IB Mathematics: Complex Numbers - Concepts and Operations | IB数学：复数的概念与运算

📚 IB Mathematics: Complex Numbers – Concepts and Operations | IB数学：复数的概念与运算

Complex numbers extend the real number system and are fundamental to higher mathematics, physics and engineering. In IB Mathematics, understanding their definitions and algebraic operations is essential for solving equations that have no real solutions.

复数将实数系扩展到了更广阔的领域，是高等数学、物理学和工程学的基础。在IB数学中，理解复数的定义及其代数运算是解无实数解方程的关键。

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

Not every quadratic equation has real roots. For example, x² + 1 = 0 has no real solution because no real number squared equals −1.

并非所有二次方程都有实数根。例如，x² + 1 = 0 没有实数解，因为没有任何实数的平方等于 −1。

To solve such equations, we introduce a new number i whose square is −1, creating the set of complex numbers.

为了求解这类方程，我们引入一个新数 i，规定其平方等于 −1，从而构建出复数集。

This extension preserves all arithmetic rules while allowing square roots of negative numbers to be handled consistently.

这种扩展保留了所有算术法则，同时使负数的平方根能够被一致地处理。

2. The Imaginary Unit i | 虚数单位 i

The imaginary unit i is defined by the property i² = −1. From this, we can derive all powers of i.

虚数单位 i 由性质 i² = −1 定义。据此可以推导出 i 的所有幂次。

i = √(−1), i² = −1, i³ = −i, i⁴ = 1

Notice that the powers of i repeat every four steps. This cyclic pattern is extremely useful when simplifying large exponents.

注意 i 的幂每四次循环一次。这一循环规律在化简高次幂时非常有用。

For example, i²⁰²³ = i^(2023 mod 4) = i³ = −i.

例如，i²⁰²³ = i^(2023 mod 4) = i³ = −i。

3. Definition of a Complex Number | 复数的定义

A complex number z is written in the form z = a + bi, where a and b are real numbers and i is the imaginary unit.

复数 z 写作 z = a + bi 的形式，其中 a 和 b 是实数，i 是虚数单位。

Here a is called the real part, denoted Re(z), and b is called the imaginary part, denoted Im(z).

其中 a 称为实部，记为 Re(z)；b 称为虚部，记为 Im(z)。

Note that the imaginary part is the coefficient of i, not the term bi itself.

注意虚部是 i 的系数，而不是 bi 这一项本身。

When b = 0, z reduces to a real number; when a = 0, z is called a purely imaginary number.

当 b = 0 时，z 退化为实数；当 a = 0 时，z 称为纯虚数。

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。

If a + bi = c + di, then a = c and b = d

This property allows us to equate coefficients when solving equations involving complex numbers.

这一性质允许我们在解复数方程时进行系数比较。

For example, if (x + y) + (x − y)i = 3 + 5i, then x + y = 3 and x − y = 5, giving x = 4 and y = −1.

例如，若 (x + y) + (x − y)i = 3 + 5i，则 x + y = 3，x − y = 5，解得 x = 4，y = −1。

5. The Complex Plane | 复平面

A complex number z = a + bi can be represented as a point (a, b) in the complex plane, often called an Argand diagram.

复数 z = a + bi 可以表示为复平面上的点 (a, b)，该平面通常称为阿甘图。

The horizontal axis is the real axis, and the vertical axis is the imaginary axis.

横轴称为实轴，纵轴称为虚轴。

This geometric interpretation connects complex numbers with vectors and enables polar representations.

这种几何解释将复数与向量联系起来，并引出了极坐标表示。

The position of a point can also be described by its distance from the origin and the angle it makes with the positive real axis.

点的位置还可以通过与原点的距离以及与正实轴所成的角来描述。

6. Addition and Subtraction of Complex Numbers | 复数的加法与减法

Addition and subtraction are performed by combining real parts and imaginary parts separately.

加法和减法通过分别合并实部和虚部来完成。

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

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

These operations follow the usual rules of algebra, treating i as a placeholder while remembering i² = −1.

这些运算遵循通常的代数法则，将 i 视为占位符，同时记住 i² = −1。

Geometrically, addition corresponds to the parallelogram law of vectors.

在几何上，加法对应向量的平行四边形法则。

7. Multiplication of Complex Numbers | 复数的乘法

Multiplying two complex numbers uses the distributive law and the fact that i² = −1.

两个复数相乘使用分配律，并利用 i² = −1。

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

Therefore the real part is ac − bd and the imaginary part is ad + bc.

因此实部为 ac − bd，虚部为 ad + bc。

For a quick check, multiply (2 + 3i)(1 − i):

快速验证：(2 + 3i)(1 − i)：

= 2 − 2i + 3i − 3i² = 2 + i + 3 = 5 + i

8. Complex Conjugates | 共轭复数

The conjugate of a complex number z = a + bi is denoted by z̄ (or z*), and is defined as z̄ = a − bi.

复数 z = a + bi 的共轭记为 z̄（或 z*），定义为 z̄ = a − bi。

The conjugate has the same real part but the opposite imaginary part.

