Introduction to Complex Numbers | 复数入门

📚 Introduction to Complex Numbers | 复数入门

In A Level Mathematics, many equations have solutions within the real number system, but some simple equations do not. For example, x² + 1 = 0 has no real solution because no real number squared can give −1. Complex numbers extend the number system so that every polynomial equation has a solution, giving us a powerful tool for algebra, geometry and applied mathematics.

在 A Level 数学中，很多方程在实数范围内有解，但有些简单的方程没有实数解。例如，x² + 1 = 0 没有实数解，因为任何实数的平方都不可能等于 −1。复数扩展了数系，使每一个多项式方程都有解，为代数、几何和应用数学提供了强大工具。

1. From Real Numbers to Complex Numbers | 从实数到复数

The real number system includes rational and irrational numbers, such as 3, ½, √2 and π. However, the square root of a negative number is not defined as a real number. Complex numbers are introduced by defining a new number whose square is negative.

实数系包括有理数和无理数，例如 3、½、√2 和 π。但是，负数的平方根在实数范围内没有定义。复数通过定义一种平方为负数的新数来引入。

Historically, mathematicians called the solution of x² = −1 an imaginary number. Although the name suggests it is not real, imaginary and complex numbers are legitimate mathematical objects with rigorous definitions and many real-world applications in physics, engineering and signal processing.

历史上，数学家把 x² = −1 的解称为虚数。虽然这个名字听起来不像真实存在，但虚数和复数是严格的数学对象，在物理、工程和信号处理中有许多实际应用。

2. The Imaginary Unit i | 虚数单位 i

The imaginary unit is denoted by i and is defined by the equation:

虚数单位记作 i，其定义为：

i² = −1

It follows that i is a square root of −1. We often write √−1 = i, but in calculations we use the defining property i² = −1 rather than treating i as an ordinary square root.

因此 i 是 −1 的一个平方根。我们常写作 √−1 = i，但在计算中应使用定义性质 i² = −1，而不是把 i 当作普通平方根来处理。

The powers of i repeat in a cycle of four. The key values are:

i 的幂以四为周期循环，关键值为：

i¹ = i	i² = −1
i³ = −i	i⁴ = 1

After i⁴ the pattern repeats, so i⁵ = i, i⁶ = −1, i⁷ = −i and i⁸ = 1. This cycle is useful when simplifying higher powers of i.

i⁴ 之后模式重复，因此 i⁵ = i，i⁶ = −1，i⁷ = −i，i⁸ = 1。这个循环在化简 i 的高次幂时非常有用。

3. Standard Form and Terminology | 标准形式与术语

A complex number is written in standard form as:

复数写成标准形式为：

z = a + bi

where a and b are real numbers. The real part of z is Re(z) = a, and the imaginary part of z is Im(z) = b. Note that the imaginary part is the real coefficient b, not bi.

其中 a 和 b 是实数。z 的实部为 Re(z) = a，虚部为 Im(z) = b。注意虚部是实数系数 b，而不是 bi。

For example, in z = 3 − 4i, the real part is 3 and the imaginary part is −4. A real number such as 5 can be written as 5 + 0i, and a pure imaginary number such as 2i can be written as 0 + 2i.

例如，在 z = 3 − 4i 中，实部是 3，虚部是 −4。实数 5 可以写成 5 + 0i，纯虚数 2i 可以写成 0 + 2i。

4. 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, if z₁ = a + bi and z₂ = c + di, then z₁ = z₂ means a = c and b = d.

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，如果 z₁ = a + bi 且 z₂ = c + di，那么 z₁ = z₂ 就意味着 a = c 且 b = d。

This rule is especially useful when solving for unknown real numbers in complex equations. For example, if x + yi = 4 + 7i, then x = 4 and y = 7.

这条规则在求解复数方程中的未知实数时特别有用。例如，如果 x + yi = 4 + 7i，那么 x = 4 且 y = 7。

When equating real and imaginary parts, always make sure both sides are expressed in standard form a + bi first.

在比较实部和虚部时，务必先把两边都写成标准形式 a + bi。

5. Adding and Subtracting Complex Numbers | 复数的加法与减法

To add complex numbers, add the real parts together and add the imaginary parts together. If z₁ = a + bi and z₂ = c + di, then:

复数相加时，实部与实部相加，虚部与虚部相加。如果 z₁ = a + bi 且 z₂ = c + di，那么：

z₁ + z₂ = (a + c) + (b + d)i

Subtraction works in the same way by subtracting the real parts and subtracting the imaginary parts:

减法同理，实部减实部，虚部减虚部：

z₁ − z₂ = (a − c) + (b − d)i

For example, (5 + 2i) + (3 − 7i) = 8 − 5i, and (4 + 6i) − (1 + 2i) = 3 + 4i. Addition and subtraction do not require the use of i² = −1.

例如，(5 + 2i) + (3 − 7i) = 8 − 5i，(4 + 6i) − (1 + 2i) = 3 + 4i。加法和减法不需要使用 i² = −1。

6. Multiplying Complex Numbers | 复数的乘法

Complex numbers are multiplied by expanding brackets normally and then replacing i² with −1. If z₁ = a + bi and z₂ = c + di, then:

复数相乘时，像普通代数式一样展开括号，然后将 i² 替换为 −1。如果 z₁ = a + bi 且 z₂ = c + di，那么：

z₁z₂ = (ac − bd) + (ad + bc)i

This formula is useful, but you can also expand directly. For example, (2 + 3i)(1 − 4i) = 2 − 8i + 3i − 12i² = 2 − 5i + 12 = 14 − 5i.

这个公式很有用，但你也可以直接展开。例如，(2 + 3i)(1 − 4i) = 2 − 8i + 3i − 12i² = 2 − 5i + 12 = 14 − 5i。

Always collect real terms and imaginary terms separately, and remember that the imaginary unit i is not a variable: its defining property is i² = −1.

