📚 Complex Numbers | 复数
are essential in AQA A-level Mathematics because they allow every quadratic equation to have a solution and they provide a powerful geometric language through Argand diagrams. This revision guide covers the core skills: arithmetic in Cartesian form, conjugate and division, modulus and argument, polar form, de Moivre's theorem, roots, solving equations and loci.
复数通过引入虚数单位 i（定义为 i² = -1）扩展了实数系。它们在 AQA A-level 数学中至关重要，因为复数使每一个二次方程都有解，并通过阿尔冈图提供了一种强大的几何语言。本复习指南涵盖核心技能：代数形式的运算、共轭与除法、模与辐角、极形式、棣莫弗定理、根、解方程以及轨迹。
1. The Imaginary Unit and Complex Numbers | 虚数单位与复数
e real numbers. Here a is called the real part, Re(z), and b is called the imaginary part, Im(z).
在实数运算中，方程 x² + 1 = 0 没有解，因为任何实数的平方都是非负数。为了解决这一问题，数学家定义虚数单位 i，满足 i² = -1。复数就是形如 z = a + bi 的数，其中 a 和 b 是实数。这里 a 称为实部，记作 Re(z)；b 称为虚部，记作 Im(z)。
3 - 5i has real part 3 and imaginary part -5. The set of all complex numbers is denoted by ℂ tten as a + 0i. When b = 0, the number is purely real; when a = 0, it is purely imaginary.
例如，z = 3 - 5i 的实部为 3，虚部为 -5。所有复数构成的集合记作 ℂ，它包含所有实数，因为实数 a 可以写成 a + 0i。当 b = 0 时，该数为纯实数；当 a = 0 时，该数为纯虚数。
2. Real and Imaginary Parts; Equality | 实部、虚部与复数相等
e equal and their imaginary parts are equal: a + bi = c + di exactly when a = c and b = d.
对于复数 z = a + bi，实部为 Re(z) = a，虚部为 Im(z) = b。注意虚部是 i 的实系数 b，而不是 bi 本身。两个复数相等，当且仅当它们的实部相等且虚部相等：a + bi = c + di 恰好意味着 a = c 且 b = d。
a + bi = c + di ⇔ a = c and b = d
nowns appear on both sides, compare real and imaginary parts separately after simplifying.
这一原理常用于求解涉及复数的方程。例如，若 x + yi = 4 - 7i，则直接得到 x = 4 且 y = -7。当未知数出现在等式两边时，化简后分别比较实部和虚部即可。
3. Addition and Subtraction in Cartesian Form | 代数形式的加法与减法
(a - c) + (b - d)i. These operations follow the same commutative, associative and distributive rules as real numbers.
复数的加法和减法分别合并实部与虚部。若 z₁ = a + bi 且 z₂ = c + di，则 z₁ + z₂ = (a + c) + (b + d)i，z₁ - z₂ = (a - c) + (b - d)i。这些运算遵循与实数相同的交换律、结合律和分配律。
(a + bi) ± (c + di) = (a ± c) + (b ± d)i
i. Addition can also be interpreted geometrically as vector addition on an Argand diagram.
例如，(5 + 2i) + (3 - 7i) = (5 + 3) + (2 - 7)i = 8 - 5i。同样，(5 + 2i) - (3 - 7i) = (5 - 3) + (2 - (-7))i = 2 + 9i。加法在阿尔冈图上也可以解释为向量的加法。
4. Multiplication and Powers of i | 乘法与 i 的幂
+ bc)i. This formula is useful but it is often safer to expand step by step in exam work.
进行复数乘法时，像通常一样展开括号，并将 i² 替换为 -1。例如，(a + bi)(c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i。这个公式很有用，但在考试中逐步展开通常更稳妥。
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
en the exponent is divided by 4. For instance, i²⁰²³ = i³ = -i because 2023 = 4 × 505 + 3.
i 的幂以 4 为周期循环：i⁰ = 1，i¹ = i，i² = -1，i³ = -i，i⁴ = 1。因此任何 i 的幂都可以利用指数除以 4 的余数进行化简。例如，i²⁰²³ = i³ = -i，因为 2023 = 4 × 505 + 3。
5. Complex Conjugate and Division | 共轭复数与除法
l number. This property is the key to division because multiplying by the conjugate of the denominator turns the denominator into a real number.
复数 z = a + bi 的共轭复数为 z* = a - bi。乘积 z z* = (a + bi)(a - bi) = a² + b² 始终是非负实数。这一性质是除法的关键，因为乘以分母的共轭复数可以将分母化为实数。
nator by the conjugate of the denominator: (a + bi)/(c + di) = [(a + bi)(c - di)]/[(c + di
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