Working with Complex Numbers | 复数的运算与应用

📚 Working with Complex Numbers | 复数的运算与应用

Complex numbers extend the familiar real number system by introducing the imaginary unit i, defined by i² = −1. In A-level mathematics, working with complex numbers involves performing arithmetic operations, understanding geometrically meaningful representations, and applying these ideas to solve equations and model periodic phenomena.

复数通过引入虚数单位 i（定义 i² = −1）扩展了我们熟悉的实数系。在 A-Level 数学中，复数的处理包括执行算术运算、理解具有几何意义的表示形式，以及应用这些概念来求解方程和模拟周期现象。

1. What Is a Complex Number? | 什么是复数？

A complex number is written in the form z = a + bi, where a and b are real numbers, and i is the imaginary unit satisfying i² = −1. The set of all complex numbers is denoted by ℂ.

复数的标准形式为 z = a + bi，其中 a 和 b 是实数，i 是满足 i² = −1 的虚数单位。全体复数构成的集合记作 ℂ。

If b = 0, then z is purely real. If a = 0, then z is purely imaginary. Every real number is therefore also a complex number with zero imaginary part.

若 b = 0，则 z 为纯实数；若 a = 0，则 z 为纯虚数。因此，每个实数也可以看作虚部为零的复数。

z = a + bi, where a = Re(z), b = Im(z)

For example, z = 3 − 2i has real part 3 and imaginary part −2. Note that the imaginary part is the coefficient of i, not the term −2i itself.

例如，z = 3 − 2i 的实部为 3，虚部为 −2。注意虚部是 i 的系数，而不是 −2i 这个整体。

2. Adding and Subtracting Complex Numbers | 复数的加法与减法

To add or subtract two complex numbers, we combine their real parts and their imaginary parts separately. This is exactly analogous to collecting like terms in algebra.

两个复数相加或相减时，我们分别合并它们的实部和虚部。这与代数中合并同类项完全类似。

If z₁ = a + bi and z₂ = c + di, then:

若 z₁ = a + bi，z₂ = c + di，则：

z₁ + z₂ = (a + c) + (b + d)i

z₁ − z₂ = (a − c) + (b − d)i

Example: (4 + 3i) + (2 − 5i) = 6 − 2i. Also, (4 + 3i) − (2 − 5i) = 2 + 8i.

例：(4 + 3i) + (2 − 5i) = 6 − 2i；(4 + 3i) − (2 − 5i) = 2 + 8i。

Addition and subtraction are both commutative and associative for complex numbers, since they rely on the same properties for real numbers.

复数的加法和减法都满足交换律和结合律，因为它们是建立在实数相应性质之上的。

3. Multiplying Complex Numbers | 复数的乘法

Multiplication of complex numbers follows the distributive law, with the key rule i² = −1 applied at the final step. You multiply each term in the first bracket by each term in the second bracket.

复数的乘法遵循分配律，并在最后一步应用关键规则 i² = −1。你需要用第一个括号中的每一项乘以第二个括号中的每一项。

If z₁ = a + bi and z₂ = c + di, then:

若 z₁ = a + bi，z₂ = c + di，则：

z₁z₂ = (ac − bd) + (ad + bc)i

This formula arises from expanding (a + bi)(c + di) and using i² = −1. It is usually simpler to multiply directly rather than memorising the result.

该公式由展开 (a + bi)(c + di) 并利用 i² = −1 得到。通常直接展开计算比死记公式更简单。

Example: (2 + 3i)(1 − 4i) = 2 − 8i + 3i − 12i² = 2 − 5i + 12 = 14 − 5i.

例：(2 + 3i)(1 − 4i) = 2 − 8i + 3i − 12i² = 2 − 5i + 12 = 14 − 5i。

Always replace i² by −1 when it appears.
永远记得将出现的 i² 替换为 −1。

4. The Complex Conjugate | 共轭复数

The conjugate of z = a + bi is written as z̄ (or z* in some texts) and is defined by z̄ = a − bi. Geometrically, it is the reflection of z across the real axis.

复数 z = a + bi 的共轭记作 z̄（某些教材也写作 z*），定义为 z̄ = a − bi。从几何上看，它是 z 关于实轴的镜像。

The conjugate is extremely useful because multiplying a complex number by its conjugate always produces a non-negative real number:

共轭极其有用，因为一个复数与其共轭相乘总得到一个非负实数：

z·z̄ = (a + bi)(a − bi) = a² + b²

This property enables us to divide complex numbers, since it allows us to rationalise denominators that contain imaginary parts.

这一性质使我们能够进行复数除法，因为它允许我们使含有虚部的分母有理化。

Note that the conjugate of a conjugate returns the original number: (z̄)̄ = z. Also, z + z̄ = 2a and z − z̄ = 2bi.

注意，共轭的共轭回到原数：(z̄)̄ = z。此外，z + z̄ = 2a，z − z̄ = 2bi。

5. Dividing Complex Numbers | 复数的除法

To divide one complex number by another, multiply both the numerator and the denominator by the conjugate of the denominator. This eliminates the imaginary part from the denominator.

