanding the conjugate allows you to turn a complex denominator into a real number, which is the key step in division.
复数的共轭是 AQA A Level 数学的核心内容之一，尤其是在化简 (3 + 4i)/(1 − 2i) 这类商式时。理解共轭可以让你把复数分母化为实数，这是复数除法的关键步骤。
1. Complex Numbers Refresher | 复数基础回顾
Im(z) is the imaginary part, and i² = −1. The set of all complex numbers is denoted by C.
复数通常写成笛卡尔形式 z = x + iy，其中 x = Re(z) 是实部，y = Im(z) 是虚部，且满足 i² = −1。全体复数构成的集合记为 C。
In AQA exam questions, a final answer is usually required in the form a + bi, where a and b are real numbers.
在 AQA 考试中，最终答案通常要求写成 a + bi 的形式，其中 a 和 b 为实数。
z = x + iy, x, y ∈ R, i² = −1
2. Defining the Complex Conjugate | 共轭复数的定义
itten as z* and is defined by z* = x − iy. It changes only the sign of the imaginary part.
若 z = x + iy，则 z 的共轭复数写作 z*，定义为 z* = x − iy。它只改变虚部的符号。
Some textbooks use the notation z̄ for the conjugate, but AQA commonly uses z*. Both notations are acceptable if you are consistent.
有些教材用 z̄ 表示共轭，但 AQA 常用 z*。只要保持一致，两种记号都可以接受。
z = x + iy ⇒ z* = x − iy
3. Geometric Interpretation of the Conjugate | 共轭复数的几何意义
On an Argand diagram, z and z* are reflections of each other in the real axis. The real part stays the same, while the imaginary part changes sign.
在阿甘特图中，z 与 z* 关于实轴互为镜像。实部保持不变，虚部改变符号。
This reflection explains why conjugation preserves the modulus, so |z*| = |z|, and why arg(z*) = −arg(z) for non-real complex numbers.
这种镜像关系解释了为什么共轭运算保持模不变，即 |z*| = |z|，并且对于非实数复数，幅角满足 arg(z*) = −arg(z)。
|z*| = |z|, arg(z*) = −arg(z) (z not real)
4. Key Properties of Conjugation | 共轭运算的核心性质
Conjugation distributes over addition, subtraction, multiplication and division. This means you can conjugate a sum, product or quotient term by term.
共轭运算对加法、减法、乘法和除法都具有分配性。这意味着你可以对和、积或商逐项取共轭。
ion is that z z* is always a non-negative real number. If z = x + iy, then z z* = x² + y².
对除法最重要的性质是：z z* 总是非负实数。若 z = x + iy，则 z z* = x² + y²。
| Property | 性质 | Formula | 公式 |
|---|---|
| Conjugate of a conjugate | 共轭的共轭 | (z*)* = z |
| Conjugate of a sum | 和的共轭 | (z + w)* = z* + w* |
| Conjugate of a product | 积的共轭 | (zw)* = z* w* |
| Conjugate of a quotient | 商的共轭 | (z/w)* = z*/w*, w ≠ 0 |
| Real product | 实数乘积 | z z* = x² + y² |
| Twice real part | 两倍实部 | z + z* = 2Re(z) |
| Twice imaginary part | 两倍虚部 | z − z* = 2i Im(z) |
5. Finding Real and Imaginary Parts Using the Conjugate | 利用共轭求实部与虚部
ecover the real and imaginary parts directly from the conjugate without reading them from the original expression.
因为 z + z* = 2x 且 z − z* = 2iy，你可以直接由共轭恢复实部和虚部，而不必从原始表达式中直接读出它们。
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
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