Complex Numbers for A-Level OCR Maths: Key Points Explained | A-Level OCR 数学：复数考点精讲

📚 Complex Numbers for A-Level OCR Maths: Key Points Explained | A-Level OCR 数学：复数考点精讲

Complex numbers form a cornerstone of the OCR A-Level Mathematics syllabus. They extend the real number system, enabling solutions to equations that have no real roots and providing powerful tools for geometry, trigonometry, and advanced calculus. This revision guide distils the essential concepts, operations, and applications you need to master — from imaginary units to De Moivre’s theorem and loci.

复数是 OCR A-Level 数学大纲中的核心内容。它们拓展了实数系，使得无实根的方程得以求解，并为几何、三角和高等微积分提供了强大工具。本复习指南凝练了从虚数单位到棣莫弗定理和轨迹等必须掌握的关键概念、运算与应用。

1. Introduction to Imaginary and Complex Numbers | 虚数与复数简介

The imaginary unit i is defined by the property i² = −1. A complex number is an expression of the form z = a + bi, where a and b are real numbers. Here, a is the real part, Re(z), and b is the imaginary part, Im(z). If b = 0, the number is purely real; if a = 0, it is purely imaginary.

虚数单位 i 由性质 i² = −1 定义。一个复数具有 z = a + bi 的形式，其中 a 和 b 为实数。a 为实部 Re(z)，b 为虚部 Im(z)。若 b = 0，此数为纯实数；若 a = 0，则为纯虚数。

Two complex numbers are equal if and only if their real and imaginary parts are respectively equal. For instance, 3 + 2i and 3 + 2i are equal; 3 + 2i ≠ 3 − 2i.

两个复数相等当且仅当它们的实部与虚部分别相等。例如，3 + 2i 与 3 + 2i 相等；而 3 + 2i ≠ 3 − 2i。

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2. Basic Operations with Complex Numbers | 复数的基本运算

Addition and subtraction are performed component‑wise: (a + bi) ± (c + di) = (a ± c) + (b ± d)i.

加法与减法按分量进行：(a + bi) ± (c + di) = (a ± c) + (b ± d)i。

Multiplication uses the distributive law and i² = −1: (a + bi)(c + di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i.

乘法利用分配律与 i² = −1：(a + bi)(c + di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i。

Division is handled by multiplying numerator and denominator by the complex conjugate of the denominator. For z₁ = a + bi and z₂ = c + di (≠ 0),

z₁ / z₂ = (a+bi)(c−di) / (c²+d²)

从而得到标准形式。

除法通过分子分母同乘分母的共轭复数实现：对于 z₁ = a + bi 和 z₂ = c + di (≠ 0)，

z₁ / z₂ = (a+bi)(c−di) / (c²+d²)

yielding the standard form.

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3. The Complex Conjugate | 共轭复数

The complex conjugate of z = a + bi is denoted z* or z and is given by z* = a − bi. Geometrically, it is the reflection of z in the real axis.

z = a + bi 的共轭复数记为 z* 或 z，为 z* = a − bi。几何上，它是 z 关于实轴的反射。

Key properties:

• z + z* = 2 Re(z) (purely real)
• z − z* = 2i Im(z)
• z z* = a² + b² = |z|² (always real and non‑negative)
• (z₁ ± z₂)* = z₁* ± z₂* ; (z₁ z₂)* = z₁* z₂* ; (z₁ / z₂)* = z₁* / z₂*

关键性质（中文对照）：

• z + z* = 2 Re(z)（纯实数）
• z − z* = 2i Im(z)
• z z* = a² + b² = |z|²（恒为非负实数）
• (z₁ ± z₂)* = z₁* ± z₂* ；(z₁ z₂)* = z₁* z₂* ；(z₁ / z₂)* = z₁* / z₂*

The conjugate is essential in division and in solving polynomial equations with real coefficients: if z is a root, so is z*.

共轭对于除法和解实系数多项式方程至关重要：若 z 是根，则 z* 也是根。

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4. Quadratic Equations with Complex Roots | 二次方程的复数根

For a quadratic equation ax² + bx + c = 0 with real coefficients, the discriminant Δ = b² − 4ac determines the nature of the roots. When Δ < 0, the roots are a conjugate pair of complex numbers:

x = (−b ± i√(4ac−b²)) / (2a)

对于实系数二次方程 ax² + bx + c = 0，判别式 Δ = b² − 4ac 决定根的性质。当 Δ < 0 时，两根为一对共轭复数：

x = (−b ± i√(4ac−b²)) / (2a)

For example, x² + 4x + 13 = 0 gives x = −2 ± 3i. These two roots are conjugates. This principle extends to higher‑degree polynomials with real coefficients.

例如，x² + 4x + 13 = 0 的解为 x = −2 ± 3i，这两个根互为共轭。此原理同样适用于更高次实系数多项式。

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5. Argand Diagrams | 阿尔冈图

An Argand diagram represents complex numbers as points or vectors in a plane with horizontal real axis and vertical imaginary axis. The number z = a + bi corresponds to the point (a, b) or the position vector from the origin to (a, b).

阿尔冈图将复数表示为平面上的点或向量，其中水平轴为实轴，垂直轴为虚轴。复数 z = a + bi 对应点 (a, b) 或从原点到 (a, b) 的位置向量。

Addition of complex numbers follows the parallelogram law for vectors, and subtraction corresponds to the vector joining the endpoints. The diagram also visualises magnitude and direction, leading to the modulus and argument.

复数加法遵循向量平行四边形法则，减法对应连接端点的向量。该图也直观展示了大小和方向，引出模和辐角的概念。

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6. Modulus and Argument of a Complex Number | 复数的模与辐角

The modulus of z = a + bi is the distance from the origin: |z| = √(a² + b²). It is always a non‑negative real number. The argument of z, denoted arg(z), is the angle θ measured from the positive real axis to the vector representing z, usually in the range (−π, π] (the principal argument).

复数 z = a + bi 的模是它到原点的距离：|z| = √(a² + b²)，是一个非负实数。辐角 arg(z) 是从正实轴到表示 z 的向量的角度 θ，通常取值在 (−π, π]（辐角主值）。

If a > 0, arg(z) = arctan(b/a); adjustments are needed for other quadrants to obtain the correct principal value.

若 a > 0，则 arg(z) = arctan(b/a)；在其他象限需进行调整以获得正确的辐角主值。

Properties: |z₁ z₂| = |z₁||z₂|, |z₁/z₂| = |z₁|/|z₂|, arg(z₁ z₂) = arg(z₁) + arg(z₂) (mod 2π).

性质：|z₁ z₂| = |z₁||z₂|，|z₁/z₂| = |z₁|/|z₂|，arg(z₁ z₂) = arg(z₁) + arg(z₂)（模 2π）。

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7. Polar Form of Complex Numbers | 复数的极坐标形式

Using modulus and argument, a complex number can be written

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