📚 Addition, Subtraction and Multiplication of Complex Numbers | 复数的加法、减法与乘法
evel Mathematics, you must be confident adding, subtracting and multiplying numbers of the form a + bi, where a and b are real numbers.
复数通过引入虚数单位 i（满足 i² = -1）扩展了实数系。在 AQA A Level 数学中，你需要熟练掌握形如 a + bi（其中 a 和 b 为实数）的复数的加法、减法和乘法。
1. Complex Numbers in AQA | AQA 中的复数
A complex number is an expression of the form z = a + bi, where a and b are real numbers and i is the imaginary unit.
复数是指形如 z = a + bi 的表达式，其中 a 和 b 是实数，i 是虚数单位。
i² = -1
The symbol i is not a variable; it is fixed by the rule above. This single rule is enough to carry out addition, subtraction and multiplication. For example, √-9 can be written as 3i because 3i × 3i = 9i² = -9.
符号 i 不是变量；它由上述规则固定。仅凭这一条规则，就足以完成加法、减法和乘法。例如，√-9 可以写成 3i，因为 3i × 3i = 9i² = -9。
2. Real and Imaginary Parts | 实部与虚部
For z = a + bi d the imaginary part is b, written Im(z) = b. Notice that the imaginary part is b, not bi.
对于 z = a + bi，实部是 a，记作 Re(z) = a；虚部是 b，记作 Im(z) = b。注意虚部是 b，而不是 bi。
Re(a + bi) = a, Im(a + bi) = b
examples: for z = 5 - 2i, Re(z) = 5 and Im(z) = -2. For z = -3i, Re(z) = 0 and Im(z) = -3.
一些例子：对于 z = 5 - 2i，Re(z) = 5，Im(z) = -2。对于 z = -3i，Re(z) = 0，Im(z) = -3。
3. 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 + bi = c + di ⇔ a = c and b = d
eal coefficients in AQA questions. For example, if x + yi = 7 - 2i, then x = 7 and y = -2.
这一事实常用于 AQA 试题中求未知实系数。例如，若 x + yi = 7 - 2i，则 x = 7，y = -2。
4. Addition of Complex Numbers | 复数加法
To add two complex numbers, add the real parts together and add the imaginary parts together.
两个复数相加时，实部与实部相加，虚部与虚部相加。
(a + bi) + (c + di) = (a + c) + (b + d)i
This is just like collecting like terms in algebra, where i is kept as the imaginary unit.
例题：(3 + 2i) + (4 - 5i) = (3 + 4) + (2 - 5)i = 7 - 3i。这与代数中的合并同类项类似，只是把 i 保留为虚数单位。
5. Subtraction of Complex Numbers | 复数减法
Subtraction works in the same way: subtract the real parts and subtract the imaginary parts. You must be careful to distribute the minus sign over both terms in the second bracket.
减法的规则相同：实部减实部，虚部减虚部。你必须注意将负号分配到第二个括号中的每一项。
(a + bi) - (c + di) = (a - c) + (b - d)i
(3 - (-4))i = 3 + 7i. A common error is to write 3 - i by forgetting to change -4i to +4i.
例题：(5 + 3i) - (2 - 4i) = (5 - 2) + (3 - (-4))i = 3 + 7i。常见的错误是忘记把 -4i 变成 +4i，从而错写为 3 - i。
6. Multiplication of Complex Numbers | 复数乘法
To multiply two complex numbers, expand the brackets exactly as in algebra, then replace i² by -1.
两个复数相乘时，像代数中一样展开括号，然后将 i² 替换为 -1。
(a + bi)(c + di) = ac + adi + bci + bdi²
= (ac - bd) + (ad + bc)i
Worked example: (2 + 3i)(1 - 4i) = 2 - 8i + 3i - 12i² = 2 - 5i + 12 = 14 - 5i.
例题：(2 + 3i)(1 - 4i) = 2 - 8i + 3i - 12i² = 2 - 5i + 12 = 14 - 5i。
7. Powers of i | i 的幂
Because i² = -1, powers of i repeat in a cycle of four.
因为 i² = -1，所以 i 的幂以四为周期循环。
| Power | Value |
| i¹ | i |
| i² | -1 |
| i³ | -i |
| i⁴ | 1 |
divide n by 4 and use the remainder. For example, 2023 = 4 × 505 + 3, so i²⁰²³ = i³ = -i.
要化简 iⁿ，可将 n 除以 4，并根据余数确定结果。例如，2023 = 4 × 505 + 3，所以 i²⁰²³ = i³ = -i。
8. Complex Conjugates | 共轭复数
z = a + bi is z* = a - bi. It is obtained by changing the sign of the imaginary part only.
复数 z = a + bi 的共轭复数是 z* = a - bi。只需改变虚部的符号。
z = a + bi ⇒ z* = a - bi
Examples: if z = 4 + 7i then z* = 4 - 7i; if z = -2 - 5i then z* = -2 + 5i.
例如：若 z = 4 + 7i，则 z* = 4 - 7i；若 z = -2 - 5i，则 z* = -2 + 5i。
9. Multiplying a Complex Number by Its Conjugate | 复数与其共轭相乘
Multiplying a complex number by its conjugate always gives a real number. This is important when dividing complex numbers and when simplifying expressions.
一个复数与其共轭复数相乘总是得到一个实数。这在复数除法和化简表达式中非常重要。
zz* = (a + bi)(a - bi) = a² - abi + abi - b²i² = a² + b²
Example: (3 + 2i)(3 - 2i) = 9 + 4 = 13. There is no imaginary part left in the answer.
例题：(3 + 2i)(3 - 2i) = 9 + 4 = 13。答案中不再含有虚部。
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