Complex Numbers

We should already be familiar with the set of real numbers $\mathbb{R}$. We will now extend further and learn the basics about the set of complex numbers $\mathbb{C}$.

Definition: A number $z$ in the form $z = a + bi$ where $a, b \in \mathbb{R}$ and $i^2 = -1$ is said to be a Complex Number. The Real Part is denoted as $\Re (z) = a$, and the Imaginary Part is denoted $\Im (z) = b$. If $\Re (z) = 0$, that is $z = 0 + bi$, then we say that $z$ is an Imaginary Number, and if $\Im (z) = 0$, that is $z = a + 0i$, then we say $z$ is a Real Number.

For example, consider the complex number $z = 2 + 3i$. We note that the real part $\Re (z) = 2$ and the imaginary part $\Im (z) = 3$. Therefore we can represent the complex number $z$ as $z = \Re(z) + \Im(z) \cdot i$.

Commonly, an imaginary number can be represented in two dimensions as a vector along both a real axis and an imaginary axis. We can plot $z = a + bi$ as the unique vector $(\Re (z), \Im (z))$ as illustrated below:

We will now look at some basic properties of complex numbers. All of these properties also hold for real numbers.

Properties of Complex Numbers

Consider the complex numbers $z_1 = a_1 + b_1i$, $z_2 = a_2 + b_2i$ and $z_3 = a_3 + b_3 i$. The following properties hold:

• Commutativity of Addition for Complex Numbers: $z_1 + z_2 = z_2 + z_1$.

• Associativity of Addition for Complex Numbers: $z_1 + (z_2 + z_3) = (z_1 + z_2) + z_3$.

• Existence of an Additive Identity for Complex Numbers: $z_1 + 0 = 0 + z_1 = z_1$.

• Existence of an Additive Inverse for Each Complex Number: $z_1 + (-z_1) = (-z_1) + z_1 = 0$.

• Commutativity of Multiplication for Complex Numbers: $z_1 \cdot z_2 = z_2 \cdot z_1$.

• Associativity of Multiplication for Complex Numbers: $z_1 \cdot (z_2 \cdot z_3) = (z_1 \cdot z_2) \cdot z_3$.

• Existence of a Multiplicative Identity for Complex Numbers: $z_1 \cdot 1 = 1 \cdot z_1 = z_1$.

• Existence of a Multiplicative Inverse for Each Complex Number: If $z \neq 0$ then $z \cdot z^{-1} = z^{-1} \cdot z = 1$.

• Distributive Property: $z_1 \cdot (z_2 + z_3) = z_1 \cdot z_2 + z_1 \cdot z_3$.

Additive and Multiplicative Inverse of a Complex Number

We note that $-z$ is the additive inverse of a complex number $z = a + bi$ if $z + (-z) = 0$. The additive inverse of a complex number is easy in fact, it is $-z = -a - bi$ since:

(1)

\begin{equation} z + (-z) = (a + bi) + (-a - bi) = (a - a) + (b - b)i = 0 + 0i = 0 \end{equation}

Finding the multiplicative inverse $z^{-1}$ is a little more complicated though, but can be calculated with the formula $z^{-1} =\frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2} \cdot i$ where both $a, b =\neq 0$ since:

(2)

\begin{align} \quad \quad z \cdot z^{-1} = (a + bi) \cdot \left ( \frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2} \cdot i \right ) = \frac{a^2 + abi}{ b^2} - \frac{ab - b^2 }{a^2 + b^2} \cdot i = \frac{a^2 + abi}{a^2 + b^2} - \frac{abi - b^2i^2}{a^2 - b^2} = \frac{a^2 - b^2i^2}{a^2 + b^2} = \frac{a^2 + b^2}{a^2 + b^2} = 1 \end{align}

Example 1

If $z = 3 + 4i$ compute the additive inverse $-z$.

We note that $-z = -3 - 4i$ since $z + (-z) = (3 + 4i) + (-3 - 4i) = (3 - 3) + (4 - 4)i = 0 + 0i = 0$.

Example 2

If $z = 5 + 3i$ compute the multiplicative inverse $z^{-1}$.

Using the formula we obtain that: $z^{-1} = \frac{5}{34} - \frac{3i}{34} = \frac{5 - 3i}{34}$.