Complex Numbers for GCSE WJEC Mathematics | GCSE WJEC 数学：复数考点精讲

📚 Complex Numbers for GCSE WJEC Mathematics | GCSE WJEC 数学：复数考点精讲

Although complex numbers are not usually part of the core GCSE WJEC Mathematics specification, understanding them can deepen your appreciation of algebra and provide a strong foundation for A-level Further Mathematics. This article introduces the essential concepts of complex numbers in a clear, exam-focused style, covering imaginary numbers, algebraic operations, the Argand diagram, and solving quadratic equations with negative discriminants.

尽管复数通常不属于 GCSE WJEC 数学核心考试范围，但理解复数可以加深你对代数的认识，并为 A-level 进阶数学打下坚实基础。本文以清晰、面向考试的风格介绍复数的基本概念，涵盖虚数、代数运算、阿根图以及判别式为负的二次方程求解。

1. The Need for Complex Numbers | 复数的由来

When solving quadratic equations like x² + 1 = 0, we find that there is no real number x that satisfies the equation, because the square of any real number is non‑negative. To overcome this, mathematicians introduced the imaginary unit i, which allows us to define solutions beyond the real numbers.

在解像 x² + 1 = 0 这样的二次方程时，我们发现没有实数 x 能满足等式，因为任何实数的平方都是非负的。为了解决这个问题，数学家引入了虚数单位 i，使我们能够定义实数之外的解。

2. The Imaginary Unit i | 虚数单位 i

The imaginary unit i is defined such that i² = −1. Using this definition, √(−1) = i, and more generally √(−n) = i√n for any positive real number n. All imaginary numbers can be written as bi where b is a real number, for example 3i, −5i, or √2 i.

虚数单位 i 定义为使得 i² = −1。根据这个定义，√(−1) = i，更一般地，对任意正实数 n，有 √(−n) = i√n。所有纯虚数都可以写成 bi 的形式，其中 b 是实数，例如 3i、−5i 或 √2 i。

3. Definition of a Complex Number | 复数的定义

A complex number z is an expression of the form z = a + bi, where a and b are real numbers. a is called the real part, denoted Re(z), and b is called the imaginary part, denoted Im(z). Note that both a and b are real numbers; the imaginary part is the coefficient of i, not bi itself.

复数 z 是形如 z = a + bi 的表达式，其中 a 和 b 为实数。a 称为实部，记作 Re(z)；b 称为虚部，记作 Im(z)。注意 a 和 b 都是实数；虚部是 i 的系数，而不是 bi 本身。

4. Equality of Complex Numbers | 复数相等

Two complex numbers a + bi and c + di are equal if and only if their real parts are equal (a = c) and their imaginary parts are equal (b = d). This simple rule underlies almost all equation-solving with complex numbers.

两个复数 a + bi 和 c + di 相等，当且仅当它们的实部相等（a = c）且虚部相等（b = d）。这一简单规则是几乎所有复数方程求解的基础。

5. Addition and Subtraction | 加法和减法

To add or subtract complex numbers, simply add or subtract the real parts and the imaginary parts separately.
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi) − (c + di) = (a − c) + (b − d)i

复数的加减运算只需分别对实部和虚部进行加减。
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi) − (c + di) = (a − c) + (b − d)i

Example: (2 + 3i) + (4 − i) = 6 + 2i
示例：(2 + 3i) + (4 − i) = 6 + 2i

6. Multiplication of Complex Numbers | 复数乘法

Multiply complex numbers as you would binomials, using the FOIL method or distribution, and then replace i² with −1.
(a + bi)(c + di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i

像展开二项式一样进行复数乘法，使用分配律，然后将 i² 替换为 −1。
(a + bi)(c + di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i

Example: (3 + 2i)(1 − 4i) = 3 − 12i + 2i − 8i² = 3 − 10i + 8 = 11 − 10i
示例：(3 + 2i)(1 − 4i) = 3 − 12i + 2i − 8i² = 3 − 10i + 8 = 11 − 10i

7. Complex Conjugate | 共轭复数

The complex conjugate of z = a + bi is denoted as z* or z̅ and is defined as a − bi. The conjugate is a mirror image across the real axis on the Argand diagram. The product of a complex number and its conjugate is always a non‑negative real number: z z* = (a + bi)(a − bi) = a² + b².

