📚 The General Complex Number | 一般复数
Complex numbers arise naturally when solving quadratic equations such as x² + 1 = 0. Since no real number has a negative square, mathematicians introduce the imaginary unit i with the defining property i² = −1. The general complex number z = x + iy combines a real part x and an imaginary part y, and it can be added, subtracted, multiplied and divided just like an algebraic expression, with i² replaced by −1 whenever it appears. This revision article covers the AQA A-Level Further Mathematics content on the general complex number, including its Cartesian form, conjugate, modulus, argument, polar form and basic loci.
复数在求解诸如 x² + 1 = 0 这样的二次方程时自然出现。由于没有任何实数的平方为负，数学家引入了虚数单位 i，其定义性质为 i² = −1。一般复数 z = x + iy 将实部 x 与虚部 y 结合起来，它可以像代数式一样进行加、减、乘、除运算，只要在出现 i² 时将其替换为 −1。本篇复习文章涵盖 AQA A-Level 高等数学中关于一般复数的内容，包括笛卡尔形式、共轭、模、辐角、极坐标形式与基本轨迹。
1. The Imaginary Unit and the General Form | 虚数单位与一般形式
The imaginary unit i is defined by i² = −1. This is not a real number, so it extends the real number system. Powers of i repeat in a cycle of length 4: i¹ = i, i² = −1, i³ = −i and i⁴ = 1. After that, the pattern repeats, so i⁵ = i, i⁶ = −1, and so on. The general complex number is written as z = x + iy, where x and y are real numbers. This is called Cartesian form because it separates the real direction from the imaginary direction.
虚数单位 i 的定义是 i² = −1。它不是一个实数，因此它扩展了实数系。i 的幂以 4 为周期循环：i¹ = i，i² = −1，i³ = −i，i⁴ = 1。之后规律重复，所以 i⁵ = i，i⁶ = −1 等等。一般复数写作 z = x + iy，其中 x 和 y 都是实数。这种形式称为笛卡尔形式，因为它把实方向与虚方向分离开来。
2. Real and Imaginary Parts | 实部与虚部
For a complex number z = x + iy, the real part is Re(z) = x and the imaginary part is Im(z) = y. Note that the imaginary part is the coefficient y, not iy. For example, if z = 3 − 4i, then Re(z) = 3 and Im(z) = −4. If z = 2i, then Re(z) = 0 and Im(z) = 2, so the number is purely imaginary. If z = 7, then Re(z) = 7 and Im(z) = 0, so the number is purely real.
对于复数 z = x + iy，实部为 Re(z) = x，虚部为 Im(z) = y。注意虚部是系数 y，而不是 iy。例如，若 z = 3 − 4i，则 Re(z) = 3，Im(z) = −4。若 z = 2i，则 Re(z) = 0，Im(z) = 2，所以该数为纯虚数。若 z = 7，则 Re(z) = 7，Im(z) = 0，所以该数为纯实数。
3. Equality of Complex Numbers | 复数相等
Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. In symbols, a + ib = c + id exactly when a = c and b = d. This principle is useful when solving equations containing an unknown real part and imaginary part. For example, if (x + y) + i(x − y) = 3 + i, then we can equate real parts: x + y = 3, and imaginary parts: x − y = 1. Solving simultaneously gives x = 2 and y = 1.
两个复数相等，当且仅当它们的实部相等且虚部相等。用符号表示，a + ib = c + id 当且仅当 a = c 且 b = d。这一原理在求解含有未知实部和虚部的方程时非常有用。例如，若 (x + y) + i(x − y) = 3 + i，则我们可以令实部相等：x + y = 3，令虚部相等：x − y = 1。联立解得 x = 2，y = 1。
4. Addition, Subtraction and Scalar Multiplication | 加法、减法与数乘
Complex numbers are added and subtracted componentwise. If z₁ = a + ib and z₂ = c + id, then z₁ + z₂ = (a + c) + i(b + d) and z₁ − z₂ = (a − c) + i(b − d). Multiplying by a real scalar k gives kz₁ = ka + ikb. For example, (2 + 3i) + (1 − 4i) = 3 − i, and 3(2 + 3i) = 6 + 9i.
复数的加法和减法按分量分别进行。若 z₁ = a + ib，z₂ = c + id，则 z₁ + z₂ = (a + c) + i(b + d)，z₁ − z₂ = (a − c) + i(b − d)。用实数 k 进行数乘得到 kz₁ = ka + ikb。例如，(2 + 3i) + (1 − 4i) = 3 − i，3(2 + 3i) = 6 + 9i。
5. Multiplication of Complex Numbers | 复数乘法
To multiply two complex numbers, expand the brackets using the distributive law and replace i² by −1. If z₁ = a + ib and z₂ = c + id, then z₁z₂ = (a + ib)(c + id) = ac + iad + ibc + i²bd = (ac − bd) + i(ad + bc). For example, (2 + 3i)(1 − 4i) = 2 − 8i + 3i − 12i² = 2 − 5i + 12 = 14 − 5i.
