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