📚 Complex Numbers for KS3 | KS3 数学：复数
Numbers are everywhere in mathematics. You already know about counting numbers, fractions, decimals, and even negative numbers. But what if we told you there is another set of numbers that can help solve equations like x² = -1? These are called complex numbers. Although complex numbers are usually studied later, this article introduces them in a simple way suitable for KS3 learners who love a challenge. Let us unlock the mystery of numbers that go beyond the real line.
数字在数学中无处不在。你已经了解计数数字、分数、小数甚至负数。但是，如果我们告诉你还有另一类数字可以帮助解决像 x² = -1 这样的方程呢？这些数字叫作复数。虽然复数通常在高年级学习，但这篇文章以简单的方式介绍给喜欢挑战的 KS3 学生。让我们一起解开超越实数直线的数字之谜。
1. What Are Imaginary Numbers? | 什么是虚数？
In ordinary arithmetic, when you square any number, the result is always positive or zero. For example, 3² = 9, (-2)² = 4, and 0² = 0. There is no real number whose square is negative. But mathematicians wanted to solve equations like x² = -1. To do this, they invented a new kind of number called an imaginary number.
在普通算术中，任何数的平方总是正数或零。例如，3² = 9，(-2)² = 4，0² = 0。没有哪个实数的平方是负数。但数学家想求解像 x² = -1 这样的方程。为此，他们发明了一种新的数，称为虚数。
2. Introducing the Imaginary Unit i | 引入虚数单位 i
The foundation of complex numbers is the symbol i, which is defined as the square root of -1. That is, i² = -1. Using i, we can write the square root of any negative number. For example, √(-4) = √(4 × -1) = 2i. The letter i stands for ‘imaginary unit’.
复数的基础是符号 i，它被定义为 -1 的平方根。即 i² = -1。利用 i，我们可以写出任何负数的平方根。例如，√(-4) = √(4 × -1) = 2i。字母 i 代表“虚数单位”。
3. Powers of i | i 的幂
The powers of i follow a repeating pattern. We already know i² = -1. Then i³ = i² × i = -1 × i = -i. And i⁴ = i² × i² = (-1) × (-1) = 1. After i⁴, the cycle repeats: i⁵ = i, i⁶ = -1, and so on. This pattern is very useful when simplifying larger powers of i.
i 的幂遵循一个循环模式。我们已经知道 i² = -1。那么 i³ = i² × i = -1 × i = -i。而 i⁴ = i² × i² = (-1) × (-1) = 1。在 i⁴ 之后，循环重复：i⁵ = i，i⁶ = -1，依此类推。这个规律在化简 i 的大幂次时非常有用。
- i¹ = i
- i² = -1
- i³ = -i
- i⁴ = 1
- i⁵ = i, and the cycle continues.
4. Complex Numbers: a + bi | 复数：形如 a + bi
A complex number is formed when you add a real number to an imaginary number. The standard form is a + bi, where a and b are real numbers. The value a is called the real part, and b is called the imaginary part. For instance, 3 + 4i is a complex number with real part 3 and imaginary part 4. Even a real number like 5 can be written as 5 + 0i, and a pure imaginary number like 2i is 0 + 2i.
当你将一个实数与一个虚数相加时，就形成了一个复数。标准形式是 a + bi，其中 a 和 b 是实数。a 称为实部，b 称为虚部。例如，3 + 4i 是一个复数，实部为 3，虚部为 4。即使是像 5 这样的实数也可以写成 5 + 0i，而像 2i 这样的纯虚数则是 0 + 2i。
5. Real and Imaginary Parts | 实部和虚部
Identifying the real and imaginary parts is straightforward. In the complex number z = a + bi, the real part is Re(z) = a, and the imaginary part is Im(z) = b (not bi). For example, if z = -2 + 5i, then Re(z) = -2 and Im(z) = 5. This separation helps us understand the structure of complex numbers and will be essential for operations.
