IB Mathematics: Complex Numbers – Definition and Operation Rules | IB数学：复数的定义与运算规则

📚 IB Mathematics: Complex Numbers – Definition and Operation Rules | IB数学：复数的定义与运算规则

Complex numbers extend the idea of our familiar one-dimensional number line to a two-dimensional plane. They are not just a mathematical abstraction; they are fundamental to fields like electrical engineering, quantum physics, and signal processing. In the IB Mathematics curriculum (Analysis and Approaches, and Applications and Interpretation), a solid grasp of complex number operations is essential for scoring top marks.

复数将我们熟悉的一维数轴延伸到了二维平面。它不仅仅是数学上的抽象概念，更是电气工程、量子物理和信号处理等领域的基础。在 IB 数学课程（分析与方法、应用与解释）中，熟练掌握复数的运算是冲击高分的必要条件。

1. The Imaginary Unit i | 虚数单位 i

The foundation of complex numbers is the imaginary unit, denoted as i. It is defined by the property that its square is -1. That is, i² = -1. This definition allows us to take square roots of negative numbers, which is impossible within the set of real numbers.

复数的基础是虚数单位，记作 i。它的定义是其平方等于 -1，即 i² = -1。这个定义使我们能够对负数开平方，这在实数范围内是不可能实现的。

i² = -1, so i = √(-1)

For example, √(-9) can be simplified as √(9 × -1) = √9 × √(-1) = 3i.

例如，√(-9) 可以化简为 √(9 × -1) = √9 × √(-1) = 3i。

2. Standard Form z = a + bi | 标准形式 z = a + bi

A complex number is typically written in the standard form z = a + bi, where a and b are real numbers. Here, a is called the real part, denoted as Re(z), and b is called the imaginary part, denoted as Im(z). It is crucial to note that the imaginary part is the coefficient of i, which is a real number.

复数通常写成标准形式 z = a + bi，其中 a 和 b 是实数。这里，a 被称为实部，记作 Re(z)；b 被称为虚部，记作 Im(z)。需要注意，虚部是 i 的系数，它本身是一个实数。

• Real part: Re(z) = a
• Imaginary part: Im(z) = b

• 实部：Re(z) = a
• 虚部：Im(z) = b

If b = 0, z is a purely real number. If a = 0 and b ≠ 0, z is a purely imaginary number.

如果 b = 0，则 z 是纯实数；如果 a = 0 且 b ≠ 0，则 z 是纯虚数。

3. Equality of Complex Numbers | 复数相等

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. If z₁ = a + bi and z₂ = c + di, then z₁ = z₂ implies a = c and b = d.

两个复数相等，当且仅当它们的实部相等且虚部相等。若 z₁ = a + bi 且 z₂ = c + di，则 z₁ = z₂ 意味着 a = c 且 b = d。

This property is frequently used to solve for unknown variables in equations involving complex numbers.

这一性质常用于求解含复数方程中的未知变量。

4. Addition and Subtraction | 加法与减法

Adding or subtracting complex numbers is straightforward: we combine the real parts and the imaginary parts separately. If z₁ = a + bi and z₂ = c + di, then z₁ ± z₂ = (a ± c) + (b ± d)i.

复数的加法与减法非常直接：我们分别合并实部和虚部。若 z₁ = a + bi 且 z₂ = c + di，则 z₁ ± z₂ = (a ± c) + (b ± d)i。

(a + bi) + (c + di) = (a + c) + (b + d)i

For example, (3 + 2i) + (5 – 4i) = 8 – 2i.

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