Equating Real and Imaginary Parts | 实部与虚部的相等

📚 Equating Real and Imaginary Parts | 实部与虚部的相等

Complex numbers are written in the form z = x + iy, where x and y are real numbers. The power of this notation lies in a simple but profound rule: two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. This technique, known as equating real and imaginary parts, transforms many complex-number problems into systems of real equations.

复数写作 z = x + iy 的形式，其中 x 和 y 是实数。这种表示法的力量源于一条简单而深刻的规则：两个复数相等当且仅当它们的实部相等且虚部相等。这一技巧称为“实部与虚部相等”，它将许多复数问题转化为实数方程组。

1. The Fundamental Rule | 基本规则

For two complex numbers a + bi and c + di, with a, b, c, d real:

对于两个复数 a + bi 和 c + di，其中 a、b、c、d 为实数：

a + bi = c + di ⇔ a = c and b = d

a + bi = c + di ⇔ a = c 且 b = d

This works because the real axis and imaginary axis are independent directions in the Argand diagram. Just as two vectors are equal only when their horizontal and vertical components match, two complex numbers coincide only when both components agree.

这一规则成立是因为实轴与虚轴在阿甘图中是相互独立的两个方向。正如两个向量只有当水平分量与垂直分量都相同时才相等，两个复数也只有当两个分量都一致时才重合。

Key point: the letters used in complex expressions are conventionally real unless stated otherwise. When we write z = x + iy, we are announcing that x and y are real parameters.

关键点：复数表达式中使用的字母在未特别说明时通常视为实数。当我们写 z = x + iy 时，就是在声明 x 和 y 是实参数。

2. Solving Simple Equations | 解简单方程

Consider the equation 2x + 3i = 8 – yi, where x and y are real. Equating real parts gives 2x = 8, so x = 4. Equating imaginary parts gives 3 = -y, so y = -3.

考虑方程 2x + 3i = 8 – yi，其中 x 和 y 为实数。比较实部得 2x = 8，因此 x = 4；比较虚部得 3 = -y，因此 y = -3。

Always separate the expression into a real part (no i) and an imaginary part (coefficient of i) before equating. For example, (a + 2) + (b – 1)i = 5 + 3i gives a + 2 = 5 and b – 1 = 3.

在比较前，务必先将表达式整理为实部（不含 i）和虚部（i 的系数）。例如 (a + 2) + (b – 1)i =

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