Expressing Complex Numbers in the Form x + iy | 将复数表示为 x + iy 的形式

📚 Expressing Complex Numbers in the Form x + iy | 将复数表示为 x + iy 的形式

In A-Level Mathematics, a complex number is usually introduced as an expression of the form x + iy, where x and y are real numbers and i is the imaginary unit with i² = -1. This article explains how to rewrite a wide range of complex expressions into this standard Cartesian form, which is essential for solving equations, performing arithmetic, and understanding geometric interpretations.

在 A-Level 数学中，复数通常被定义为形如 x + iy 的表达式，其中 x 和 y 是实数，i 是虚数单位，满足 i² = -1。本文讲解如何将各种复数表达式改写为这一标准笛卡尔形式，这是解方程、进行运算以及理解几何意义的关键。

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1. Definition and Notation | 定义与记号

A complex number is written as z = x + iy, where x = Re(z) is the real part and y = Im(z) is the imaginary part. Both x and y must be real numbers. The symbol i satisfies i² = -1.

复数写作 z = x + iy，其中 x = Re(z) 是实部，y = Im(z) 是虚部。x 和 y 都必须是实数。符号 i 满足 i² = -1。

• Example: 3 + 2i has real part 3 and imaginary part 2.
• 示例： 3 + 2i 的实部为 3，虚部为 2。
• If y = 0, the number is real; if x = 0, it is purely imaginary.
• 若 y = 0，该数为实数；若 x = 0，则为纯虚数。

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

To add or subtract complex numbers, combine their real parts and their imaginary parts separately. The result is already in the form x + iy.

加减复数时，分别合并实部和虚部。结果自然就是 x + iy 的形式。

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

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

• Example: (2 + 3i) + (4 - 5i) = 6 - 2i.
• 示例：(2 + 3i) + (4 - 5i) = 6 - 2i。
• Example: (7 - i) - (2 + 3i) = 5 - 4i.
• 示例：(7 - i) - (2 + 3i) = 5 - 4i。

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3. Multiplying Complex Numbers | 复数的乘法

Use the distributive law (FOIL) and replace i² with -1. Collect the real and imaginary terms to express the product in the form x + iy.

使用分配律（FOIL）展开，并将 i² 替换为 -1。合并实部和虚部项，即可将乘积表示为 x + iy。

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

• Example: (2 + 3i)(1 - i) = 2 - 2i + 3i - 3i² = 2 + i + 3 = 5 + i.
• 示例：(2 + 3i)(1 - i) = 2 - 2i + 3i - 3i² = 2 + i + 3 = 5 + i。
• Example: (1 + i)² = 1 + 2i + i² = 2i. Notice the real part is 0.
• 示例：(1 + i)² = 1 + 2i + i² = 2i。注意实部为 0。

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4. Complex Conjugate and Division | 共轭复数与除法

The conjugate of z = x + iy is z̄ = x - iy. Multiplying a complex number by its conjugate gives a real number: zz̄ = x² + y². To divide two complex numbers, multiply the numerator and denominator by the conjugate of the denominator.

复数 z = x + iy 的共轭为 z̄ = x - iy。复数与其共轭相乘得到实数：zz̄ = x² + y²。两个复数相除时，将分子分母同时乘以分母的共轭。

a + bic + di = (a + bi)(c - di)(c + di)(c - di) = ^{ac + bd}⁄_{c² + d²} + ^{bc - ad}⁄_{c² + d²}i

• Example: 1+i2-i = (1+i)(2+i)(2-i)(2+i) = ^{2 + i + 2i + i2}⁄₅ = 1 + 3i5 = 15 + 35i.
• 示例：1+i2-i = (1+i)(2+i)(2-i)(2+i) = ^{2 + i + 2i + i2}⁄₅ = 1 + 3i5 = 15 + 35i。

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5. Powers of i | i 的幂

Because i² = -1, higher powers repeat in a cycle of length 4. To simplify iⁿ, reduce n modulo 4.

因为 i² = -1，较高的幂以 4 为周期循环。要化简 iⁿ，将指数 n 对 4 取余。

n bmod 4	0	1	2	3
in	1	i	−1	−i

• Example: i²⁷ = i²⁴⁺³ = i³ = -i.
• 示例：i²⁷ = i²⁴⁺³ = i³ = -i。
• This pattern is useful when simplifying expressions involving powers of complex numbers.
• 这一规律在化简含复数幂的表达式时非常有用。

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6. Solving Quadratic Equations | 解二次方程

When a quadratic equation has a negative discriminant, its solutions are complex conjugates. Using the quadratic formula and writing results in the form x + iy is a key skill.

当二次方程的判别式为负时，其解是共轭复数对。使用求根公式并将结果写成 x + iy 的形式是一项关键技能。

z = ^{-b ± √b2 - 4ac}⁄_2a

• Example: Solve z² + 2z + 5 = 0.
• 示例：解方程 z² + 2z + 5 = 0。
• z = -2 ± √4 - 202 = -2 ± √-162 = -2 ± 4i2 = -1 ± 2i.
• z = -2 ± √4 - 202 = -2 ± √-162 = -2 ± 4i2 = -1 ± 2i。

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7. Equating Real and Imaginary Parts | 实部与虚部相等

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. This principle allows us to convert a complex equation into two real equations.

