📚 Complex Numbers of the Form x + iy | 复数 x + iy 形式
l numbers and i is the imaginary unit. This form is the foundation for simplifying expressions, solving equations, and representing complex numbers on an Argand diagram.
在 AQA A-Level 数学中，复数通常写成笛卡尔形式 z = x + iy，其中 x 和 y 是实数，i 是虚数单位。这一形式是化简表达式、解方程以及在 Argand 图上表示复数的基础。
1. What Does x + iy Mean? | x + iy 是什么意思？
A complex number is an expression of the form x + iy, where x, y ∈ ℝ is called the imaginary unit. In this notation, x is the real part and y is the imaginary part.
复数是指形如 x + iy 的表达式，其中 x, y ∈ ℝ，i 由 i² = −1 定义。符号 i 称为虚数单位。在这种写法中，x 是实部，y 是虚部。
z = x + iy, x, y ∈ ℝ, i² = −1
For example, 3 + 4i has real part 3 and imaginary part 4, while −2 − 5i has real part −2 and imaginary part −5. Every expression involving i can ultimately be simplified into this standard form.
例如，3 + 4i 的实部是 3，虚部是 4；而 −2 − 5i 的实部是 −2，虚部是 −5。任何含有 i 的表达式最终都可以化简到这个标准形式。
2. Real and Imaginary Parts | 实部与虚部
oefficient of i, not iy itself. This distinction is a common source of mistakes in exam questions.
给定 z = x + iy，我们记 Re(z) = x，Im(z) = y。请注意，Im(z) 是 i 的实数系数，而不是 iy 本身。这一区别是考试题中常见的错误来源。
Re(x + iy) = x, Im(x + iy) = y
A real number a can be written as a + 0i, and a pure imaginary number bi with b ≠ 0 can be written as 0 + bi. This is useful when subtracting or adding expressions that combine real and imaginary terms.
实数 a 可以写成 a + 0i，纯虚数 bi（其中 b ≠ 0）可以写成 0 + bi。这在加减同时含有实部和虚部的表达式时很有用。
. Many equations ask you to identify these two parts before solving a system of equations.
例如，Re(7 − 9i) = 7，Im(7 − 9i) = −9。许多方程题会先要求你识别这两个部分，然后再解方程组。
3. Equality of Complex Numbers | 复数的相等条件
Two complex numbers a + bi and c + di are equal exactly when their real parts are equal and their imaginary parts are equal. This principle lets us compare coefficients to find unknown real numbers.
两个复数 a + bi 和 c + di 相等，当且仅当它们的实部相等且虚部相等。这一原则让我们可以通过比较系数来求未知实数。
a + bi = c + di ⇔ a = c and b = d
y)i = 5 + 3i, then x + y = 5 and x − y = 3. Solving simultaneously gives x = 4 and y = 1.
例如：若 (x + y) + (x − y)i = 5 + 3i，则 x + y = 5 且 x − y = 3。联立求解得到 x = 4，y = 1。
This method is often used when a complex number is set equal to a known expression, especially when finding real constants from an identity.
当一个复数与已知表达式相等时，尤其是在由恒等式求实常数的问题中，常使用这个方法。
4. Addition and Subtraction | 加法与减法
To add or subtract complex numbers, combine the real parts together and the imaginary parts together. This works because the real and imaginary directions are independent.
复数的加减法只需分别合并实部和虚部。这是因为实部方向和虚部方向相互独立。
(x₁ + iy₁) + (x₂ + iy₂) = (x₁ + x₂) + i(y₁ + y₂)
Always give the final answer in the form x + iy, with real and imaginary parts simplified.
例如：(3 + 2i) + (5 − 4i) = 8 − 2i，而 (3 + 2i) − (5 − 4i) = −2 + 6i。最终答案一定要写成 x + iy 的形式，并化简实部和虚部。
When a negative sign appears in front of a bracket, remember to apply it to both the real and the imaginary part before collecting terms.
当括号前出现负号时，记得先将其
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