Products and Quotients of Complex Numbers in Polar Form | 复数极坐标形式的乘法与除法

📚 Products and Quotients of Complex Numbers in Polar Form | 复数极坐标形式的乘法与除法

Many A-Level candidates can multiply complex numbers in Cartesian form, but polar form turns multiplication and division into a short, structured process: combine moduli and arguments. This article reviews the rules, derivations, geometric meaning and exam-ready examples for AQA Mathematics.

许多 A-Level 学生习惯用笛卡尔形式做复数乘法，但极坐标形式把乘法和除法变成简单而有结构的过程：只需处理模和辐角。本文为 AQA 数学考生梳理极坐标形式的乘法与除法法则、推导、几何意义和例题。

1. Recap: The Polar Form of a Complex Number | 回顾：复数的极坐标形式

A complex number z = x + iy has modulus r = |z| = √(x² + y²) and argument θ = arg z, where cos θ = x/r and sin θ = y/r. The polar form is z = r(cos θ + i sin θ), often abbreviated as z = r cis θ.

复数 z = x + iy 的模为 r = |z| = √(x² + y²)，辐角 θ = arg z 满足 cos θ = x/r 与 sin θ = y/r。其极坐标形式为 z = r(cos θ + i sin θ)，常简写为 z = r cis θ。

z = x + iy = r(cos θ + i sin θ) = r cis θ

For AQA, the principal argument is usually taken as -π < θ ≤ π. If a polar form does not use the principal argument, it is still valid, but answers often require the principal value.

在 AQA 考试中，主辐角通常取 -π < θ ≤ π。如果极坐标形式没有使用主辐角，它仍然有效，但答案通常要求给出主值。

2. Multiplication Rule: Multiply Moduli, Add Arguments | 乘法法则：模相乘，辐角相加

Suppose z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂). Then their product has modulus r₁r₂ and argument θ₁ + θ₂:

设 z₁ = r₁(cos θ₁ + i sin θ₁)、z₂ = r₂(cos θ₂ + i sin θ₂)。则它们的乘积的模为 r₁r₂，辐角为 θ₁ + θ₂：

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

In words: multiply the moduli and add the arguments. If θ₁ + θ₂ lies outside the principal range, add or subtract 2π to give the principal argument.

简言之：模相乘，辐角相加。如果 θ₁ + θ₂ 超出主辐角范围，需要加上或减去 2π 来得到主辐角。

3. Geometric Interpretation of Multiplication | 乘法的几何意义

Geometrically, multiplying z₁ by z₂ stretches z₁ by factor r₂ and rotates it anticlockwise through angle θ₂. If r₂ = 1, the transformation is a pure rotation; if θ₂ = 0, it is a pure enlargement by scale factor r₂.

从几何角度看，z₁ 乘以 z₂ 会以原点为中心将 z₁ 的模拉长为原来的 r₂ 倍，并绕原点逆时针旋转 θ₂。若 r₂ = 1，变换是纯旋转；若 θ₂ = 0，则是纯缩放，比例因子为 r₂。

Multiplying by i is the same as rotating anticlockwise by π/2, because i = 1(cos π/2 + i sin π/2).

乘以 i 等同于逆时针旋转 π/2，因为 i = 1(cos π/2 + i sin π/2)。

4. Division Rule: Divide Moduli, Subtract Arguments | 除法法则：模相除，辐角相减

For division, provided r₂ ≠ 0, the quotient has modulus r₁/r₂ and argument θ₁ – θ₂:

对于除法，只要 r₂ ≠ 0，商的模为 r₁/r₂，辐角为 θ₁ – θ₂：

z₁ ÷ z₂ = r₁/r₂ [cos(θ₁ – θ₂) + i sin

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