AS Mathematics: Complex Numbers Key Concepts | AS 数学：复数考点精讲

📚 AS Mathematics: Complex Numbers Key Concepts | AS 数学：复数考点精讲

Complex numbers form a cornerstone of AS-level Mathematics, extending the real number system to include solutions to equations like x² = -1. Mastering their algebra, geometry, and applications is essential for success in topics ranging from quadratics to further pure mathematics. This revision guide walks you through the key concepts, each explained with clear English explanations followed by their Chinese counterparts.

复数是 AS 数学的基石之一，它将实数系扩展到能够求解 x² = -1 这类方程的根。掌握复数的代数运算、几何意义及其应用，对于二次方程乃至进阶纯数都至关重要。本考点精讲通过先英文后中文的配对讲解，带你系统梳理所有核心考点。

1. Definition and the Imaginary Unit i | 复数定义与虚数单位 i

A complex number is any number that can be written in the form z = a + bi, where a and b are real numbers. The symbol i denotes the imaginary unit, defined purely by the relation i² = -1. The real part of z is Re(z) = a, and the imaginary part is Im(z) = b.

复数是可以表示为 z = a + bi 形式的数，其中 a 与 b 为实数。符号 i 表示虚数单位，其唯一定义为 i² = -1。z 的实部记为 Re(z) = a，虚部记为 Im(z) = b。

i² = -1

It is crucial to remember that i is not a variable but a constant with this fixed property. When b = 0, z is purely real; when a = 0, z is purely imaginary, such as 3i or -2i.

务必牢记 i 并非变量，而是具有这一固定性质的常量。当 b = 0 时，z 为纯实数；当 a = 0 时，z 为纯虚数，例如 3i 或 -2i。

2. Addition, Subtraction, and Scalar Multiplication | 加法、减法与数乘运算

Addition and subtraction of complex numbers are performed by handling the real and imaginary parts separately. For z₁ = a + bi and z₂ = c + di, we have z₁ + z₂ = (a + c) + (b + d)i and z₁ – z₂ = (a – c) + (b – d)i. Scalar multiplication with a real number k simply scales both components: kz = ka + kbi.

复数的加法与减法只需分别对实部和虚部进行组合。设 z₁ = a + bi，z₂ = c + di，则 z₁ + z₂ = (a + c) + (b + d)i，z₁ – z₂ = (a – c) + (b – d)i。与实数 k 的数乘则对两部分同时缩放：kz = ka + kbi。

These operations mirror the algebra of vectors in two dimensions, which is why the Argand diagram becomes such a powerful tool. You can think of complex numbers as vectors, with addition following the parallelogram law.

这些运算与二维向量的代数规则一致，这正是 Argand 图极具威力的原因。你可以把复数看作向量，加法满足平行四边形法则。

3. Multiplication of Complex Numbers | 复数乘法

Multiplication uses the distributive law, exactly as with algebraic binomials, and then simplifies by applying i² = -1. For two complex numbers z₁ = a + bi and z₂ = c + di, the product is z₁z₂ = (ac – bd) + (ad + bc)i.

复数乘法利用分配律展开，就像代数中的二项式相乘，然后利用 i² = -1 进行化简。对于 z₁ = a + bi 与 z₂ = c + di，乘积为 z₁z₂ = (ac – bd) + (ad + bc)i。

z₁z₂ = (a + bi)(c + di) = ac + adi + bci + bdi² = (ac – bd) + (ad + bc)i

It helps to memorise the pattern: the real part is the product of the reals minus the product of the imaginaries, while the imaginary part is the sum of the cross products. Multiply carefully and always replace i² with -1.

记住这个模式非常有帮助：实部为“实乘实减虚乘虚”，虚部为“交叉相乘再相加”。计算时务必仔细，并始终将 i² 替换为 -1。

4. Complex Conjugate and Division | 共轭复数与除法

The complex conjugate of z = a + bi is denoted by z̅ or z* and is defined as a – bi. Conjugation reflects a point across the real axis on the Argand diagram and has the crucial property that z z̅ = a² + b², a non-negative real number.