共轭复数的实部相同，虚部相反。

Key properties of conjugation include:

共轭的主要性质包括：

(z̄)̄ = z — the conjugate of a conjugate returns the original number.

(z̄)̄ = z —— 共轭的共轭回到原数。
z + z̄ = 2Re(z) — the sum of a number and its conjugate is twice the real part.

z + z̄ = 2Re(z) —— 一个数与其共轭之和等于两倍实部。
z − z̄ = 2b i — the difference is twice the imaginary part times i.

z − z̄ = 2b i —— 之差等于两倍虚部乘以 i。
(z₁ ± z₂)̄ = z̄₁ ± z̄₂ — conjugation distributes over addition and subtraction.

(z₁ ± z₂)̄ = z̄₁ ± z̄₂ —— 共轭对加减法可分配。
(z₁ z₂)̄ = z̄₁ z̄₂ — conjugation distributes over multiplication.

(z₁ z₂)̄ = z̄₁ z̄₂ —— 共轭对乘法可分配。

9. Division of Complex Numbers | 复数的除法

Division of complex numbers is performed by multiplying the numerator and denominator by the conjugate of the denominator.

复数除法通过将分子和分母同时乘以分母的共轭来完成。

(a + bi) / (c + di) = [(a + bi)(c − di)] / [(c + di)(c − di)]

The denominator becomes a real number because (c + di)(c − di) = c² + d².

分母变成实数，因为 (c + di)(c − di) = c² + d²。

Hence the result is:

因此结果为：

[(ac + bd) + (bc − ad)i] / (c² + d²)

For example, (1 + 2i) / (3 − 4i):

例如，(1 + 2i) / (3 − 4i)：

= [(1 + 2i)(3 + 4i)] / (9 + 16) = (3 + 4i + 6i + 8i²) / 25 = (−5 + 10i) / 25 = −0.2 + 0.4i

10. Modulus of a Complex Number | 复数的模

The modulus of z = a + bi is its distance from the origin in the complex plane, denoted |z|.

复数 z = a + bi 的模是它在复平面上到原点的距离，记为 |z|。

|z| = √(a² + b²)

Important properties of the modulus include:

模的重要性质包括：

|z| = |z̄| — a number and its conjugate have the same modulus.

|z| = |z̄| —— 一个数与其共轭具有相同的模。
z z̄ = |z|² — the product of a number and its conjugate equals the square of the modulus.

z z̄ = |z|² —— 一个数与其共轭之积等于模的平方。
|z₁ z₂| = |z₁| |z₂| — the modulus of a product is the product of the moduli.

|z₁ z₂| = |z₁| |z₂| —— 乘积的模等于模的乘积。
|z₁ / z₂| = |z₁| / |z₂| (z₂ ≠ 0) — the modulus of a quotient is the quotient of the moduli.

|z₁ / z₂| = |z₁| / |z₂| (z₂ ≠ 0) —— 商的模等于模的商。

The modulus also enables the polar form of a complex number, which is frequently used in IB Mathematics.

模还促成了复数的极坐标形式，这是IB数学中常用的表示。

11. Powers and Roots of i | i 的幂与根

Because i repeats in cycles of four, any power iⁿ can be simplified using n mod 4.

由于 i 以四为周期循环，任何 iⁿ 都可以用 n mod 4 化简。

i¹ = i, i² = −1, i³ = −i, i⁴ = 1

For n ≥ 0, if n ≡ 0 mod 4, iⁿ = 1; if n ≡ 1 mod 4, iⁿ = i; if n ≡ 2 mod 4, iⁿ = −1; if n ≡ 3 mod 4, iⁿ = −i.

对于 n ≥ 0，若 n ≡ 0 mod 4，则 iⁿ = 1；若 n ≡ 1 mod 4，则 iⁿ = i；若 n ≡ 2 mod 4，则 iⁿ = −1；若 n ≡ 3 mod 4，则 iⁿ = −i。

This pattern also applies to negative exponents using the reciprocal, for example i⁻¹ = −i.

该模式也适用于负指数，例如 i⁻¹ = −i。

Square roots of negative numbers can be expressed using i: √(−9) = 3i.

负数的平方根可以用 i 表示：√(−9) = 3i。

12. Solving Equations with Complex Solutions | 求解含复数解的方程

Many quadratic equations have complex conjugate roots when the discriminant is negative.

许多判别式为负的二次方程具有一对共轭复根。

For example, solve x² + 2x + 5 = 0 using the quadratic formula:

例如，用求根公式解 x² + 2x + 5 = 0：

x = [−2 ± √(4 − 20)] / 2 = [−2 ± √(−16)] / 2 = [−2 ± 4i] / 2 = −1 ± 2i

Thus the solutions are −1 + 2i and −1 − 2i, which are conjugates of each other.

因此解为 −1 + 2i 和 −1 − 2i，它们互为共轭。

This result is guaranteed by the Fundamental Theorem of Algebra: every non-constant polynomial has at least one complex root.

这一结果由代数基本定理保证：每个非常数多项式至少有一个复数根。

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IB Mathematics: Complex Numbers – Concepts and Operations | IB数学：复数的概念与运算