# Definition of imaginary and complex numbers

## Imaginary numbers

In order to solve equations like $$x^2 + x + 1 = 0$$ (where we need to take square roots of negative numbers), we “invent” a new type of number.

Definition: Define the imaginary unit $$i$$ as the number that satisfies the equation $$x^2 = -1$$, i.e. $$i$$ is a square root of $$-1$$. Any real multiple of the imaginary unit is called an imaginary number.

Example: The numbers $$2i, -6i, \pi i, 0$$ are all imaginary numbers.

We can use the fact that $$i^2 = -1$$ to work out powers of $$i$$.

Worked exercise: Evaluate $$i^4$$ and $$i^5$$.

Solution: $$i^4 = i^2 \times i^2 = (-1)^2 = 1$$ and $$i^5 = i^4 \times i = 1 \times i = i$$.

To avoid confusion, it is best not to write $$\sqrt{-1}$$, because there are two distinct square roots of $$-1$$. If you square both $$i$$ and $$-i$$, you get $$-1$$.

## Complex numbers

Definition: A complex number is a number that can be expressed in the form $$a + bi$$, where $$a$$ and $$b$$ are real numbers. In simple terms, it is what we get when we add a real number and an imaginary number. We denote the set of complex numbers as $$\mathbb{C}$$.

For a complex number $$z$$, we denote the real part of $$z$$ as $$\text{Re}(z)$$ or $$\Re(z)$$ and the imaginary part of $$z$$ as $$\text{Im}(z)$$ or $$\Im(z)$$.

Note that the imaginary part of $$z$$ does not include $$i$$, i.e. the imaginary part of $$x + yi$$ is $$y$$, not $$yi$$.

## Problems

1. Evaluate:
1. $$i^2$$
2. $$i^3$$
3. $$i^4$$
4. $$i^5$$
5. $$i^6$$
6. $$i^{10}$$
7. $$i^{27}$$
8. $$i^{100}$$
9. $$i^{2020}$$
10. $$i^{-1}$$
11. $$i^{-11}$$
12. $$i^{-37}$$
2. Let $$k$$ be an integer. Evaluate $$i^{4k}$$, $$i^{4k+1}$$, $$i^{4k+2}$$ and $$i^{4k+3}$$.
3. Write down $$\text{Re}(z)$$ and $$\text{Im}(z)$$ of:
1. $$z = 3 + 4i$$
2. $$z = -6i$$
3. $$z = -2$$
4. $$z = 0$$