复数 z = a + bi 的共轭复数记为 z̅ 或 z*，定义为 a – bi。共轭相当于将点关于实轴作镜面反射，其关键性质为 z z̅ = a² + b²，这是一个非负实数。

z z̅ = (a + bi)(a – bi) = a² – (bi)² = a² + b²

Division of complex numbers makes use of the conjugate. To compute (z₁)/(z₂), multiply numerator and denominator by the conjugate of the denominator: (z₁/z₂) = (z₁ z̅₂)/(z₂ z̅₂). This turns the denominator into a real number, allowing separation into real and imaginary parts.

复数的除法需要借助共轭。计算 z₁/z₂ 时，将分子分母同乘以分母的共轭：z₁/z₂ = (z₁ z̅₂)/(z₂ z̅₂)。这样分母变成实数，从而可将结果写成 a + bi 的形式。

5. The Argand Diagram | 复平面 (Argand 图)

The Argand diagram represents complex numbers as points or position vectors in a plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. A complex number z = a + bi corresponds to the coordinates (a, b).

Argand 图将复数表示为平面上的点或位置向量。横轴为实轴，纵轴为虚轴。复数 z = a + bi 对应于坐标 (a, b)。

This visualisation transforms algebraic operations into geometry. Addition becomes vector addition, and conjugation becomes a reflection in the real axis. The distance from the origin and the angle with the positive real axis give rise to the modulus and argument, respectively.

这一可视化将代数运算转化为几何图形。加法变为向量加法，共轭变为关于实轴的反射。点到原点的距离以及点与正实轴的夹角分别引出模与辐角的概念。

6. Modulus of a Complex Number | 复数的模

The modulus of a complex number z = a + bi, denoted |z|, is the distance from the origin to the point (a, b) on the Argand diagram. By the Pythagorean theorem, |z| = √(a² + b²). It is always non-negative and is closely linked to the conjugate: |z|² = z z̅.

复数 z = a + bi 的模记作 |z|，是 Argand 图上原点到点 (a, b) 的距离。根据勾股定理，|z| = √(a² + b²)。模总是非负的，并且与共轭紧密相关：|z|² = z z̅。

|z| = √(a² + b²)

Properties of the modulus include |z₁z₂| = |z₁| |z₂| and |z₁/z₂| = |z₁|/|z₂| for z₂ ≠ 0. The modulus is fundamental for finding the polar form and solving geometric problems involving circles and lines in the complex plane.

模的性质有 |z₁z₂| = |z₁| |z₂| 以及 |z₁/z₂| = |z₁|/|z₂|（z₂ ≠ 0）。模是求极形式以及解决复平面上圆与直线等几何问题的基础。

7. Argument of a Complex Number | 复数的辐角

The argument of a non-zero complex number z, written arg(z), is the angle θ measured from the positive real axis to the line segment joining the origin to z. It is usually given in radians, with the principal argument falling in the interval (-π, π] or (-180°, 180°]. You can find θ using tan θ = b/a, but careful consideration of the quadrant is essential.

非零复数 z 的辐角记为 arg(z)，是从正实轴逆时针旋转到原点到 z 连线所成的角 θ。通常以弧度给出，其主值落在区间 (-π, π] 或 (-180°, 180°]。可利用 tan θ = b/a 求角，但必须注意所在象限。

For example, z = 1 + i lies in the first quadrant, so arg(z) = π/4. But for z = -1 – i, which is in the third quadrant, arg(z) = -3π/4 (not π/4). Always sketch the point to avoid sign errors.

例如，z = 1 + i 位于第一象限，arg(z) = π/4。而对于第三象限的 z = -1 – i，其辐角主值为 -3π/4 而非 π/4。务必绘出点的位置以避免符号错误。

8. Polar Form (Modulus-Argument Form) | 极形式 (模-辐角形式)

A complex number can be expressed in polar form using its modulus r = |z| and its argument θ = arg(z). The polar form is z = r (cos θ + i sin θ). This representation is particularly powerful for multiplication, division, and finding powers and roots.