复数 z = a + bi 的共轭复数记为 z* 或 z̅，定义为 a − bi。共轭复数在阿根图中是关于实轴的镜像。一个复数与其共轭的乘积始终是一个非负实数：z z* = (a + bi)(a − bi) = a² + b²。

8. Division of Complex Numbers | 复数除法

To divide two complex numbers, multiply the numerator and denominator by the conjugate of the denominator. This eliminates the imaginary part from the denominator.
(a + bi) / (c + di) = (a + bi)(c − di) / (c + di)(c − di) = (a + bi)(c − di) / (c² + d²)

进行复数除法时，将分子和分母同时乘以分母的共轭。这样可以消去分母中的虚部。
(a + bi) / (c + di) = (a + bi)(c − di) / (c + di)(c − di) = (a + bi)(c − di) / (c² + d²)

Example: (2 + 3i) / (1 − i) = (2 + 3i)(1 + i) / (1 − i)(1 + i) = (2 + 2i + 3i + 3i²) / (1 + 1) = (−1 + 5i) / 2 = −½ + (5/2)i
示例：(2 + 3i) / (1 − i) = (2 + 3i)(1 + i) / (1 − i)(1 + i) = (2 + 2i + 3i + 3i²) / (1 + 1) = (−1 + 5i) / 2 = −½ + (5/2)i

9. The Argand Diagram | 阿根图（复平面）

An Argand diagram represents complex numbers as points or vectors in a plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. The complex number z = a + bi corresponds to the point (a, b). This visualisation helps with understanding modulus and argument.

阿根图将复数表示为平面上的点或向量。横轴为实轴，纵轴为虚轴。复数 z = a + bi 对应于点 (a, b)。这种可视化有助于理解复数的模和辐角。

10. Modulus of a Complex Number | 复数的模

The modulus (or absolute value) of a complex number z = a + bi is denoted |z| and is defined as |z| = √(a² + b²). It represents the distance from the origin to the point (a, b) on the Argand diagram. The modulus is always a non‑negative real number.

复数 z = a + bi 的模（或绝对值）记作 |z|，定义为 |z| = √(a² + b²)。它表示从原点到阿根图上点 (a, b) 的距离。模始终是一个非负实数。

Example: |3 + 4i| = √(3² + 4²) = √25 = 5
示例：|3 + 4i| = √(3² + 4²) = √25 = 5

11. Solving Quadratic Equations with Negative Discriminant | 解判别式为负的二次方程

When a quadratic equation ax² + bx + c = 0 has discriminant Δ = b² − 4ac < 0, it has no real roots but two complex conjugate roots given by the formula x = (−b ± i√(−Δ)) / (2a). This is the standard quadratic formula extended to complex numbers.

当二次方程 ax² + bx + c = 0 的判别式 Δ = b² − 4ac < 0 时，它没有实根，但有两个共轭复根，由公式 x = (−b ± i√(−Δ)) / (2a) 给出。这是将标准求根公式推广到复数。

Example: Solve x² − 2x + 5 = 0. Δ = 4 − 20 = −16. Roots: x = (2 ± √(16)i) / 2 = 1 ± 2i.
示例：解方程 x² − 2x + 5 = 0。Δ = 4 − 20 = −16。根为：x = (2 ± √(16)i) / 2 = 1 ± 2i。

12. Key Summary and Exam Tips | 要点总结与考试技巧

Always remember that i² = −1. For operations, treat i like a variable, then simplify using i² = −1. Complex conjugates are your tool for division and for finding real‑valued products. When working with Argand diagrams, the x‑axis is real and the y‑axis is imaginary. In WJEC GCSE, you may not be directly examined on complex numbers, but these concepts enrich your algebraic thinking and prepare you for advanced topics.

始终牢记 i² = −1。进行运算时，将 i 当作变量处理，然后用 i² = −1 化简。共轭复数是进行除法以及得到实值乘积的工具。在处理阿根图时，x 轴为实轴，y 轴为虚轴。在 WJEC GCSE 中，你可能不会直接考到复数，但这些概念能丰富你的代数思维，为后续高阶内容做好准备。

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Complex Numbers for GCSE WJEC Mathematics | GCSE WJEC 数学：复数 考点精讲