📚 A-Level Mathematics: Complex Numbers | 复数考点精讲
Complex numbers extend the real number system by introducing the imaginary unit i, where i2 = −1. They are essential for solving equations like x2 + 1 = 0 and underpin advanced topics in algebra, calculus, and physics. This revision guide covers all key A-Level concepts, from Cartesian form to De Moivre’s theorem and roots of polynomial equations.
复数通过引入虚数单位 i（i2 = −1）扩展了实数系统。它们是求解 x2 + 1 = 0 这类方程的基础，并支撑着代数、微积分和物理学中的高级主题。本复习指南涵盖从笛卡尔形式到棣莫弗定理、多项式方程求根等所有 A-Level 核心概念。
1. Introduction to Complex Numbers | 复数入门
A complex number is an expression of the form z = x + iy, where x and y are real numbers and i is the imaginary unit satisfying i2 = −1.
复数是一种形如 z = x + iy 的表达式，其中 x 和 y 是实数，i 是满足 i2 = −1 的虚数单位。
The real part is Re(z) = x, and the imaginary part is Im(z) = y (not iy).
实部记作 Re(z) = x，虚部记作 Im(z) = y（不是 iy）。
If y = 0, the number is purely real; if x = 0 but y ≠ 0, it is purely imaginary.
若 y = 0，该数是实数；若 x = 0 而 y ≠ 0，则为纯虚数。
Complex numbers allow us to solve any polynomial equation, including those with negative discriminants.
复数使我们能够求解任何多项式方程，包括那些判别式为负的方程。
2. The Imaginary Unit i and Cartesian Form | 虚数单位 i 与笛卡尔形式
The defining identity is i2 = −1, which leads to i3 = −i, i4 = 1, and so on.
定义恒等式为 i2 = −1，由此可得 i3 = −i, i4 = 1 等。
In Cartesian form, a complex number is written as z = a + ib, where a, b ∈ ℝ.
在笛卡尔形式中，复数写作 z = a + ib，其中 a, b ∈ ℝ。
This representation separates the number into horizontal (real) and vertical (imaginary) components, exactly like coordinates.
这种表示将数分为水平（实）分量和垂直（虚）分量，完全类似于坐标。
Manipulating powers of i is straightforward: in repeats in cycles of 4.
处理 i 的幂很直接：in 以 4 为周期循环。
3. Argand Diagram and Geometric Interpretation | Argand 图与几何解释
An Argand diagram plots the real part on the x‑axis and the imaginary part on the y‑axis, turning complex numbers into points or vectors.
Argand 图将实部画在 x 轴，虚部画在 y 轴，从而把复数变成点或向量。
The complex number z = x + iy is represented by the vector from the origin to (x, y).
复数 z = x + iy 用从原点到 (x, y) 的向量表示。
Addition of complex numbers corresponds to vector addition on the Argand diagram.
复数的加法在 Argand 图上对应向量的加法。
Subtraction can be viewed as adding the negative of the subtrahend vector.
减法可视为加上减数向量的反向向量。
4. Modulus and Argument of a Complex Number | 复数的模与辐角
The modulus of z = x + iy is |z| = √(x2 + y2), representing the distance from the origin.
复数 z = x + iy 的模是 |z| = √(x2 + y2)，表示到原点的距离。
The argument, denoted arg(z), is the angle θ between the positive real axis and the vector, measured in radians. It is usually taken in (−π, π] or [0, 2π).
辐角记作 arg(z)，是正实轴与向量之间的夹角 θ，以弧度度量，通常取区间 (−π, π] 或 [0, 2π)。
For a given z, tan θ = y/x, but you must consider the quadrant to determine the correct angle.
对于给定的 z，有 tan θ = y/x，但必须考虑象限才能确定正确的角度。
The principal argument Arg(z) is the unique value of arg(z) lying in (−π, π].
主辐角 Arg(z) 是 arg(z) 在 (−π, π] 内的唯⼀值。
The modulus satisfies |z1z2| = |z1||z2| and |z1/z2| = |z1|/|z2|.