两个复数相除时，将分子和分母同时乘以分母的共轭，从而消除分母中的虚部。

Example: divide (3 + 4i) by (1 − 2i).

例：计算 (3 + 4i) ÷ (1 − 2i)。

(3 + 4i) / (1 − 2i) = (3 + 4i)(1 + 2i) / [(1 − 2i)(1 + 2i)]

Numerator: 3 + 6i + 4i + 8i² = 3 + 10i − 8 = −5 + 10i. Denominator: 1 + 4 = 5. So the result is −1 + 2i.

分子：3 + 6i + 4i + 8i² = 3 + 10i − 8 = −5 + 10i；分母：1 + 4 = 5。因此结果为 −1 + 2i。

Always simplify your answer to the form a + bi. Division is not defined when the denominator is zero.

始终将答案化简为 a + bi 的形式。当分母为零时，除法无定义。

6. The Argand Diagram | 阿尔冈图（复平面）

A complex number z = a + bi can be represented as a point (a, b) on a plane called the Argand diagram. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.

复数 z = a + bi 可以表示为平面上的点 (a, b)，这个平面称为阿尔冈图。横轴是实轴，纵轴是虚轴。

This geometric representation allows us to visualise addition and subtraction as vector operations. The vector from the origin to z has horizontal component a and vertical component b.

这种几何表示使我们能够将加法和减法视为向量运算。从原点到 z 的向量，其水平分量为 a，垂直分量为 b。

For example, z = 3 + 2i is the point (3, 2), and its conjugate z̄ = 3 − 2i is the reflection across the real axis, at (3, −2).

例如，z = 3 + 2i 对应点 (3, 2)，其共轭 z̄ = 3 − 2i 是实轴下的镜像，位于 (3, −2)。

The distance from the origin to the point z is called the modulus, and the angle measured anticlockwise from the positive real axis is called the argument.

从原点到点 z 的距离称为模，从正实轴逆时针方向测量的角度称为辐角。

7. Modulus of a Complex Number | 复数的模

The modulus of z = a + bi is denoted by |z| and is defined as the distance from the origin to the point (a, b) in the Argand diagram. By Pythagoras’ theorem:

复数 z = a + bi 的模记作 |z|，定义为阿尔冈图中点 (a, b) 到原点的距离。由勾股定理：

|z| = √(a² + b²)

The modulus is always a non-negative real number. It satisfies the important identity |z|² = z·z̄.

模始终是非负实数。它满足重要恒等式 |z|² = z·z̄。

Example: if z = 5 − 12i, then |z| = √(25 + 144) = √169 = 13.

例：若 z = 5 − 12i，则 |z| = √(25 + 144) = √169 = 13。

The modulus obeys the multiplicative property |z₁z₂| = |z₁||z₂|, and also |z₁/z₂| = |z₁|/|z₂| for z₂ ≠ 0.

模满足乘法性质 |z₁z₂| = |z₁||z₂|，以及当 z₂ ≠ 0 时，|z₁/z₂| = |z₁|/|z₂|。

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

The argument of a non-zero complex number z, denoted arg(z), is the angle θ between the positive real axis and the vector representing z, measured anticlockwise in radians.

非零复数 z 的辐角记作 arg(z)，是正实轴与表示 z 的向量之间的夹角 θ，以弧度逆时针方向测量。

The principal argument, usually written Arg(z) with a capital A, lies in the interval (−π, π]. For any z, the general argument is Arg(z) + 2kπ, where k is any integer.

主辐角通常用大写 Arg(z) 表示，取值范围为 (−π, π]。对任意 z，一般辐角为 Arg(z) + 2kπ，其中 k 为任意整数。

For z = a + bi, the principal argument satisfies tan θ = b/a, but you must consider the quadrant of the point (a, b).

对于 z = a + bi，主辐角满足 tan θ = b/a，但必须考虑点 (a, b) 所在的象限。

First quadrant: Arg(z) = arctan(b/a)
Second quadrant: Arg(z) = π − arctan(|b/a|)
Third quadrant: Arg(z) = −π + arctan(|b/a|) (or π + arctan(b/a) if you use a positive angle)
Fourth quadrant: Arg(z) = −arctan(|b/a|)

第一象限：Arg(z) = arctan(b/a)
第二象限：Arg(z) = π − arctan(|b/a|)
第三象限：Arg(z) = −π + arctan(|b/a|)（若用正角则为 π + arctan(b/a)）
第四象限：Arg(z) = −arctan(|b/a|)

For example, z = −1 + i lies in the second quadrant, and its principal argument is 3π/4.

例如，z = −1 + i 位于第二象限，其主辐角为 3π/4。

9. Polar Form and the Exponential Form | 极坐标形式与指数形式

Using the modulus r = |z| and the argument θ, any non-zero complex number can be written in polar form:

利用模 r = |z| 和辐角 θ，任何非零复数都可以写成极坐标形式：

z = r(cos θ + i sin θ)

This is often abbreviated as r cis θ. Euler’s formula gives the equivalent exponential form z = r·e^{iθ}, which is extremely powerful for multiplication and powers because exponents add when bases multiply.

