Addition, subtraction and multiplication of complex numbers of the form x + iy | x+iy 形式复数的加减乘

📚 Addition, subtraction and multiplication of complex numbers of the form x + iy | x+iy 形式复数的加减乘

In AQA A-Level Mathematics, you will work with complex numbers written in Cartesian form z = x + iy, where x and y are real numbers and i is the imaginary unit satisfying i² = -1. This article focuses on the three core operations: addition, subtraction and multiplication. Mastering these operations is essential before moving on to conjugates, division, modulus-argument form and polynomial equations with complex roots.

在 AQA A-Level 数学中，你需要处理笛卡儿形式 z = x + iy 的复数，其中 x 和 y 是实数，i 是满足 i² = -1 的虚数单位。本文重点讲解加法、减法和乘法三种核心运算。掌握这些运算是进一步学习共轭复数、除法、模-辐角形式以及含复数根的多项式方程的基础。

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1. Complex Numbers in Standard Form x + iy | 标准形式 x + iy 的复数

A complex number in standard form is written as z = x + iy. Here x is called the real part, Re(z), and y is called the imaginary part, Im(z). Both x and y are real numbers. For example, in 3 – 4i, the real part is 3 and the imaginary part is -4, not 4. The sign in front of the imaginary unit belongs to the imaginary part.

标准形式的复数写作 z = x + iy。其中 x 称为实部，记作 Re(z)，y 称为虚部，记作 Im(z)。x 和 y 都是实数。例如，在 3 – 4i 中，实部为 3，虚部为 -4，而不是 4。虚数单位前面的符号属于虚部。

When rewriting a complex number, always aim to express it with exactly two terms: a real term and an imaginary term. For instance, 2 – 5i is already in standard form, while -7i can be written as 0 – 7i, showing that the real part is zero.

在改写复数时，始终要把它表示成恰好两项：一个实数项和一个虚数项。例如，2 – 5i 已经是标准形式，而 -7i 可以写作 0 – 7i，表明它的实部为零。

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2. Real and Imaginary Parts as Separate Channels | 实部与虚部作为独立通道

When adding or subtracting complex numbers, it is useful to think of the real part and imaginary part as separate channels. The symbol i acts as a label for the imaginary component; it is not a variable that you need to solve for. This separation makes addition and subtraction very similar to working with vectors in two dimensions.

在做复数加法或减法时，把实部和虚部分成两个独立通道会非常有用。符号 i 只是虚部的标记，并不是需要求解的变量。这种分离使得复数的加减法与二维向量的运算非常相似。

Keeping the real and imaginary parts separate also helps you check your final answer. A correct answer in Cartesian form should have no i attached to the real part and no real number mixed into the coefficient of i.

把实部和虚部分开还有助于检查最终答案。笛卡儿形式下的正确答案中，实部不应带有 i，虚部系数中也不应混入实数。

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3. Addition of Complex Numbers | 复数的加法

To add two complex numbers (x₁ + iy₁) and (x₂ + iy₂), add the real parts together and add the imaginary parts together separately. This gives the rule:

要将两个复数 (x₁ + iy₁) 和 (x₂ + iy₂) 相加，需要分别把实部相加、虚部相加。得到如下规则：

(x₁ + iy₁) + (x₂ + iy₂) = (x₁ + x₂) + i(y₁ + y₂)

For example, (2 + 3i) + (4 – 5i) = (2 + 4) + i(3 – 5) = 6 – 2i. Notice that the i is factored out from the imaginary part addition, so the answer remains in the form a + bi.

例如，(2 + 3i) + (4 – 5i) = (2 + 4) + i(3 – 5) = 6 – 2i。注意 i 在虚部相加时被提取出来，因此答案仍然保持 a + bi 的形式。

Addition of complex numbers is commutative and associative, meaning the order in which you add them does not change the result. This is useful when simplifying longer expressions.

复数加法满足交换律和结合律，也就是说相加的顺序不会改变结果。这在化简较长表达式时非常有用。

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4. Subtraction of Complex Numbers | 复数的减法

Subtraction works in the same way as addition, but you must subtract both the real and imaginary parts. The rule is:

减法的思路与加法相同，但必须同时减去实部和虚部。规则如下：

(x₁ + iy₁) – (x₂ + iy₂) = (x₁ – x₂) + i(y₁ – y₂)

Be especially careful with signs when the second complex number has a negative imaginary part. For example, (3 + 2i) – (1 – 4i) = (3 – 1) + i(2 – (-4)) = 2 + 6i. Subtracting -4i is the same as adding 4i.

当第二个复数的虚部为负数时，要特别小心符号。例如，(3 + 2i) – (1 – 4i) = (3 – 1) + i(2 – (-4)) = 2 + 6i。减去 -4i 等价于加上 4i。

A common mistake is to write (3 + 2i) – (1 – 4i) as 2 – 2i by incorrectly computing 2 – 4i as -2i. Always place the subtracted imaginary part in brackets before expanding the minus sign.