识别实部和虚部很简单。在复数 z = a + bi 中，实部是 Re(z) = a，虚部是 Im(z) = b（不是 bi）。例如，如果 z = -2 + 5i，那么 Re(z) = -2，Im(z) = 5。这种分离帮助我们理解复数的结构，对运算至关重要。
6. Adding and Subtracting Complex Numbers | 复数的加减法
To add two complex numbers, simply add their real parts and add their imaginary parts separately. For (a + bi) + (c + di), the result is (a + c) + (b + d)i. Subtraction works the same way: (a + bi) – (c + di) = (a – c) + (b – d)i. For instance, (2 + 3i) + (1 + 4i) = 3 + 7i.
两个复数相加，只需分别将实部相加、虚部相加。对于 (a + bi) + (c + di)，结果是 (a + c) + (b + d)i。减法同理：(a + bi) – (c + di) = (a – c) + (b – d)i。例如，(2 + 3i) + (1 + 4i) = 3 + 7i。
Let us see another example: (5 – 2i) – (3 – 6i). First, subtract the real parts: 5 – 3 = 2. Then subtract the imaginary parts: -2 – (-6) = -2 + 6 = 4. So the answer is 2 + 4i. Always treat the imaginary term with its sign.
再看一个例子：(5 – 2i) – (3 – 6i)。首先，实部相减：5 – 3 = 2。然后虚部相减：-2 – (-6) = -2 + 6 = 4。所以答案是 2 + 4i。始终要带着符号处理虚部。
7. Multiplying Complex Numbers | 复数的乘法
Multiplying complex numbers is like expanding brackets in algebra. Use the FOIL method: (a + bi)(c + di) = ac + adi + bci + bdi². Since i² = -1, the term bdi² becomes -bd. So the product simplifies to (ac – bd) + (ad + bc)i. For example, (3 + 2i)(1 + 4i) = 3×1 + 3×4i + 2i×1 + 2i×4i = 3 + 12i + 2i + 8i² = 3 + 14i – 8 = -5 + 14i.
复数乘法就像代数中的展开括号。使用 FOIL 方法：(a + bi)(c + di) = ac + adi + bci + bdi²。因为 i² = -1，项 bdi² 变为 -bd。因此乘积简化为 (ac – bd) + (ad + bc)i。例如，(3 + 2i)(1 + 4i) = 3×1 + 3×4i + 2i×1 + 2i×4i = 3 + 12i + 2i + 8i² = 3 + 14i – 8 = -5 + 14i。
When multiplying by a real number, you just scale both parts. Also, multiplying a complex number by its conjugate (which we will meet next) gives a real number. This trick is very handy.
当一个实数乘复数时，只需将实部和虚部分别缩放。此外，将一个复数乘以它的共轭（接下来会介绍）会得到一个实数。这个技巧非常实用。
8. The Complex Conjugate | 共轭复数
The complex conjugate of a number a + bi is a – bi. It is denoted by a bar over the number or a star: z* = a – bi if z = a + bi. Conjugates reflect a complex number across the real axis. The product of a complex number and its conjugate is always a real number: (a + bi)(a – bi) = a² + b², because the imaginary parts cancel out.
复数 a + bi 的共轭复数是 a – bi。它用上方横杠或星号表示：如果 z = a + bi，则 z* = a – bi。共轭反映了复数关于实轴的对称。一个复数与其共轭的乘积总是实数：(a + bi)(a – bi) = a² + b²，因为虚部相互抵消。
For example, the conjugate of 4 + 3i is 4 – 3i. Their product is 4² + 3² = 16 + 9 = 25. This property is essential for division.
例如，4 + 3i 的共轭是 4 – 3i。它们的乘积为 4² + 3² = 16 + 9 = 25。这个性质对于除法非常关键。
9. Dividing Complex Numbers | 复数的除法
To divide one complex number by another, we multiply the numerator and denominator by the conjugate of the denominator. This turns the denominator into a real number. For example, (2 + i) ÷ (1 – i) = (2 + i)/(1 – i). Multiply top and bottom by (1 + i). The denominator becomes (1 – i)(1 + i) = 1² + 1² = 2. The numerator is (2 + i)(1 + i) = 2 + 2i + i + i² = 2 + 3i – 1 = 1 + 3i. So the result is (1 + 3i)/2 = 0.5 + 1.5i.
