## What is a complex number?

The number $$i$$ is defined to be such that $$i^2 = -1$$. It is called an imaginary number because, unlike the natural numbers and the real numbers, it has no immediate physical interpretation. (Note that some books use $$j$$ instead of $$i$$. So beware!)

Even though $$i$$ is imaginary, it is useful in many engineering applications.

We can mix $$i$$ with the real numbers using addition and multiplication to form the set of complex numbers, denoted by $$\mathbb{C}$$, with the special properties $$i \cdot i = -1$$ and $$0\cdot i = 0$$.

Just as irrational numbers were invented to solve equations such as $$x^2-2=0$$, the number $$i$$ was invented to solve equations that have no real solutions, such as $$x^2+1=0$$. For a more detailed motivation for complex numbers, click here.

A complex number is written in the form $$a+bi$$ where $$a,b\in \mathbb{R}$$. Here, $$a$$ is the real part of the complex number and $$b$$ is the imaginary part. (Note that $$i$$ is not included when we refer to the imaginary part.) We also use the following notation to extract the real and imaginary parts:

Where $$z = a+bi$$, $$\textbf{Re}(z) = a$$ and $$\textbf{Im}(z) = b$$.

For convenience, if $$b=1$$, we simply write $$a + i$$ instead of $$a + 1 i$$. If $$a = 0$$, we simply write $$bi$$ instead of $$0 + bi$$. Of course, when $$a = 0$$ and $$b = 1$$, we get just the imaginary number $$i$$.

## Examples

1. If $$z = 3 - 4i$$, then $$\textbf{Re}(z) = 3$$ and $$\textbf{Im}(z) = -4$$.

2. If $$w = 2i$$, then $$\textbf{Re}(w) = 0$$ and $$\textbf{Im}(w) = 2$$.

3. If $$u = -5$$, then $$\textbf{Re}(u) = -5$$ and $$\textbf{Im}(u) = 0$$.

## Exercises

For each of the following complex numbers, give its real part and imaginary part.

1. $$1+3i$$

2. $$\sqrt{2}i$$

3. $$-\pi - 4i$$

4. $$2$$