📚 The Modulus and Argument of a Complex Number | 复数的模与辐角
In AQA A-Level Mathematics, complex numbers are often manipulated in both Cartesian form and modulus-argument form. Understanding the modulus and argument is essential for multiplication, division, powers, and many geometric applications.
在 AQA A-Level 数学中，复数常以直角坐标形式和模-辐角形式进行运算。理解模和辐角对于乘法、除法、幂以及许多几何应用都至关重要。
1. Complex Numbers on the Argand Diagram | 阿尔冈图上的复数
) and y is the imaginary part Im(z). On an Argand diagram, z is represented by the point (x, y), so the complex plane works like a two-dimensional coordinate system.
复数通常写作 z = x + iy，其中 x 是实部 Re(z)，y 是虚部 Im(z)。在阿尔冈图上，z 表示为点 (x, y)，因此复平面就像一个二维坐标系。
Although Cartesian coordinates give one useful description, many AQA questions focus on the polar description: how far the point is from the origin and what angle it makes with the positive real axis. These two quantities are the modulus and the argument.
虽然直角坐标是一种有用的描述方式，但许多 AQA 题目关注极坐标描述：点离原点有多远，以及与正实轴所成的角度。这两个量就是模和辐角。
2. Definition of the Modulus | 模的定义
y is the distance from the origin to the point (x, y). It is always a non-negative real number and is written |z|.
复数 z = x + iy 的模是原点到点 (x, y) 的距离。它始终是一个非负实数，记作 |z|。
|z| = √x² + y²
For example, if z = 3 + 4i, then |z| = √3² + 4² = √25 = 5.
例如，若 z = 3 + 4i，则 |z| = √3² + 4² = √25 = 5。
If z is real, say z = a, then |z| = √a² = |a|, which matches the usual absolute value. If z is purely imaginary, say z = bi, then |z| = |b|. This shows that the modulus generalises the idea of absolute value to two dimensions.
如果 z 是实数，例如 z = a，则 |z| = √a² = |a|，这与通常的绝对值一致。如果 z 是纯虚数，例如 z = bi，则 |z| = |b|。这说明模将绝对值的概念推广到了二维情形。
3. Definition of the Argument | 辐角的定义
The argument of z, denoted arg z, is the angle θ measured from the positive real axis to the line joining the origin to the point (x, y). It is usually measured in radians.
复数 z 的辐角记作 arg z，是从正实轴到连接原点与点 (x, y) 的直线所成的角 θ。它通常以弧度为单位。
For z = x + iy, the argument θ must satisfy the relationship:
对于 z = x + iy，辐角 θ 必须满足以下关系：
tan θ = y / x, where x ≠ 0
However, the argument is not defined for z = 0, because the zero vector has no direction.
极坐标关系 x = r cos θ 和 y = r sin θ 给出同一个角。不过，z = 0 时辐角没有定义，因为零向量没有方向。
4. Principal Argument and Conventions | 主辐角与约定
e principal argument is the value of θ in a chosen interval. AQA normally uses −π < θ ≤ π.
相差 2π 的角度描述同一方向，因此一个复数可以有无穷多个辐角。主辐角是 θ 在一个选定区间内的值。AQA 通常使用 −π < θ ≤ π。
When finding the principal argument from x and y, avoid quoting arctan(y / x) blindly: the arctan function only returns values between −π/2 and π/2. You must adjust for the quadrant.
在由 x 和 y 求主辐角时，不要盲目地写成 arctan(y / x)：反正切函数只返回 −π/2 到 π/2 之间的值。你必须根据象限进行调整。
Let α = arctan(|y / x|) be the acute reference angle. The principal argument is then determined as follows:
设 α = arctan(|y / x|) 为锐角参考角。主辐角则按下表确定：
| Quadrant / 象限 | Sign of x, y / x, y 的符号 | Principal argument / 主辐角 |
|---|---|---|
| First / 第一象限 | x > 0, y > 0 | θ = α |
| Second / 第二象限 | x < 0, y > 0 | θ = π − α |
| Third / 第三象限 | x < 0, y < 0 | θ = −π + α |
| Fourth / 第四象限 | x > 0, y < 0 | θ = −α |
On the axes, the principal argument is 0 for a positive real number, π for a negative real number, π/2 for a positive imaginary number, and −π/2 for a negative imaginary number.
在坐标轴上，正实数的主辐角为 0，负实数的主辐角为 π，正虚数的主辐角为 π/2，负虚数的主辐角为 −π/2。
5. From Cartesian to Modulus-Argument Form | 从直角坐标到模-辐角形式
To convert z = x + iy into modulus-argument form, first calculate r = √x² + y². Then calculate the acute reference angle α = arctan(|y / x| ly identify the quadrant from the signs of x and y and use the correct principal argument.
要将 z = x + iy 转换为模-辐角形式，首先计算 r = √x² + y²。然后计算锐角参考角 α = arctan(|y / x|)。最后根据 x 和 y 的符号确定象限，并使用正确的主辐角。
Example: z = −√3 + i. Here x = −√3 and y = 1, so r = √3 + 1 = 2. The reference angle is α = arctan(1/√3 π/6. Since x < 0 and y > 0, the point lies in the second quadrant, so θ = π − π/6 = 5π/6.
例题：z = −√3 + i。这里 x = −√3，y = 1，因此 r = √3 + 1 = 2。参考角为 α = arctan(1/√3) = π/6。由于 x < 0 且 y > 0，该点位于第二象限，所以 θ = π − π/6 = 5π/6。
z = −√3 + i → r = 2, θ = 5π/6
Always write r in exact form where possible, such as √2 or 2√3, rather than rounding to a decimal. The argument should also be given as an exact multiple or fraction of π when the question expects exact values.
在可能的情况下，r 应写成精确形式，如 √2 或 2√3，而不是四舍五入为小数。当题目要求精确值时，辐角也应写成 π 的精确倍数或分数。
6. Writing a Complex Number in Modulus-Argument Form | 用模-辐角形式表示复数
he modulus-argument form, also called polar form, writes z as r(cos θ + i sin θ). Here r = |z| and θ = arg z. This form is especially useful for multiplication, division and powers.
模-辐角形式也称为极坐标形式，将 z 写作 r(cos θ + i sin θ)。这里 r = |z|，θ = arg z。这种形式对于乘法、除法和幂运算特别有用。
z = r (cos θ + i sin θ)
cis θ, where cis θ means cos θ + i sin θ. In AQA exams, you should be comfortable with both notations.
一些教材使用简写 z = r cis θ，其中 cis θ 表示 cos θ + i sin θ。在 AQA 考试中，你应当熟悉这两种记号。
From the example
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