📚 Introduction to Complex Numbers | 复数入门
Complex numbers extend the real number system by introducing the imaginary unit i, defined by i² = -1. In AQA A-Level Mathematics, this topic builds the foundation for solving equations that have no real solutions and for representing numbers in a plane.
复数通过引入虚数单位 i（定义为 i² = -1）扩展了实数系。在 AQA A-Level 数学中，本主题为求解无实数解的方程以及在平面上表示数奠定了基础。
1. Why We Need Complex Numbers | 为什么需要复数
In AQA A-Level Mathematics, complex numbers first appear when a quadratic equation has a negative discriminant. For example, x² + 1 = 0 gives x² = -1, which has no real solution.
在 AQA A-Level 数学中，当一个二次方程的判别式为负时，就会遇到复数。例如，x² + 1 = 0 得到 x² = -1，这在实数范围内无解。
By defining a new number whose square is -1, we can solve these equations and extend the real number system into the complex number system.
通过定义一种平方等于 -1 的新数，我们可以求解这类方程，并将实数系扩展为复数系。
2. The Imaginary Unit i | 虚数单位 i
The imaginary unit is written as i. It is defined by the equation i² = -1.
虚数单位记作 i。它由方程 i² = -1 定义。
i = √(-1) and i² = -1
Although i is not a real number, it can be combined with real numbers using ordinary algebra.
虽然 i 不是实数，但它可以按照通常的代数规则与实数结合。
3. Complex Number Form and Terminology | 复数的形式与术语
A complex number has the standard form z = x + iy, where x and y are real numbers.
复数具有标准形式 z = x + iy，其中 x 和 y 是实数。
z = x + iy
The real part is Re(z) = x. The imaginary part is Im(z) = y, not iy. If x = 0, z is purely imaginary; if y = 0, z is real.
实部为 Re(z) = x。虚部为 Im(z) = y，而不是 iy。若 x = 0，则 z 为纯虚数；若 y = 0，则 z 为实数。
4. Equality of Complex Numbers | 复数的相等
Two complex numbers a + bi and c + di are equal exactly when their real parts are equal and their imaginary parts are equal.
两个复数 a + bi 和 c + di 相等，当且仅当它们的实部相等且虚部相等。
a + bi = c + di ⇔ a = c and b = d
This means a single complex equation can be split into two real equations, which is often used to find unknown real or imaginary parts.
这意味着一个复数方程可以拆成两个实数方程，常用于求未知实部或虚部。
5. Addition and Subtraction | 加法与减法
To add or subtract complex numbers, combine the real parts and combine the imaginary parts separately.
复数的加减法要分别合并实部和虚部。
(a + bi) ± (c + di) = (a ± c) + (b ± d)i
For example, (3 + 2i) + (1 – 5i) = 4 – 3i.
例如，(3 + 2i) + (1 – 5i) = 4 – 3i。
6. Multiplication and Powers of i | 乘法与 i 的幂
Multiplication follows the usual binomial expansion, then replace i² by -1.
乘法按照通常的二项式展开，然后将 i² 替换为 -1。
(a + bi)(c + di) = ac + adi + bci + bdi² = (ac – bd) + (ad +
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