Complex Numbers

We know how to find the square root of any positive real number. In a similar way, we can find the square root of any negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number. The imaginary number $i$ is defined as the square root of $-1$.

$\sqrt{-1}=i$

So, using properties of radicals,

$i^{2}=(\sqrt{-1})^{2}=-1$

We can write the square root of any negative number as a multiple of $i$. Consider the square root of $-49$.

$\begin{aligned}\sqrt{-49} &=\sqrt{49 \cdot(-1)} \\&=\sqrt{49} \sqrt{-1} \\&=7 i\end{aligned}$

We use $7 i$ and not $-7 i$ because the principal root of $49$ is the positive root.

A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written $a+b i$ where $a$ is the real part and $b$ is the imaginary part. For example, $5+2 i$ is a complex number. So, too, is $3+4 i \sqrt{3}$.

Imaginary numbers differ from real numbers in that a squared imaginary number produces a negative real number. Recall that when a positive real number is squared, the result is a positive real number and when a negative real number is squared, the result is also a positive real number. Complex numbers consist of real and imaginary numbers.