Complex Numbers Examples 1

Complex Numbers Examples 1

Recall from the Complex Numbers page that numbers in the form $z = a + bi$ where $a, b \in \mathbb{R}$ and $i = \sqrt{-1}$ are called complex numbers and the set of all complex numbers is denoted by $\mathbb{C}$. We will now look at some examples regarding complex numbers.

Example 1

Graph the numbers $-2 - i$ and $2 + i$ in the complex plane.

The graph of these two complex numbers is given below:

Example 2

Determine the values of $i^1$, $i^2$, … $i^n$ for $n \in \mathbb{N}$.

We first have that $i^1 = i$. So $i^2 = \sqrt{(-1)}^2 = -1$, and multiplying by $i$ again gives us that $i^3 = -i$. Multiplying by $i$ once more gives us $-i^2 = -(-1) = 1$.

Its not hard to see that:

(1)
\begin{align} \quad i^n = \left\{\begin{matrix} i & \mathrm{if \: n = 1, 5, 9, ...}\\ -1& \mathrm{if \: n = 2, 6, 10, ...} \\ -i & \mathrm{if \: n = 3, 7, 11, ...} \\ 1 & \mathrm{if \: n = 4, 8, 12, ...} \end{matrix}\right. \end{align}

Example 3

Verify that the sequence $\{ \frac{i^{2n}}{n} \}_{n=1}^{\infty}$ converges to zero.

Note that $i^{2n} = -1$ if $n$ is odd and $i^{2n} = 1$ if $n$ is even. The first few terms of this sequence is $\left \{ -1, \frac{1}{2}, -\frac{1}{3}, \frac{1}{4}, ... \right \}$.

We note that the numerator of this sequence is bounded, that is $-1 ≤ i^{2n} ≤ 1$ for all $n \in \mathbb{N}$ and the denominator is unbounded as $\lim_{n \to \infty} n = \infty$ and so $\lim_{n \to \infty} \frac{i^{2n}}{n} = 0$.

Example 4

Simplify the product $(2i - 3i^2 + 4)(-i + 2)$. Is this a real number?

When we expand this product we get that:

(2)
\begin{align} \quad -2i^2 + 4i + 3i^3 - 6i^2 - 4i + 8 \\ \quad = -2(-1) -3i + 6 + 8 \\ \quad = 16 - 3i \end{align}

So this number is not a real number. Alterantively, we could have simplified the product from that start by noting that $-3i^2 = 3$, and so $(2i - 3i^2 + 4)(-i + 2) = (2i + 7)(-i + 2)$ and thus:

(3)
\begin{align} \quad (2i + 7)(-i + 2) \\ \quad = -2i^2 + 4i - 7i + 14 \\ \quad 2 -3i + 14 \\ \quad 16 - 3i \end{align}