Introduction to Complex Numbers | 复数入门

📚 Introduction to Complex Numbers | 复数入门

Complex numbers extend the real number system by introducing the imaginary unit i, defined by i² = -1. They are a fundamental topic in A-Level mathematics, enabling us to solve equations that have no real solutions and providing powerful tools across pure and applied mathematics.

复数通过在实数系中引入虚数单位 i（定义 i² = -1）来扩展数系。它们是 A-Level 数学的基础课题，使我们能够求解没有实数解的方程，并为纯数学与应用数学提供强大工具。

1. Why Do We Need Complex Numbers? | 为什么需要复数？

The equation x² = -1 has no real solution, since the square of any real number is non-negative. To solve such equations, we define a new number i such that i² = -1.

方程 x² = -1 没有实数解，因为任何实数的平方都是非负的。为了求解这样的方程，我们定义一个新数 i，使得 i² = -1。

With this single definition, we can solve any quadratic equation, even when the discriminant is negative. For example, x² = -9 gives x = ±3i.

凭借这一定义，我们可以求解任何二次方程，即使判别式为负。例如，x² = -9 给出 x = ±3i。

2. The Imaginary Unit i | 虚数单位 i

The powers of i follow a repeating pattern that is essential for simplifying expressions:

i 的幂遵循一个重要的循环规律，可用于化简表达式：

i¹ = i, i² = -1, i³ = -i, i⁴ = 1

The pattern repeats every four powers:

该循环每四个幂重复一次：

n	1	2	3	4	5	6
iⁿ	i	-1	-i	1	i	-1

To simplify a high power such as i²³, divide the exponent by 4: 23 = 5×4 + 3, so i²³ = i³ = -i.

要化简高次幂如 i²³，将指数除以 4：23 = 5×4 + 3，因此 i²³ = i³ = -i。

3. Standard Form: a + bi | 标准形式：a + bi

A complex number is written in standard form as z = a + bi, where a and b are real numbers. Here a is called the real part, written Re(z), and b is called the imaginary part, written Im(z).

复数以标准形式写作 z = a + bi，其中 a 和 b 是实数。a 称为实部，记作 Re(z)；b 称为虚部，记作 Im(z)。

For example, in z = 3 – 4i, we have Re(z) = 3 and Im(z) = -4. Note that the imaginary part is a real number multiplied by i.

例如，对于 z = 3 – 4i，Re(z) = 3，Im(z) = -4。注意虚部是乘以 i 的实数。

If b = 0, the complex number is purely real; if a = 0, it is purely imaginary.

若 b = 0，该复数为纯实数；若 a = 0，则为纯虚数。

4. Equality of Complex Numbers | 复数的相等

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal:

两个复数相等当且仅当它们的实部相等且虚部相等：

a + bi = c + di ⇔ a = c and b = d

This property allows us to equate coefficients. For example, if (x + 2) + (y – 3)i = 5 – i, then x + 2 = 5 and y – 3 = -1, giving x = 3 and y = 2.

这一性质使我们能够比较系数。例如，若 (x + 2) + (y – 3)i = 5 – i，则 x + 2 = 5，y – 3 = -1，解得 x = 3，y = 2。

5. Adding and Subtracting Complex Numbers | 复数的加减法

To add or subtract complex numbers, simply add or subtract their real parts and imaginary parts separately:

复数相加减时，只需分别对实部和虚部进行加减：

(a + bi) + (c + di) = (a + c) + (b + d)i

Example: (3 + 2i) + (1 – 4i) = 4 – 2i, and (3 + 2i) – (1 – 4i) = 2 + 6i.

示例：(3 + 2i) + (1 – 4i) = 4 – 2i；(3 + 2i) – (1 – 4i) = 2 + 6i。

6. Multiplying Complex Numbers | 复数的乘法

Multiplication uses the distributive law (expanding brackets), with i² replaced by -1:

乘法使用分配律（展开括号），并将 i² 替换为 -1：

(a + bi)(c + di) = ac + adi + bci + bdi² = (ac – bd) + (ad + bc)i

Example: (2 + 3i)(1 – 4i) = 2 – 8i + 3i – 12i² = 2 – 5i + 12 = 14 – 5i.