两个复数相乘时，先用分配律展开括号，再将 i² 替换为 −1。若 z₁ = a + ib，z₂ = c + id，则 z₁z₂ = (a + ib)(c + id) = ac + iad + ibc + i²bd = (ac − bd) + i(ad + bc)。例如，(2 + 3i)(1 − 4i) = 2 − 8i + 3i − 12i² = 2 − 5i + 12 = 14 − 5i。
6. The Complex Conjugate | 复共轭
The complex conjugate of z = x + iy is written as z* and is defined by z* = x − iy. Geometrically, the conjugate is the reflection of z in the real axis on an Argand diagram. Key properties include z + z* = 2Re(z), z − z* = 2i Im(z), and zz* = x² + y². The last result is particularly important because it is equal to |z|², the square of the modulus of z. The conjugate of a sum is the sum of the conjugates, the conjugate of a product is the product of the conjugates, and the conjugate of a quotient is the quotient of the conjugates.
复数 z = x + iy 的共轭记作 z*，定义为 z* = x − iy。从几何上看，共轭是 z 在阿尔冈图中关于实轴的反射。关键性质包括 z + z* = 2Re(z)，z − z* = 2i Im(z)，以及 zz* = x² + y²。最后一个结果特别重要，因为它等于 |z|²，即 z 的模的平方。和的共轭等于共轭的和，积的共轭等于共轭的积，商的共轭等于共轭的商。
7. Division of Complex Numbers | 复数除法
To divide one complex number by another, multiply both the numerator and the denominator by the conjugate of the denominator. This makes the denominator real. In general, (a + ib) ÷ (c + id) = [(a + ib)(c − id)] ÷ [(c + id)(c − id)] = [(ac + bd) + i(bc − ad)] ÷ (c² + d²). For example, (3 + 2i) ÷ (1 − i) = [(3 + 2i)(1 + i)] ÷ [(1 − i)(1 + i)] = (1 + 5i) ÷ 2 = 1/2 + (5/2)i.
两个复数相除时，把分子和分母同时乘以分母的共轭。这样可以使分母变为实数。一般地，(a + ib) ÷ (c + id) = [(a + ib)(c − id)] ÷ [(c + id)(c − id)] = [(ac + bd) + i(bc − ad)] ÷ (c² + d²)。例如，(3 + 2i) ÷ (1 − i) = [(3 + 2i)(1 + i)] ÷ [(1 − i)(1 + i)] = (1 + 5i) ÷ 2 = 1/2 + (5/2)i。
8. The Argand Diagram and Modulus | 阿尔冈图与模
An Argand diagram represents complex numbers as points in a plane. The horizontal axis is the real axis and the vertical axis is the imaginary axis. The complex number z = x + iy is plotted as the point (x, y). The modulus of z, written |z|, is the distance from the origin to that point. It is calculated using |z| = √(x² + y²). For example, |3 − 4i| = √(3² + (−4)²) = √25 = 5. The modulus is always a non-negative real number.
阿尔冈图把复数表示为平面上的点。横轴是实轴，纵轴是虚轴。复数 z = x + iy 画作点 (x, y)。z 的模记作 |z|，是原点到该点的距离，计算公式为 |z| = √(x² + y²)。例如，|3 − 4i| = √(3² + (−4)²) = √25 = 5。模始终是非负实数。
Important modulus properties are |z₁z₂| = |z₁||z₂| and |z₁ ÷ z₂| = |z₁| ÷ |z₂| whenever z₂ ≠ 0. These are used to simplify expressions and to solve equations involving moduli.
重要的模性质有 |z₁z₂| = |z₁||z₂|，以及当 z₂ ≠ 0 时，|z₁ ÷ z₂| = |z₁| ÷ |z₂|。这些性质可用于化简表达式和求解含有模的方程。
9. Argument and Principal Argument | 辐角与主辐角
The argument of a non-zero complex number z, written arg z, is the angle measured from the positive real axis to the line joining the origin to the point z on the Argand diagram. If z = x + iy, then tan θ = y/x, but the quadrant of (x, y) must be considered to determine the correct angle. The principal argument, often written Arg z with a capital A, is usually chosen in the interval (−π, π]. For example, arg(1 + i) = π/4, arg(−1 + i) = 3π/4, and arg(−1 − i) = −3π/4 under this convention.
非零复数 z 的辐角记作 arg z，是从正实轴到阿尔冈图中原点到点 z 的连线所成的角。若 z = x + iy，则 tan θ = y/x，但必须考虑点 (x, y) 所在的象限才能确定正确的角。主辐角通常记作 Arg z（首字母大写），一般取在区间 (−π, π] 内。例如，arg(1 + i) = π/4，arg(−1 + i) = 3π/4，在该约定下 arg(−1 − i) = −3π/4。
10. Modulus-Argument Form and Euler’s Formula | 模-辐角形式与欧拉公式
Any non-zero complex number can be written in modulus-argument form, also called
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