📚 Complex Numbers as 2-D Vectors | 复数作为二维向量
At AS level, a complex number is often first met as an expression a + bi, but algebra alone misses the picture. Treating a complex number as a two-dimensional vector on the Argand diagram turns many operations into simple geometric transformations.
在 AS 阶段，复数最初常常被当作 a + bi 这样的表达式，但仅靠代数会错过整体图像。把复数看作阿尔冈图上的二维向量，可以将许多运算变成简单的几何变换。
1. From Real Line to Complex Plane | 从实数轴到复平面
The real numbers lie on a one-dimensional number line. Introducing i, where i² = -1, creates a new number a + bi that needs two real coordinates: a on the real axis and b on the imaginary axis.
实数位于一维数轴上。引入满足 i² = -1 的 i 后，得到新的数 a + bi，它需要两个实坐标：实轴上的 a 和虚轴上的 b。
This plane is called the Argand diagram or complex plane. The real part Re(z) is the horizontal coordinate, and the imaginary part Im(z) is the vertical coordinate.
这个平面称为阿尔冈图或复平面。实部 Re(z) 是水平坐标，虚部 Im(z) 是垂直坐标。
2. Representing z = x + iy as a Position Vector | 用位置向量表示 z=x+iy
A complex number z = x + iy can be written as the vector (x, y) from the origin to the point (x, y). This is exactly the same idea as a two-dimensional position vector in mechanics or pure geometry.
复数 z = x + iy 可以写成从原点到点 (x, y) 的向量 (x, y)。这与力学或纯几何中的二维位置向量完全相同。
For example, z = 3 + 4i corresponds to the vector 3i + 4j, or the column vector (3, 4). The i in the vector basis and the imaginary unit i are different objects, so keep the notation clear.
例如，z = 3 + 4i 对应向量 3i + 4j，即列向量 (3, 4)。向量基中的 i 与虚数单位 i 是不同的对象，因此要保持记号清晰。
3. Addition as Vector Addition | 加法即向量加法
If z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂, then z₁ + z₂ = (x₁ + x₂) + i(y₁ + y₂). This is the same as adding the two position vectors component by component.
若 z₁ = x₁ + iy₁，z₂ = x₂ + iy₂，则 z₁ + z₂ = (x₁ + x₂) + i(y₁ + y₂)。这与对两个位置向量逐分量相加完全一致。
Geometrically, addition follows the parallelogram rule or the triangle rule. The sum z₁ + z₂ is the fourth vertex of the parallelogram built on the vectors of z₁ and z₂.
从几何上看，加法遵循平行四边形法则或三角形法则。和 z₁ + z₂ 是以 z₁ 和 z₂ 的向量为邻边所作平行四边形的第四个顶点。
4. Scalar Multiplication and Stretching | 标量乘法与伸缩
Multiplying a complex number by a real number k gives kz = kx + iky. In vector language this is scalar multiplication, which stretches the vector by a factor |k| and reverses its direction if k < 0.
将一个复数乘以实数 k 得到 kz = kx + iky。用向量语言来说，这就是标量乘法，它以 |k| 为倍数伸长向量；若 k < 0，则方向反转。
For instance, 2(1 + 3i) = 2 + 6i doubles both coordinates, while -1(2 + i) = -2 – i sends the vector to the opposite quadrant.
例如，2(1 + 3i) = 2 + 6i 使两个坐标都加倍，而 -1(2 + i) = -2 – i 把向量送到相反的象限。
5. Modulus as Vector Length | 模作为向量长度
The modulus of z = x + iy is |z| = √(x² + y²). This is exactly the length of the position vector (x, y) from the origin to the point representing z.
z = x + iy 的模是 |z| = √(x² + y²)。这正是从原点到表示 z 的点处位置向量 (x, y) 的长度。
For example, |3 – 4i| = √(3² + (-4)²) = 5. The modulus is always real and non-negative, just like a distance.
例如，|3 – 4i| = √(3² + (-4)²) = 5。模总是实数且非负，就像距离一样。
In vector notation, this is equivalent to the magnitude of the vector, often written as ||(x, y)||.
用向量记号，这等价于向量的模长，通常写成 ||(x, y)||。
6. Conjugate as Reflection in the Real Axis | 共轭作为关于实轴的镜像
The complex conjugate of z = x + iy is z* = x – iy. Geometrically, this changes the sign of the imaginary part, so the point (x, y) is reflected to (x, -y) across the real axis.
z = x + iy 的共轭复数是 z* = x – iy。从几何上看，它改变了虚部的符号，因此点 (x, y) 关于实轴反射到 (x, -y)。
This reflection property helps explain why z + z* = 2x is real and z – z* = 2iy is purely imaginary: the reflected vector cancels or doubles one component.
这个反射性质有助于解释为什么 z + z* = 2x 是实数，而 z – z* = 2iy 是纯虚数：反射后的向量抵消或加倍了其中一个分量。
7. Argument as Direction Angle | 辐角作为方向角
The argument of a non-zero complex number, arg(z), is the angle θ measured from the positive real axis to the position vector, usually in the interval -π < θ ≤ π.
非零复数的辐角 arg(z) 是从正实轴到位置向量的角 θ，通常取区间 -π < θ ≤ π。
If z = x + iy, then tan θ = y/x, but the quadrant matters. For example, z = -1 + i has arg(z) = 3π/4, not -π/4, because the vector lies in the second quadrant.
若 z = x + iy，则 tan θ = y/x，但所在象限很重要。例如 z = -1 + i 的辐角为 3π/4，而不是 -π/4，因为该向量位于第二象限。
Together, the modulus r and argument θ give the polar form z = r(cos θ + i sin θ), which is the vector specified by length and direction.
模 r 与辐角 θ 共同给出极坐标形式 z = r(cos θ + i sin θ)，也就是由长度和方向确定的向量。
8. Subtraction and Distance | 减法与距离
Subtraction z₁ – z₂ = (x₁ – x₂) + i(y₁ – y₂) corresponds to the vector from the point z₂ to the point z₁. It is the displacement vector, not a position vector from the origin.
减法 z₁ – z₂ = (x₁ – x₂) + i(y₁ – y₂) 对应从点 z₂ 到点 z₁ 的向量。它是位移向量，而不是从原点出发的位置向量。
The modulus |z₁ – z₂| is therefore the distance between the two points in the complex plane. For example, the distance between 1 + 2i and 4 + 6i is |(1+2i)-(4+6i)| = |-3-4i| = 5.
因此，|z₁ – z₂| 是复平面上两点之间的距离。例如，1 + 2i 与 4 + 6i 之间的距离为 |(1+2i)-(4+6i)| = |-3-4i| = 5。
This distance formula is widely used to describe circles: |z – a| = r represents all points z at a fixed distance r from the centre a.
这个距离公式被广泛用于描述圆：|z – a| = r 表示所有到中心 a 的距离为固定值 r 的点 z。
9. Multiplication by i as 90° Rotation | 乘以 i 作为 90° 旋转
Multiplying z = x + iy by i gives iz = i(x + iy) = -y + ix. In vector
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