示例：(2 + 3i)(1 – 4i) = 2 – 8i + 3i – 12i² = 2 – 5i + 12 = 14 – 5i。

Notice that multiplying two complex numbers generally produces another complex number, and the process is exactly the same as expanding two linear brackets.

注意，两个复数相乘通常得到另一个复数，其过程与展开两个线性括号完全相同。

7. Complex Conjugates | 共轭复数

The conjugate of z = a + bi is z̄ = a – bi, where the imaginary part changes sign while the real part stays the same.

z = a + bi 的共轭复数为 z̄ = a – bi，即虚部变号，实部保持不变。

Two important results are that their sum and product are real numbers:

两个重要结论是：共轭复数的和与积都是实数：

z + z̄ = 2a, z·z̄ = a² + b²

Example: if z = 3 + 4i, then z̄ = 3 – 4i, z + z̄ = 6, and z·z̄ = 9 + 16 = 25.

示例：若 z = 3 + 4i，则 z̄ = 3 – 4i，z + z̄ = 6，z·z̄ = 9 + 16 = 25。

8. Dividing Complex Numbers | 复数的除法

To divide by a complex number, multiply the numerator and denominator by the conjugate of the denominator to make the denominator real:

复数相除时，将分子分母同时乘以分母的共轭复数，使分母变为实数：

(1 + 2i)/(3 – i) = (1 + 2i)(3 + i)/((3 – i)(3 + i)) = (1 + 7i)/10 = 1/10 + 7i/10

This process is called rationalising the denominator, and the answer should always be expressed in standard form a + bi.

这一过程称为分母有理化，最终答案应始终表示为标准形式 a + bi。

9. The Argand Diagram | 阿尔冈图

A complex number z = a + bi can be represented as a point on the Argand diagram, where the horizontal axis is the real axis and the vertical axis is the imaginary axis. The number a + bi corresponds to the point (a, b).

复数 z = a + bi 可以表示为阿尔冈图上的一个点，其中水平轴为实轴，垂直轴为虚轴。数 a + bi 对应坐标 (a, b)。

For example, z₁ = 3 + 2i is plotted at (3, 2), and z₂ = -1 – i is plotted at (-1, -1). The diagram also allows complex numbers to be treated as position vectors from the origin.

例如，z₁ = 3 + 2i 对应点 (3, 2)，z₂ = -1 – i 对应点 (-1, -1)。在图中，复数还可视为从原点出发的位置向量。

10. Modulus and Argument | 模和辐角

The modulus of z = a + bi is its distance from the origin, given by:

z = a + bi 的模是它到原点的距离：

|z| = √(a² + b²)

The argument, denoted arg(z), is the angle θ between the positive real axis and the line joining the origin to the point, measured in radians. For z = a + bi, tan θ = b/a, with θ adjusted for the correct quadrant.

辐角记作 arg(z)，是正实轴与连接原点和该点的线段之间的夹角 θ，以弧度为单位。对于 z = a + bi，tan θ = b/a，需根据象限调整 θ。

Example: for z = 1 + i, |z| = √2, and since the point lies in the first quadrant, arg(z) = π/4.

示例：对于 z = 1 + i，|z| = √2，由于点在第一象限，arg(z) = π/4。

11. Solving Quadratic Equations with Complex Roots | 求解含复数根的二次方程

When the discriminant of a quadratic equation is negative, the equation has two complex conjugate roots. For example, solve x² + 4x + 13 = 0 using the quadratic formula:

当二次方程的判别式为负时，方程有两个共轭复根。例如，用求根公式解 x² + 4x + 13 = 0：

x = (-4 ± √(16 – 52))/2 = (-4 ± √(-36))/2 = (-4 ± 6i)/2 = -2 ± 3i

The two solutions are x = -2 + 3i and x = -2 – 3i, which are complex conjugates. This is a general result: any quadratic equation with real coefficients and a negative discriminant has two roots that are complex conjugates of each other.

两个解为 x = -2 + 3i 和 x = -2 – 3i，它们互为共轭复数。这是一般结论：实系数二次方程若判别式为负，则其两根互为共轭复数。

Complex numbers also appear in higher-degree polynomial equations, electrical engineering, and quantum mechanics, making them an indispensable tool throughout advanced mathematics.

复数还出现在高次多项式方程、电气工程和量子力学等领域，使其成为高等数学中不可或缺的工具。

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Introduction to Complex Numbers | 复数入门