一个常见错误是把 (3 + 2i) – (1 – 4i) 写成 2 – 2i，因为错误地计算了 2 – 4i = -2i。在展开负号之前，一定要先给被减的虚部加上括号。

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5. Multiplication: Expanding Brackets and Using i² = -1 | 乘法：展开括号并利用 i² = -1

To multiply two complex numbers, expand the brackets as you would with binomials. Then replace i² by -1 and collect real and imaginary terms. The full expansion is:

两个复数相乘时，像展开二项式一样展开括号。然后把 i² 替换为 -1，并合并实部和虚部。完整展开如下：

(x₁ + iy₁)(x₂ + iy₂) = x₁x₂ + x₁iy₂ + iy₁x₂ + iy₁iy₂

Since iy₁iy₂ = i²y₁y₂ = -y₁y₂, the result can be collected as:

因为 iy₁iy₂ = i²y₁y₂ = -y₁y₂，结果可以合并为：

(x₁ + iy₁)(x₂ + iy₂) = (x₁x₂ – y₁y₂) + i(x₁y₂ + y₁x₂)

For example, (2 + 3i)(1 – 4i) = 2 – 8i + 3i – 12i² = 2 – 5i + 12 = 14 – 5i. The real part is 14 and the imaginary part is -5.

例如，(2 + 3i)(1 – 4i) = 2 – 8i + 3i – 12i² = 2 – 5i + 12 = 14 – 5i。实部为 14，虚部为 -5。

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6. The Role of i² = -1 | i² = -1 的作用

The single most important simplification in complex multiplication is the identity i² = -1. This identity turns a product of two imaginary terms into a real term. For instance, i(3i) = 3i² = -3, which is purely real.

复数乘法中最重要的化简依据是恒等式 i² = -1。这个恒等式把两个虚数项的乘积变成了实数项。例如，i(3i) = 3i² = -3，结果是一个纯实数。

It is also useful to recognise the first four powers of i, because they repeat in cycles:

认识 i 的前四次幂也很有用，因为它们会循环出现：

Power	Value	中文
i¹	i	i
i²	-1	-1
i³	-i	-i
i⁴	1	1

This cycle can simplify higher powers such as i⁶ = i⁴ × i² = 1 × (-1) = -1. Being confident with these results reduces mistakes in longer multiplications.

这个循环可以化简更高的幂，例如 i⁶ = i⁴ × i² = 1 × (-1) = -1。熟练掌握这些结果能减少较长乘法中的错误。

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7. Multiplying by Real Numbers and Pure Imaginary Numbers | 乘以实数与纯虚数

Multiplying a complex number by a real number k simply scales both the real and imaginary parts. The rule is k(x + iy) = kx + iky. For example, 2(3 – i) = 6 – 2i.

复数乘以实数 k 只是对实部和虚部同时进行缩放。规则为 k(x + iy) = kx + iky。例如，2(3 – i) = 6 – 2i。

Multiplying by a pure imaginary number bi introduces a factor of i and then an i² term when necessary. For instance, i(4 + 2i) = 4i + 2i² = -2 + 4i. Always rewrite the final answer with the real part first.

乘以纯虚数 bi 会引入一个 i 因子，并在需要时出现 i² 项。例如，i(4 + 2i) = 4i + 2i² = -2 + 4i。最终答案始终要把实部写在前面。

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8. Common Mistakes and How to Avoid Them | 常见错误及避免方法

A common error is to forget that i² is -1, or to write the result with an i attached to the real part. Another common error is mishandling subtraction of a negative imaginary part. Always write the final answer in the form a + bi, where a and b are real numbers.

一个常见错误是忘记 i² = -1，或把实部也带上了 i。另一个常见错误是在减去负虚部时处理不当。最终答案一定要写成 a + bi 的形式，其中 a 和 b 都是实数。

When multiplying, do not skip the step of replacing i². If you expand (a + bi)(c + di) and leave it as ac + adi + bci + bdi², you have not finished the simplification. Replace bdi² with -bd and then collect like terms.

做乘法时，不要跳过替换 i² 的步骤。如果你把 (a + bi)(c + di) 展开后写成 ac + adi + bci + bdi²，化简还没有完成。要把 bdi² 替换为 -bd，然后合并同类项。

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9. Worked Examples | 例题

Below are two complete examples showing the three operations in sequence. First, add and subtract:

下面是两个完整例题，按顺序展示三种运算。首先进行加法和减法：

(2 + 3i) + (4 – i) = 6 + 2i

(2 + 3i) – (4 – i) = -2 + 4i

Then multiply the same pair of complex numbers:

然后对同一对复数进行乘法：

(2 + 3i)(4 – i) = 8 – 2i + 12i – 3i² = 8 + 10i + 3 = 11 + 10i

A second example is (1 + 2i)(3 – 4i). Expand to get 3 – 4i + 6i – 8i² = 3 + 2i + 8 = 11 + 2i. The real part is 11 and the imaginary part is 2.

第二个例子是 (1 + 2i)(3 – 4i)。展开得到 3 – 4i + 6i – 8i² = 3 + 2i + 8 = 11 + 2i。实部为 11，虚部为 2。

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10. Exam Tips and Summary | 考试技巧与总结

In AQA exam questions, show clear steps for multiplication: expand the brackets, replace i² with -1, and collect real and imaginary terms. Write complex numbers in the required form a + bi. If a question says ‘where a and b are real’, make sure no i remains in the real part.

在 AQA 考试题中，乘法要写出清晰的步骤：展开括号、用 -1 替换 i²、合并实部和虚部。把复数写成题目要求的 a + bi 形式。如果题目说“其中 a 和 b 为实数”，要确保实部中没有 i。

Finally, practise recognising when an expression is fully simplified. A final answer such as -2 + 4i is correct, while 4i – 2 is also acceptable but should ideally be written with the real part first. Consistent practice with these operations will make complex number algebra automatic and secure.

最后，要练习判断一个表达式是否已经彻底化简。像 -2 + 4i 这样的最终答案是正确的，而 4i – 2 也可以接受，但最好把实部写在前面。坚持练习这些运算会让复数代数变得熟练而稳定。

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