Products and Quotients of Complex Numbers in Polar Form | 极坐标形式下复数的乘除运算

📚 Products and Quotients of Complex Numbers in Polar Form | 极坐标形式下复数的乘除运算

In this article, we explore the elegant rules for multiplying and dividing complex numbers in polar form. This is a key topic in AQA A-Level Mathematics, appearing in the Pure Mathematics paper, and understanding these rules will save you significant time in exams.

本文将深入探讨极坐标形式下复数乘除运算的优雅法则。这是 AQA A-Level 数学的重点内容，出现在纯数试卷中。掌握这些规则将为你节省大量考试时间。

1. The Polar Form | 极坐标形式

The polar form, also called the modulus-argument form, of a complex number is written as z = r(cos θ + i sin θ), where r is the modulus and θ is the argument. For a complex number z = a + bi, the modulus is calculated as r = √(a² + b²), and the argument satisfies tan θ = b/a.

复数的极坐标形式（又称模幅式）写作 z = r(cos θ + i sin θ)，其中 r 是模长，θ 是幅角。对于复数 z = a + bi，模长计算为 r = √(a² + b²)，幅角满足 tan θ = b/a。

The value of θ must be determined according to the quadrant in which the point (a, b) lies. In A-Level mathematics, the principal argument is conventionally taken to lie in the interval -π < θ ≤ π.

θ 的取值必须依据点 (a, b) 所在的象限确定。在 A-Level 数学中，主幅角通常取在区间 -π < θ ≤ π 内。

For example, the complex number -1 + i has modulus √((-1)² + 1²) = √2 and principal argument 3π/4, not -π/4, because it lies in the second quadrant. Always sketch the Argand diagram when converting to avoid this mistake.

例如，复数 -1 + i 的模长为 √((-1)² + 1²) = √2，主幅角为 3π/4 而非 -π/4，因为它位于第二象限。转换时务必画出阿甘图（复平面图）以避免此类错误。

2. The Product Rule | 乘法法则

Let z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂). The product is given by:

设 z₁ = r₁(cos θ₁ + i sin θ₁)，z₂ = r₂(cos θ₂ + i sin θ₂)。它们的乘积为：

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

The key insight is that moduli multiply and arguments add. When multiplying two complex numbers in polar form, we simply multiply the two moduli and add the two arguments.

关键规律是模长相乘，幅角相加。两个极坐标形式的复数相乘时，只需将两个模长相乘，并将两个幅角相加。

To verify this rule, we expand the product and apply the compound angle identities:

为验证这一法则，我们展开乘积并应用两角和公式：

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