这常简写为 r cis θ。欧拉公式给出了等价指数形式 z = r·e^{iθ}，它在乘法与幂运算中极为强大，因为同底数幂相乘时指数相加。

Given z = a + bi, we have r = √(a² + b²) and θ = arg(z). Conversely, a = r cos θ and b = r sin θ.

已知 z = a + bi 时，有 r = √(a² + b²)，θ = arg(z)。反过来，a = r cos θ，b = r sin θ。

Example: z = 1 + √3 i has r = 2 and θ = π/3, so z = 2(cos π/3 + i sin π/3).

例：z = 1 + √3 i 的模为 2，辐角为 π/3，因此 z = 2(cos π/3 + i sin π/3)。

10. Multiplying and Dividing in Polar Form | 极坐标形式下的乘法与除法

If z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), then multiplication and division are much simpler than in Cartesian form:

若 z₁ = r₁(cos θ₁ + i sin θ₁)，z₂ = r₂(cos θ₂ + i sin θ₂)，则乘法和除法比直角坐标形式简单得多：

z₁z₂ = r₁r₂ [cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]

z₁ / z₂ = (r₁/r₂) [cos(θ₁ − θ₂) + i sin(θ₁ − θ₂)]

Thus, when multiplying complex numbers, multiply the moduli and add the arguments. When dividing, divide the moduli and subtract the arguments.

因此，复数相乘时，模相乘、辐角相加；复数相除时，模相除、辐角相减。

Example: if z₁ = 2(cos 30° + i sin 30°) and z₂ = 3(cos 45° + i sin 45°), then z₁z₂ = 6(cos 75° + i sin 75°).

例：若 z₁ = 2(cos 30° + i sin 30°)，z₂ = 3(cos 45° + i sin 45°)，则 z₁z₂ = 6(cos 75° + i sin 75°)。

These rules follow directly from the exponential form z = re^{iθ} and the laws of exponents.

这些规则直接来源于指数形式 z = re^{iθ} 和指数运算法则。

11. De Moivre’s Theorem and Powers | 棣莫弗定理与幂

De Moivre’s theorem states that for any integer n and any real angle θ:

棣莫弗定理指出，对任意整数 n 和任意实数 θ：

(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)

Equivalently, if z = r(cos θ + i sin θ), then zⁿ = rⁿ [cos(nθ) + i sin(nθ)]. This provides a fast route to computing high powers of complex numbers.

等价地，若 z = r(cos θ + i sin θ)，则 zⁿ = rⁿ [cos(nθ) + i sin(nθ)]。这为计算复数的高次幂提供了捷径。

Example: Compute (1 + i)¹⁰. First write 1 + i in polar form: r = √2, θ = π/4. Then (1 + i)¹⁰ = (√2)¹⁰ [cos(10·π/4) + i sin(10·π/4)] = 32(cos(5π/2) + i sin(5π/2)) = 32(0 + i·1) = 32i.

例：计算 (1 + i)¹⁰。先将 1 + i 写成极坐标形式：r = √2，θ = π/4。于是 (1 + i)¹⁰ = (√2)¹⁰ [cos(10·π/4) + i sin(10·π/4)] = 32(cos(5π/2) + i sin(5π/2)) = 32(0 + i·1) = 32i。

The theorem also holds for negative and rational exponents in appropriately restricted settings, making it a central tool for solving equations such as zⁿ = w.

在适当限制下，该定理对负指数和有理指数也成立，因此它是求解 zⁿ = w 这类方程的核心工具。

12. Solving Quadratic Equations with Complex Roots | 含复数根的二次方程

When a quadratic equation ax² + bx + c = 0 has discriminant Δ = b² − 4ac < 0, its solutions are complex conjugates of each other. The quadratic formula still works, but we must accept the appearance of negative numbers under the square root.

当二次方程 ax² + bx + c = 0 的判别式 Δ = b² − 4ac < 0 时，其解是一对共轭复数。求根公式仍然适用，只是根号下会出现负数。

Example: Solve x² + 2x + 5 = 0.

例：解方程 x² + 2x + 5 = 0。

x = [−2 ± √(4 − 20)] / 2 = [−2 ± √(−16)] / 2 = −1 ± 2i

If a polynomial with real coefficients has a non-real root a + bi, then its conjugate a − bi is also a root. This is called the conjugate root theorem.

若一个实系数多项式有一个非实数根 a + bi，则其共轭 a − bi 也必是一个根。这称为共轭根定理。

This theorem is essential when factorising polynomials over the real numbers, because every quadratic factor corresponds to a pair of complex conjugate roots.

该定理在实数范围内因式分解多项式时至关重要，因为每个二次因式对应一对共轭复数根。

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Working with Complex Numbers | 复数的运算与应用