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. $z = 0$