The Polar Form of a Complex Number | 复数的极坐标形式

📚 The Polar Form of a Complex Number | 复数的极坐标形式

The polar form of a complex number is one of the most powerful representations in A-Level mathematics. Instead of writing a complex number as z = x + yi, we describe it by its distance from the origin and the angle it makes with the positive real axis. This article provides a thorough, exam-focused guide to everything you need to know about the polar form for AQA A-Level Mathematics.

复数的极坐标形式是 A-Level 数学中最强大的表示方法之一。我们不再将复数写作 z = x + yi，而是通过它到原点的距离以及与正实轴所成的角度来描述它。本文将提供一份针对 AQA A-Level 数学考试、全面的极坐标形式学习指南。

1. The Modulus and Argument | 模与辐角

Before we can write a complex number in polar form, we must understand its two components: the modulus and the argument.

在将复数写成极坐标形式之前，我们必须理解它的两个组成部分：模与辐角。

The modulus of a complex number z = x + yi is its distance from the origin on the Argand diagram. It is denoted by |z| and is always non-negative.

复数 z = x + yi 的模是它在阿甘图（复平面）上到原点的距离，记作 |z|，始终为非负数。

|z| = √(x² + y²)

The argument of z is the angle θ that the line from the origin to z makes with the positive real axis, measured anticlockwise. The principal argument is the unique value of θ such that -π < θ ≤ π.

复数 z 的辐角是原点与 z 连线与正实轴之间的夹角 θ，按逆时针方向测量。主辐角是满足 -π < θ ≤ π 的唯一 θ 值。

Example: For z = 1 + i, we have |z| = √(1² + 1²) = √2, and arg(z) = π/4.

例如：对于 z = 1 + i，有 |z| = √(1² + 1²) = √2，且 arg(z) = π/4。

Special cases to memorise: the modulus of a real number a is its absolute value |a|; if a is positive, arg(a) = 0, and if a is negative, arg(a) = π. Pure imaginary numbers have argument π/2 for positive yi and -π/2 for negative yi.

需要记忆的特殊情况：实数 a 的模是其绝对值 |a|；若 a 为正，则 arg(a) = 0；若 a 为负，则 arg(a) = π。纯虚数中，正 yi 的辐角为 π/2，负 yi 的辐角为 -π/2。

2. Expressing a Complex Number in Polar Form | 用极坐标形式表达复数

Once the modulus r and argument θ are known, we can write the complex number in polar form (also called modulus-argument form).

一旦已知模 r 与辐角 θ，我们就可以将复数写成极坐标形式（也称为模-辐角形式）。

z = r(cos θ + i sin θ)

This is derived directly from the Cartesian coordinates: x = r cos θ and y = r sin θ. Substituting these into z = x + yi gives the formula above.

这可以直接从笛卡尔坐标推导得出：x = r cos θ，y = r sin θ。将这些代入 z = x + yi 即可得到上述公式。

For example, the complex number with modulus 2 and argument π/3 can be written as:

例如，模为 2、辐角为 π/3 的复数可以写为：

z = 2(cos(π/3) + i sin(π/3)) = 2(½ + i√3/2) = 1 + i√3

A compact notation commonly used is r cis θ, where cis θ = cos θ + i sin θ. Remember this abbreviation; it is widely used in examination solutions.

一种常用的紧凑写法是 r cis θ，其中 cis θ = cos θ + i sin θ。请记住这个缩写，它在考试解答中被广泛使用。

3. Converting Between Cartesian and Polar Forms | 笛卡尔形式与极坐标形式的相互转换

Converting from Cartesian to polar form requires care, especially when determining the argument. The modulus is straightforward, but the argument must be adjusted for the correct quadrant.

从笛卡尔形式转换为极坐标形式需要特别小心，尤其是在确定辐角时。模的计算很直接，但辐角必须根据所在象限进行调整。

r = √(x² + y²), θ = arctan(y/x) (adjusted for the correct quadrant)

When using θ = arctan(y/x), the calculator gives a value in the range (-π/2, π/2). However, the correct argument may differ by π or 2π depending on which quadrant the complex number lies in. The following table is essential.

当使用 θ = arctan(y/x) 时，计算器给出的值在 (-π/2, π/2) 范围内。然而，正确的辐角可能相差 π 或 2π，具体取决于复数所在的象限。下表至关重要。