要将一个复数除以另一个复数，我们将分子和分母同时乘以分母的共轭复数。这样分母就变成了实数。例如，(2 + i) ÷ (1 – i) = (2 + i)/(1 – i)。将分子和分母同乘以 (1 + i)。分母变为 (1 – i)(1 + i) = 1² + 1² = 2。分子为 (2 + i)(1 + i) = 2 + 2i + i + i² = 2 + 3i – 1 = 1 + 3i。因此结果为 (1 + 3i)/2 = 0.5 + 1.5i。
Always remember to write the final answer in the form a + bi. This method works for any division with complex numbers.
务必记住将最终答案写成 a + bi 的形式。此方法适用于任何复数除法。
10. The Complex Plane | 复平面
Complex numbers can be visualised on a diagram called the complex plane or Argand diagram. The horizontal axis represents the real part, and the vertical axis represents the imaginary part. Each complex number is a point on this plane. For example, 3 + 4i is located at coordinates (3, 4). This geometric view helps in understanding addition as vector addition and multiplication as rotation and scaling.
复数可以在一个叫作复平面或阿尔冈图的图上可视化。横轴表示实部，纵轴表示虚部。每个复数都是这个平面上的一个点。例如，3 + 4i 位于坐标 (3, 4) 处。这种几何观点有助于将加法理解为向量加法，将乘法理解为旋转和缩放。
The distance from the origin to the point is called the modulus, denoted |z| = √(a² + b²). This is the length of the vector. For instance, |3 + 4i| = √(3² + 4²) = 5. This connects algebra with geometry beautifully.
从原点到该点的距离称为模，记作 |z| = √(a² + b²)。这是向量的长度。例如，|3 + 4i| = √(3² + 4²) = 5。这优雅地将代数与几何联系起来。
11. Why Are Complex Numbers Useful? | 复数为什么有用？
Complex numbers are not just abstract playthings. They are used in engineering, physics, signal processing, and control systems. For example, alternating current electricity is described using complex numbers. Fractal patterns like the Mandelbrot set are generated using complex iteration. Although you may not use them daily, they are a powerful tool for many scientific fields.
复数不仅仅是抽象的玩物。它们被用于工程、物理、信号处理和控制系统。例如，交流电用复数描述。像曼德勃罗集这样的分形图案就是通过复数迭代生成的。虽然你可能在日常生活中不会用到它们，但它们对许多科学领域来说是一个强大的工具。
12. Summary | 总结
In this article, we learned that i = √(-1) opens the door to complex numbers of the form a + bi. We can add, subtract, multiply, and divide these numbers using simple rules. The complex conjugate is a key idea for division, and the complex plane gives us a geometric interpretation. With these basics, you are ready to explore more advanced topics when the time comes.
在本文中，我们学到了 i = √(-1) 打开了形如 a + bi 的复数之门。我们可以使用简单规则对这些数进行加减乘除。共轭复数是除法的关键概念，而复平面则给了我们几何解释。有了这些基础知识，你将来就可以进一步探索更高级的主题了。
| Operation | Formula |
| Addition | (a+bi)+(c+di) = (a+c)+(b+d)i |
| Subtraction | (a+bi)-(c+di) = (a-c)+(b-d)i |
| Multiplication | (a+bi)(c+di) = (ac-bd)+(ad+bc)i |
| Division | (a+bi)/(c+di) = [(a+bi)(c-di)]/(c²+d²) |
Remember: i² = -1, and the powers of i repeat i, -1, -i, 1. Practice these skills with simple numbers, and gradually you will feel comfortable with complex numbers.
记住：i² = -1，i 的幂依次重复 i、-1、-i、1。用简单的数字练习这些技巧，你会逐渐对复数感到得心应手。
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