Expressing Complex Numbers in the Form x + iy | 将复数化为 x + iy 的形式

📚 Expressing Complex Numbers in the Form x + iy | 将复数化为 x + iy 的形式

Complex numbers are a central topic in AQA A-level Mathematics. The form x + iy is the standard way to write a complex number: the real part first, then the imaginary unit i multiplied by a real number y. This article explains the notation, shows how to perform arithmetic with complex numbers, and gives AQA-style examples so that you can confidently express any result in the form x + iy.

复数是 AQA 高等数学的核心内容。形式 x + iy 是书写复数的标准方式：先写实部，再写虚数单位 i 与实数 y 的乘积。本文将解释这种记号，演示复数的四则运算，并提供 AQA 风格例题，帮助你自信地把任何复数结果化为 x + iy 的形式。

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

The equation x² + 1 = 0 has no real solution because the square of any real number is non-negative. If we introduce a new number i with the property i² = −1, then the equation x² + 1 = 0 has two solutions: x = i and x = −i.

方程 x² + 1 = 0 没有实数解，因为任何实数的平方都非负。如果我们引入一个新的数 i，并规定 i² = −1，那么方程 x² + 1 = 0 就有两个解：x = i 与 x = −i。

This single new number extends the number system from the real line to a plane of complex numbers. Every complex number can be written uniquely as x + iy, where x and y are ordinary real numbers.

这一个新数把数系从实数轴扩展到了复数平面。每个复数都可以唯一地写成 x + iy 的形式，其中 x 和 y 都是普通的实数。

2. The Imaginary Unit and the Form x + iy | 虚数单位 i 与 x + iy 的形式

In AQA A-level Mathematics, the imaginary unit is written as i and is defined by the key identity:

在 AQA 高等数学中，虚数单位写作 i，并由以下关键恒等式定义：

i² = −1

From this definition, a complex number z in standard form is written as:

根据这个定义，复数 z 的标准形式写作：

z = x + iy

Here x and y are real numbers. The value x is called the real part of z, and y is called the imaginary part of z. Notice that the imaginary part is the real number y, not the term iy.

其中 x 与 y 是实数。x 称为 z 的实部，y 称为 z 的虚部。注意，虚部是实数 y，而不是 iy 这一项。

3. Real Part, Imaginary Part and the Argand Diagram | 实部、虚部与阿甘图

For a complex number z = x + iy, we use the notation:

对于复数 z = x + iy，我们使用记号：

Re(z) = x, Im(z) = y

On an Argand diagram, the complex number x + iy is represented by the point with coordinates (x, y). The horizontal axis is the real axis, and the vertical axis is the imaginary axis.

在阿甘图中，复数 x + iy 用坐标为 (x, y) 的点表示。横轴称为实轴，纵轴称为虚轴。

For example, the number 3 + 4i is at position (3, 4), while −2 − 7i is at position (−2, −7). A purely real number such as 5 is written as 5 + 0i, and a purely imaginary number such as 2i is written as 0 + 2i.

例如，复数 3 + 4i 位于点 (3, 4)，而 −2 − 7i 位于点 (−2, −7)。纯实数 5 可写成 5 + 0i，纯虚数 2i 可写成 0 + 2i。

4. Equality of Complex Numbers | 复数相等

Two complex numbers are equal exactly when 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 + ib = c + id ⇔ a = c and b = d

This is a powerful tool. For example, if x + iy = 3 − 4i, then x = 3 and y = −4. You cannot solve an equation involving complex numbers without first matching real and imaginary parts separately.

这是一个非常有用的工具。例如，若 x + iy = 3 − 4i，则 x = 3，y = −4。解含复数的方程时，必须先分别比较实部与虚部。

5. Addition and Subtraction | 复数的加减法

To add or subtract complex numbers, combine the real parts and the imaginary parts separately:

复数的加减运算就是把实部与实部、虚部与虚部分别相加减：

(a + ib) + (c + id) = (a + c) + i(b + d)

(a + ib) − (c + id) = (a − c) + i(b − d)

Worked example: (3 + 2i) + (5 − 7i) = 8 − 5i. Subtracting gives (3 + 2i) − (5 − 7i) = −2 + 9i, because 2i − (−7i) = 9i.

例题：(3 + 2i) + (5 − 7i) = 8 − 5i。相减得 (3 + 2i) − (5 − 7i) = −2 + 9i，因为 2i − (−7i) = 9i。

6. Multiplication of Complex Numbers | 复数的乘法

To multiply two complex numbers, expand the brackets as you would with a binomial, then replace i² with −1:

两个复数相乘时，先像二项式一样展开括号，再把 i² 替换为 −1：

(a + ib)(c + id) = ac + iad + ibc + i²bd = (ac − bd) + i(ad + bc)

Worked example: (3 + 2i)(4 − i) = 12 − 3i + 8i − 2i² = 12 + 5i + 2 = 14 + 5i. The answer is already in the form x + iy.

例题：(3 + 2i)(4 − i) = 12 − 3i + 8i − 2i² = 12 + 5i + 2 = 14 + 5i。答案已经化为 x + iy 的形式。

7. The Complex Conjugate | 共轭复数

For a complex number z = x + iy, its complex conjugate is written as z̄ and is defined by:

对于复数 z = x + iy，它的共轭复数记作 z̄，定义为：

z̄ = x − iy

The product of a complex number and its conjugate is always a real number:

一个复数与其共轭复数的乘积总是一个实数：

z z̄ = (x + iy)(x − iy) = x² + y²

This property is essential for dividing complex numbers and for rewriting expressions in the form x + iy.

这个性质对于复数的除法以及把表达式化为 x + iy 的形式至关重要。

8. Division: Quotients in the Form x + iy | 除法：化为 x + iy 的形式

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

除以一个复数时，把分子分母同时乘以分母的共轭复数：

(a + ib) / (c + id) = ((a + ib)(c − id)) / (c² + d²)

Worked example: express (3 + i) / (2 − i) in the form x + iy.

例题：将 (3 + i) / (2 − i) 化为 x + iy 的形式。

(3 + i) / (2 − i) = ((3 + i)(2 + i)) / ((2 − i)(2 + i)) = (6 + 3i + 2i + i²) / 5 = (5 + 5i) / 5 = 1 + i

Therefore the quotient is 1 + i, so x = 1 and y = 1. You should

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Expressing Complex Numbers in the Form x + iy | 将复数化为 x + iy 的形式