始终分别合并实部项和虚部项，并记住虚数单位 i 不是变量：它的定义性质是 i² = −1。

7. Complex Conjugates | 共轭复数

The complex conjugate of z = a + bi is denoted by z* and is defined as:

复数 z = a + bi 的共轭复数记作 z*，其定义为：

z* = a − bi

In other words, the conjugate is obtained by changing the sign of the imaginary part. The conjugate has two important properties. The sum z + z* = 2a is real, and the product zz* = a² + b² is real and non-negative.

换句话说，共轭复数就是把虚部的符号改变。共轭复数有两个重要性质：和 z + z* = 2a 是实数，积 zz* = a² + b² 是实数且非负。

For example, if z = 3 + 4i, then z* = 3 − 4i, z + z* = 6 and zz* = 9 + 16 = 25. The product of a complex number and its conjugate is especially important for division.

例如，如果 z = 3 + 4i，那么 z* = 3 − 4i，z + z* = 6，zz* = 9 + 16 = 25。复数与其共轭的乘积对除法尤其重要。

8. Dividing Complex Numbers | 复数的除法

To divide one complex number by another, multiply both the numerator and the denominator by the conjugate of the denominator. This converts the denominator into a real number.

要将一个复数除以另一个复数，将分子和分母同时乘以分母的共轭复数。这样可以把分母化为实数。

For example, to calculate (3 + 2i) ÷ (1 − i), multiply by the conjugate of 1 − i, which is 1 + i:

例如，计算 (3 + 2i) ÷ (1 − i)，乘以 1 − i 的共轭 1 + i：

(3 + 2i)(1 + i) ÷ (1 − i)(1 + i) = (1 + 5i) ÷ 2 = ½ + (5/2)i

In general, if z₁ = a + bi and z₂ = c + di, then:

一般来说，如果 z₁ = a + bi 且 z₂ = c + di，那么：

z₁ ÷ z₂ = (ac + bd)/(c² + d²) + (bc − ad)/(c² + d²) i

You do not need to memorise this formula if you can multiply by the conjugate confidently.

只要你能熟练地乘以共轭复数，就不需要记忆这个公式。

9. Solving Quadratic Equations | 解二次方程

Complex numbers allow quadratic equations with negative discriminants to have solutions. For ax² + bx + c = 0, the quadratic formula is:

复数使得判别式为负的二次方程也有解。对于 ax² + bx + c = 0，二次公式为：

x = (−b ± √(b² − 4ac)) ÷ 2a

If the discriminant b² − 4ac is negative, the square root becomes an imaginary number written in terms of i.

如果判别式 b² − 4ac 为负，平方根就变成用 i 表示的虚数。

For example, solve x² + 4x + 13 = 0. The discriminant is 4² − 4 × 1 × 13 = 16 − 52 = −36, so:

例如，解方程 x² + 4x + 13 = 0。判别式为 4² − 4 × 1 × 13 = 16 − 52 = −36，因此：

x = (−4 ± √−36) ÷ 2 = (−4 ± 6i) ÷ 2 = −2 ± 3i

When a polynomial has real coefficients, any complex roots always occur in conjugate pairs. Here the roots are −2 + 3i and −2 − 3i.

当多项式具有实系数时，任何复数根总是成共轭对出现。这里根为 −2 + 3i 和 −2 − 3i。

10. Argand Diagrams and Modulus-Argument Form | 阿尔冈图与模-辐角形式

Complex numbers can be represented geometrically on an Argand diagram. The horizontal axis is the real axis and the vertical axis is the imaginary axis. The complex number z = a + bi is plotted at the point (a, b).

复数可以在阿尔冈图上用几何方式表示。横轴是实轴，纵轴是虚轴。复数 z = a + bi 对应点 (a, b)。

The modulus of z is the distance from the origin to the point, given by:

z 的模是原点到该点的距离，公式为：

|z| = √(a² + b²)

The argument of z, denoted arg z, is the angle θ from the positive real axis to the line joining the origin to the point. It satisfies tan θ = b/a, but you must choose the correct quadrant.

z 的辐角记作 arg z，是从正实轴到原点与该点连线的角 θ。它满足 tan θ = b/a，但必须选择正确的象限。

The modulus-argument form is:

模-辐角形式为：

z = r(cos θ + i sin θ)

For example, z = 1 + i√3 has modulus |z| = √(1 + 3) = 2 and argument θ = π/3, so z = 2(cos π/3 + i sin π/3).

例如，z = 1 + i√3 的模为 |z| = √(1 + 3) = 2，辐角为 θ = π/3，所以 z = 2(cos π/3 + i sin π/3)。

11. Summary of Key Skills | 关键技能总结

When working with complex numbers, always identify the real and imaginary parts first. Use i² = −1 to simplify products, use the conjugate to divide, and represent solutions geometrically on an Argand diagram when needed.

处理复数时，始终先确定实部和虚部。利用 i² = −1 化简乘积，利用共轭进行除法，并在需要时用阿尔冈图几何表示解。

Key skills include simplifying powers of i, adding, subtracting, multiplying and dividing in standard form, solving quadratic equations with negative discriminants, and converting between standard form and modulus-argument form.

关键技能包括化简 i 的幂，以标准形式进行加减乘除，解判别式为负的二次方程，以及在标准形式和模-辐角形式之间转换。

With these foundations, complex numbers become a natural extension of the real number system and prepare you for more advanced topics such as de Moivre’s theorem and complex roots of unity.

有了这些基础，复数就自然成为实数系的扩展，并为进一步学习棣莫弗定理和单位复数根等更高级主题做好准备。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Introduction to Complex Numbers | 复数入门