What are complex numbers?

First, let's talk about imaginary units.

An imaginary unit is a number that is -1 when squared.

The imaginary unit is denoted as \( i \) and satisfies the following equation

\[ i^{2}=-1 \]

The following numbers, which are imaginary units multiplied by real numbers, are called imaginary numbers.

\[ i,3i,\frac{1}{2}i \]

A number that can be expressed using \( a \) and \( b \) as follows is called a complex number.

\[ a+bi \]

\( a \) is called the real part and \( b \) is called the imaginary part.

Example

Example 1

Calculate \( \left(1+2i\right)+\left(3+4i\right) \).

Solution

\begin{align} &\left(1+2i\right)+\left(3+4i\right)\notag\\ =&\left(1+3\right)+\left(2+4\right)i\notag\\ =&4+6i\notag \end{align}

Example 2

Calculate \( \left(1+2i\right)-\left(3+4i\right) \).

Solution

\begin{align} &\left(1+2i\right)-\left(3+4i\right)\notag\\ =&\left(1-3\right)+\left(2-4\right)i\notag\\ =&-2-2i\notag \end{align}

Example 3

Calculate \( \left(1+2i\right)\times\left(3+4i\right) \).

Solution

\begin{align} &\left(1+2i\right)\times\left(3+4i\right)\notag\\ =&3+4i+6i+8\left(i\right)^{2}\notag\\ =&-5+10i\notag \end{align}

Example 4

Calculate \( \left(1+2i\right)\div\left(3+4i\right) \).

Solution

\begin{align} &\left(1+2i\right)\div\left(3+4i\right)\notag\\[+5pt] =&\frac{1+2i}{3+4i}\notag\\[+10pt] =&\frac{\left(1+2i\right)\left(3-4i\right)}{\left(3+4i\right)\left(3-4i\right)}\notag\\[+10pt] =&\frac{3-4i+6i-8\left(i\right)^{2}}{9+16}\notag\\[+10pt] =&\frac{11+2i}{25}\notag\\[+10pt] =&\frac{11}{25}+\frac{2}{25}i\notag \end{align}