模满足 |z1z2| = |z1||z2| 及 |z1/z2| = |z1|/|z2|。
5. Polar Form and Exponential Form | 极坐标形式与指数形式
Using modulus and argument, we can write z = r(cos θ + i sin θ), where r = |z| and θ = arg(z).
利用模和辐角，可将复数写作 z = r(cos θ + i sin θ)，其中 r = |z|，θ = arg(z)。
Euler’s formula states eiθ = cos θ + i sin θ, giving the exponential form z = reiθ.
欧拉公式给出 eiθ = cos θ + i sin θ，从而得到指数形式 z = reiθ。
Polar form makes multiplication and division much simpler: multiply moduli, add arguments; divide moduli, subtract arguments.
极坐标形式使乘除变得简单：模相乘，辐角相加；模相除，辐角相减。
For z1 = r1eiθ1, z2 = r2eiθ2, we have z1z2 = r1r2 ei(θ1+θ2).
对于 z1 = r1eiθ1, z2 = r2eiθ2，有 z1z2 = r1r2 ei(θ1+θ2)。
6. Complex Conjugate and Its Properties | 共轭复数及其性质
The conjugate of z = x + iy is z = x − iy, obtained by changing the sign of the imaginary part.
z = x + iy 的共轭是 z = x − iy，只需改变虚部的符号。
Geometrically, conjugation reflects a point across the real axis on the Argand diagram.
几何上，共轭相当于将点关于实轴反射。
Key properties: z1 + z2 = z1 + z2, z1z2 = z1 z2, z = z, and zz = |z|2 = x2 + y2, a real number.
关键性质：z1 + z2 = z1 + z2，z1z2 = z1 z2，z = z，以及 zz = |z|2 = x2 + y2，是一个实数。
The conjugate is extremely useful in division and in finding real and imaginary parts.
共轭在除法运算以及求实部和虚部时极为有用。
If a polynomial with real coefficients has a complex root, its conjugate is also a root.
若实系数多项式有一个复数根，则其共轭也必定是根。
7. Addition, Subtraction, Multiplication of Complex Numbers | 复数的加、减、乘法
Addition and subtraction are performed component‑wise: (x1 + iy1) ± (x2 + iy2) = (x1 ± x2) + i(y1 ± y2).
加法和减法按分量进行：(x1 + iy1) ± (x2 + iy2) = (x1 ± x2) + i(y1 ± y2)。
Multiplication uses the distributive law and i2 = −1: (x1 + iy1)(x2 + iy2) = (x1x2 − y1y2) + i(x1y2 + x2y1).
乘法运用分配律和 i2 = −1：(x1 + iy1)(x2 + iy2) = (x1x2 − y1y2) + i(x1y2 + x2y1)。
Always replace i2 with −1 when simplifying products; forgetting this step is a common mistake.
化简乘积时一定要用 i2 = −1 替换；忘记这⼀步是常见错误。
Multiplication in polar form is more elegant: multiply moduli and add arguments.
极坐标形式的乘法更优雅：模相乘，辐角相加。
8. Division of Complex Numbers | 复数的除法
To divide two complex numbers, multiply numerator and denominator by the conjugate of the denominator:
两个复数相除，需将分子分母同乘以分母的共轭：
(x1 + iy1)/(x2 + iy2) = [(x1 + iy1)(x2 − iy2)] / (x22 + y22)
这个技巧使得分母实数化，从而很容易分离实部和虚部。
The result is always expressed in the form a + ib, with real a and b.
结果总是表达为 a + ib 的形式，其中 a, b 为实数。
In polar form, division is even simpler: r1/r2 ei(θ1−θ2).
在极坐标中，除法更简单：r1/r2 ei(θ1−θ2)。
You should be comfortable with both Cartesian and polar division, as exam questions may specify the method.
