# $i$ (imaginary unit)

Consider the following sequence of statements.

$\sqrt{9}=3$ since $3×3=9$

$\sqrt{4}=2$ since $2×2=4$

$\sqrt{1}=1$ since $1×1=1$

$\sqrt{0}=0$ since $0×0=0$

$\sqrt{-1}=\underset{_}{\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}}$ since $\underset{_}{\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}}×\underset{_}{\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}}=-1$

There is no real number which can be filled in the blanks in the last line. (Remember that any negative number multiplied by itself is positive.)

So, we say that the required number is imaginary , and we call it $i$ .

The imaginary unit $i$ is defined by the equation ${i}^{2}=-1$ . This enables us to define the square root of any negative number. The term “imaginary” is used because there is no real number that has a negative square.

### Powers of $i$

Note that $i=\sqrt{-1}$ and ${i}^{2}=-1$ .

As any other quantity, $i$ raised to power of zero is $1$ . That is, ${i}^{0}=1$ .

We can use these equations to find the higher powers of $i$ .

$\begin{array}{l}{i}^{3}={i}^{2}\cdot i\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left(-1\right)\cdot i\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=-i\\ {i}^{4}={i}^{2}\cdot {i}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left(-1\right)\left(-1\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=1\end{array}$

You can see that the cycle repeats every four powers after this.

Example 1:

What is the value of $\sqrt{-4}$ ?

$\begin{array}{l}\sqrt{-4}=\sqrt{4\left(-1\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\sqrt{4}\cdot \sqrt{-1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=±2i\end{array}$

Example 2:

Solve the quadratic equation ${x}^{2}+9=0$ .

Subtract $9$ from both sides.

$\begin{array}{l}{x}^{2}+9=0\\ {x}^{2}=-9\end{array}$

Take the square root on both sides.

$\begin{array}{l}x=\sqrt{-9}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\sqrt{9\left(-1\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\sqrt{9}\cdot \sqrt{-1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=±3i\end{array}$

A complex number is a number with a real part and an imaginary part – that is, it is the sum of a real number and a multiple of $i$ . The general form of a complex number $z$ is $a+bi$ where $a$ and $b$ are real numbers and $i$ is the principal square root of $-1$ .