Argument and Polar Form of Complex Numbers | 复数的辐角与极坐标形式

📚 Argument and Polar Form of Complex Numbers | 复数的辐角与极坐标形式

Complex numbers are far more than points on a plane; they encode rotation and scaling. The argument of a complex number — the angle it makes with the positive real axis — unlocks elegant formulas for multiplication, powers, and roots. This revision guide explains the argument and polar form step by step, exactly as required by the IB Analysis and Approaches syllabus.

复数远不止是平面上的点，它们还体现了旋转与缩放。复数的辐角——即它与正实轴之间的夹角——为乘法、幂和开方等运算提供了优美的公式。本复习指南将按 IB 数学分析与方法（AA）大纲的要求，逐步讲解辐角与极坐标形式。

1. The Modulus | 模长

For z = a + bi, the modulus is the distance from the origin to (a, b) in the Argand diagram.

对于 z = a + bi，模长是阿尔冈图中从原点到点 (a, b) 的距离。

|z| = √(a² + b²)

For example, if z = 3 + 4i, then |z| = √(3² + 4²) = 5. The modulus is always a non-negative real number.

例如，若 z = 3 + 4i，则 |z| = √(3² + 4²) = 5。模长总是非负实数。

2. The Argument | 辐角

The argument is the angle θ, measured anticlockwise from the positive real axis to the position vector of z.

辐角是自正实轴逆时针转到 z 的位置向量所经过的角度 θ。

Because a full rotation of 2π does not change the point on the Argand diagram, the argument is only defined modulo 2π. The principal argument Arg(z) is therefore chosen in the interval (-π, π].

因为旋转整周 2π 不改变点在阿尔冈图中的位置，辐角只能按 2π 取模来定义。因此主辐角 Arg(z) 通常取在区间 (-π, π] 内。

3. Finding the Principal Argument | 求主辐角

Let z = a + bi and let α = arctan(|b/a|) be the reference angle. The value of Arg(z) depends on the quadrant in which the point lies.

设 z = a + bi，令 α = arctan(|b/a|) 为参考角。Arg(z) 的具体取值取决于点所在的象限。

Special cases on the axes: if a > 0 and b = 0, then Arg(z) = 0; if a < 0 and b = 0, then Arg(z) = π; if a = 0 and b > 0, then Arg(z) = π/2; if a = 0 and b < 0, then Arg(z) = –π/2.

坐标轴上的特殊情况：若 a > 0 且 b = 0，则 Arg(z) = 0；若

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