📚 Complex Conjugates and Division of Complex Numbers | 复数的共轭与除法
Complex numbers are a cornerstone of A-Level Mathematics, and two essential skills are finding the conjugate of a complex number and dividing complex numbers written in the form x + iy. These techniques appear across AQA papers, from pure mathematics questions to applications in solving equations. In this article, we will build a solid understanding of the conjugate, explore its algebraic properties, and master division through clear, exam-focused examples.
复数是 A-Level 数学的核心内容，而两项基本技能是：求复数的共轭，以及计算形如 x + iy 的复数的除法。这些技巧在 AQA 试卷中频繁出现，从纯数学题到解方程的应用题都会涉及。在本文中，我们将牢固理解共轭的概念，探索其代数性质，并通过紧扣考点的例题掌握除法运算。
1. What is a Complex Conjugate? | 什么是复共轭？
n as z = x + iy, where x and y are real numbers and i² = -1. The conjugate of z, denoted z̄, is formed by changing the sign of the imaginary part only.
每个复数都可以写成 z = x + iy，其中 x 和 y 是实数，且 i² = -1。z 的共轭记作 z̄，只需改变虚部的符号即可得到。
z = x + iy ⇒ z̄ = x - iy
For example, if z = 3 + 4i, then z̄ = 3 - 4i. If z = -2 - 7i, then z̄ = -2 + 7i.
例如，若 z = 3 + 4i，则 z̄ = 3 - 4i；若 z = -2 - 7i，则 z̄ = -2 + 7i。
Notice that the real part remains unchanged. Some textbooks also use z* for the conjugate, but AQA uses z̄.
注意实部保持不变。有些教材也使用 z* 表示共轭，但 AQA 使用 z̄。
2. Key Properties of Conjugates | 共轭的基本性质
Conjugation behaves very nicely under algebraic operations. For any complex numbers z₁ and z₂, the following properties hold.
共轭运算在代数运算中表现非常良好。对于任意复数 z₁ 和 z₂，以下性质成立。
| Property | Formula |
| Conjugate of a sum | (z₁ + z₂)̄ = z̄₁ + z̄₂ |
| Conjugate of a difference | (z₁ − z₂)̄ = z̄₁ − z̄₂ |
| Conjugate of a product | (z₁ z₂)̄ = z̄₁ z̄₂ |
| Conjugate of a quotient | (z₁ / z₂)̄ = z̄₁ / z̄₂, if z₂ ≠ 0 |
| Conjugate of a conjugate | (z̄)̄ = z |
| Real number test | z = z̄ ⇔ z is real |
These properties mean that conjugation preserves addition, subtraction, multiplication, and division. They also form the theoretical basis for solving equations with real coefficients.
这些性质表明共轭运算保持加法、减法、乘法和除法。它们也是解实系数方程的理论基础。
3. Conjugates and Modulus | 共轭与模
The modulus of z = x + iy is defined as |z| = √x² + y². A key identity links the modulus to the conjugate:
复数 z = x + iy 的模定义为 |z| = √x² + y²。一个关键恒等式将模与共轭联系起来：
z z̄ = x² + y² = |z|²
This is because (x + iy)(x - iy) = x² - i²y² = x² + y². Since |z| ve, multiplying by the conjugate is the standard way to rationalise a complex denominator.
这是因为 (x + iy)(x - iy) = x² - i²y² = x² + y²。由于 |z|² 是非负实数，乘以其共轭是化去复数分母的标准方法。
Also note that |z₁ z₂| = |z₁| |z₂| and, provided z₂ ≠ 0, |z₁ / z₂| = |z₁| / |z₂|.
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