你应该熟练运用笛卡尔和极坐标两种除法，因为考题可能会指定方法。
9. De Moivre’s Theorem | 棣莫弗定理
For any integer n, (cos θ + i sin θ)n = cos(nθ) + i sin(nθ). Equivalently, (eiθ)n = einθ.
对任意整数 n，有 (cos θ + i sin θ)n = cos(nθ) + i sin(nθ)。等价地，(eiθ)n = einθ。
This theorem can be proved by mathematical induction for positive integers and extended to negative integers using e−iθ = cos θ − i sin θ.
该定理可用数学归纳法对正整数证明，并利用 e−iθ = cos θ − i sin θ 扩展到负整数。
De Moivre’s theorem is a powerful tool for deriving trigonometric identities, such as cos 2θ = cos2θ − sin2θ.
棣莫弗定理是推导三角恒等式的有力工具，例如 cos 2θ = cos2θ − sin2θ。
It also enables the computation of powers of complex numbers without expanding brackets repeatedly.
它也使得无需反复展开就能计算复数的乘方。
When n is rational, the theorem gives multiple values (roots), which we explore next.
当 n 为有理数时，定理给出多个值（方根），我们将在下一节探讨。
10. Roots of Complex Numbers | 复数的根
To find the n distinct n‑th roots of a complex number z = r(cos θ + i sin θ), use the formula:
要求复数 z = r(cos θ + i sin θ) 的 n 个不同 n 次方根，使用公式：
zk = r1/n [ cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n) ], k = 0, 1, …, n−1
弧度制下，通过加上 2kπ 来产生所有不同的根。
The n roots lie on a circle of radius r1/n centred at the origin, equally spaced by angle 2π/n.
这 n 个根位于以原点为圆心、半径为 r1/n 的圆上，且相邻根夹角为 2π/n。
For example, the cube roots of unity (n=3, r=1, θ=0) are 1, e2πi/3, e4πi/3.
例如，单位元的立方根（n=3, r=1, θ=0）是 1, e2πi/3, e4πi/3。
Always give all distinct roots in Cartesian form unless the question asks otherwise.
除非题目另有要求，否则必须用笛卡尔形式给出所有不同的根。
Knowing the geometric pattern helps you check that your roots are symmetric.
了解几何规律有助于检验根的对称性。
11. Solving Polynomial Equations | 解多项式方程
In A‑Level exams, you will often solve cubic or quartic equations with real coefficients given one complex root.
在 A‑Level 考试中，常遇到给定一个复数根来求解实系数三次或四次方程。
By the conjugate root theorem, if z = a + ib is a root, then its conjugate a − ib is also a root.
根据共轭根定理，若 z = a + ib 是根，则其共轭 a − ib 也是根。
You can then construct a quadratic factor (z − (a+ib))(z − (a−ib)) = z2 − 2a z + (a2+b2).
然后构建二次因式 (z − (a+ib))(z − (a−ib)) = z2 − 2a z + (a2+b2)。
Divide the original polynomial by this quadratic to find the remaining real factor(s).
将原多项式除以这个二次因式，即可得到剩余的实因式。
Equations like zn = w are solved by writing w in polar form and finding the n‑th roots.
形如 zn = w 的方程可通过将 w 写成极坐标形式并求 n 次方根来求解。
12. Common Pitfalls and Exam Tips | 常见错误与考试技巧
Forgetting that i2 = −1 during multiplication leads to sign errors. Always double‑check your expansions.
乘法过程中忘记 i2 = −1 会导致符号错误。务必仔细检查展开式。
Confusing the modulus and argument when moving between forms: sketch an Argand diagram to verify.
在形式转换时混淆模和辐角：画一个 Argand 图来验证。
Giving only one root when asked for all n‑th roots; remember that there are always n distinct values.
题目要求所有 n 次方根时只给出一个根；记住总是有 n 个不同的值。
Using degrees instead of radians for the argument, unless the question explicitly uses degrees.
辐角使用度数而非弧度，除非
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