复数可以用模 r = |z| 与辐角 θ = arg(z) 表示为极形式 z = r (cos θ + i sin θ)。这一表示法在乘除运算以及求幂和方根时尤为强大。

z = r (cos θ + i sin θ) with r = |z|, θ = arg(z)

Converting between Cartesian form (a + bi) and polar form is a standard skill. Given r and θ, you find a = r cos θ and b = r sin θ. Given a and b, you compute r = √(a² + b²) and determine θ appropriately by quadrant.

直角形式 (a + bi) 与极形式之间的相互转化是基本技能。已知 r 与 θ，可得 a = r cos θ，b = r sin θ。已知 a 与 b，则计算 r = √(a² + b²) 并根据象限确定 θ。

Complex Number	Polar Form
1 + i	√2 (cos(π/4) + i sin(π/4))
-2	2 (cos π + i sin π)
3i	3 (cos(π/2) + i sin(π/2))

9. Multiplication and Division in Polar Form | 极形式下的乘除

Multiplying two complex numbers in polar form is elegantly simple: multiply their moduli and add their arguments. If z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), then z₁z₂ = r₁r₂ [cos(θ₁+θ₂) + i sin(θ₁+θ₂)].

在极形式下进行复数乘法非常简洁：模相乘，辐角相加。若 z₁ = r₁(cos θ₁ + i sin θ₁)，z₂ = r₂(cos θ₂ + i sin θ₂)，则 z₁z₂ = r₁r₂ [cos(θ₁+θ₂) + i sin(θ₁+θ₂)]。

Division follows a similar pattern: divide the moduli and subtract the arguments. z₁/z₂ = (r₁/r₂) [cos(θ₁-θ₂) + i sin(θ₁-θ₂)], provided z₂ ≠ 0. These results are easily proved by expanding the trigonometric expressions and using addition formulas.

除法则遵循相似的模式：模相除，辐角相减。z₁/z₂ = (r₁/r₂) [cos(θ₁-θ₂) + i sin(θ₁-θ₂)]，其中 z₂ ≠ 0。通过展开三角函数并使用和角公式可轻松证明这些结果。

|z₁z₂| = |z₁| |z₂|, arg(z₁z₂) = arg(z₁) + arg(z₂)

These geometric interpretations mean that multiplication by i corresponds to a rotation of π/2 counterclockwise, since |i| = 1 and arg(i) = π/2. This insight often simplifies complex geometry problems.

这些几何意义意味着乘以 i 相当于逆时针旋转 π/2，因为 |i| = 1 且 arg(i) = π/2。这一见解常能简化复平面几何问题。

10. Solving Quadratic Equations and the Conjugate Root Theorem | 解二次方程与共轭根定理

In AS Mathematics, complex numbers often first appear when solving quadratic equations whose discriminant is negative. For ax² + bx + c = 0, if Δ = b² – 4ac < 0, the formula x = [-b ± √(-Δ) i] / (2a) yields a pair of complex conjugate roots.

在 AS 数学中，复数常首次出现于求解判别式为负的二次方程。对于 ax² + bx + c = 0，若 Δ = b² – 4ac < 0，则公式 x = [-b ± √(-Δ) i] / (2a) 将给出一对共轭复根。

x = [-b ± i√(4ac – b²)] / (2a) when b² – 4ac < 0

Furthermore, the Conjugate Root Theorem states that for any polynomial equation with real coefficients, non-real complex roots always occur in conjugate pairs. This means if a + bi is a root, then a – bi must also be a root, which is extremely useful when factorising polynomials or verifying solutions.

进一步，共轭根定理指出：对于任何实系数多项式方程，非实数复根总是以共轭对形式出现。这意味着若 a + bi 是一个根，则 a – bi 也必为根，这在因式分解多项式或验证解时极为有用。

You will often be asked to form a quadratic equation given one complex root, or to find the remaining roots of a cubic knowing one complex root. Simply use the conjugate pair and the fact that sum and product of roots relate to the coefficients.

考试常要求根据已知的一个复根构造二次方程，或已知一个复根求三次方程的其余根。只需利用共轭对以及根与系数的和与积关系即可解决。

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AS Mathematics: Complex Numbers Key Concepts | AS 数学：复数 考点精讲