两个复数相等当且仅当它们的实部相等且虚部相等。这一原理可以将一个复数方程转化为两个实数方程。

a + bi = c + di iff a = c and b = d

• Example: If (x + 2) + (y - 3)i = 4 + 5i, then x + 2 = 4 and y - 3 = 5, so x = 2, y = 8.
• 示例：若 (x + 2) + (y - 3)i = 4 + 5i，则 x + 2 = 4，y - 3 = 5，因此 x = 2，y = 8。
• This method is frequently used to find unknown real constants in complex identities.
• 此方法常用于在复数恒等式中求未知实常数。

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8. Expressing Rationalised Denominators | 有理化分母

An expression such as 1a + bi can be written in the form x + iy by multiplying numerator and denominator by the conjugate a - bi.

形如 1a + bi 的表达式可以通过将分子分母同时乘以共轭 a - bi 来写成 x + iy。

1a + bi = ^{a - bi}⁄_{a² + b²} = ^a⁄_{a² + b²} - ^b⁄_{a² + b²}i

• Example: 13 + 4i = 3 - 4i25 = 325 - 425i.
• 示例：13 + 4i = 3 - 4i25 = 325 - 425i。
• Always simplify fully: both real and imaginary parts should be written as single fractions if required.
• 始终彻底化简：如果需要，实部和虚部都应写成单一分数形式。

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9. Modulus and Argument in Cartesian Form | 模与辐角的笛卡尔形式

The modulus of z = x + iy is |z| = √x² + y². The argument is arg(z) = tan^-1(y/x), adjusted for the correct quadrant. These are often used together to convert between polar and Cartesian forms.

复数 z = x + iy 的模为 |z| = √x² + y²。辐角为 arg(z) = tan^-1(y/x)，需要根据象限调整。二者常用于极坐标与笛卡尔形式之间的转换。

• Example: For z = 1 + i, |z| = √2 and arg(z) = π/4.
• 示例：对于 z = 1 + i，|z| = √2，arg(z) = π/4。
• If z = -1 + i, |z| = √2 but arg(z) = 3π/4, not -π/4.
• 若 z = -1 + i，|z| = √2，但 arg(z) = 3π/4，而不是 -π/4。

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10. From Polar / Exponential Form to x + iy | 从极坐标或指数形式到 x + iy

Given a modulus r and argument θ, Euler's formula e^iθ = cosθ + isinθ gives the Cartesian form directly:

给定模 r 和辐角 θ，欧拉公式 e^iθ = cosθ + isinθ 可直接给出笛卡尔形式：

r e^iθ = rcosθ + i rsinθ

• Example: 2e^iπ/3 = 2cos(π/3) + i2sin(π/3) = 1 + √3i.
• 示例：2e^iπ/3 = 2cos(π/3) + i2sin(π/3) = 1 + √3i。
• Remember to express the final answer with i written before the imaginary term, e.g. x + iy, not x + yi unless clearly conventional.
• 注意最终答案通常写成 x + iy，虚部放在 i 后面，例如 x + iy，除非约定俗成也可写作 x + yi。

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11. Square Roots of a Complex Number | 复数的平方根

To find √a + bi in the form x + iy, set (x + iy)² = a + bi. Equating real and imaginary parts gives a system of equations for x and y.

要求 √a + bi 并将其写成 x + iy，令 (x + iy)² = a + bi。比较实部和虚部可得关于 x 和 y 的方程组。

x² - y² = a, 2xy = b

• Also use x² + y² = √a² + b² to help solve.
• 同时利用 x² + y² = √a² + b² 来辅助求解。
• Example: (1 + i)² = 2i, so √2i = 1 + i or -1 - i.
• 示例：(1 + i)² = 2i，所以 √2i = 1 + i 或 -1 - i。

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12. Common Pitfalls and Exam Tips | 常见错误与考试提示

When writing complex numbers in the form x + iy, students often make sign errors or forget to simplify i². Practice rewriting expressions step by step.

在写成 x + iy 形式时，学生常犯符号错误或忘记化简 i²。应逐步练习改写表达式。

• Always check that your final answer has the structure (real number) + (real number)i.
• 始终检查最终答案是否具有“实数 + 实数·i”的结构。
• Never leave i in the denominator of a fraction in the final answer.
• 最终答案中切勿将 i 留在分母里。
• Use the quadrant diagram when converting from polar to Cartesian form to determine the sign of sinθ and cosθ.
• 从极坐标转换为笛卡尔形式时，使用象限图判断 sinθ 和 cosθ 的正负。
• For AQA exam questions, always show the equating of real and imaginary parts when solving for unknowns.
• 对于 AQA 考试题目，在求解未知数时务必展示实部与虚部的比较过程。

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