# What is a pure imaginary number?

## Question:

What is a pure imaginary number?

## Imaginary Numbers:

An imaginary number is any number that gives a negative result when we take its square. This is opposed to the real numbers we are used to working with, which always end up as positive when squared. Imaginary numbers are always written in terms of the imaginary number i, which itself equals {eq}\sqrt{-1} {/eq}. For example, the imaginary number {eq}\sqrt{-16} {/eq} written in terms of i becomes 4i as follows.

{eq}\sqrt{-16} = \sqrt{16} \times \sqrt{-1} = 4 \times i = 4i {/eq}

A pure imaginary number is any complex number whose real part is equal to 0.

A complex number is a number with both real and imaginary parts written in the form a + bi. In this form, a is the real part and bi is the imaginary part. Both a and b are real numbers, but b multiplied by the imaginary number i makes bi together an imaginary number. The following four examples show what a complex number you might run into in a math course can look like.

{eq}1)\;2 + 3i \\ 2)\;-1 + \sqrt{7}i \\ 3)\;12 + (-i) \\ 4)\;\frac{2}{3} + 221i {/eq}

Note that there is no restriction on what real numbers a and b can be. This means that it is possible for both a and b to be equal to 0. When the real part of a complex number (a) happens to equal 0, all we are left with is the imaginary part (bi) of the complex number.

{eq}0 + bi = bi {/eq}

Because there is no real part remaining, we call these numbers pure imaginary numbers. Conversely, if b equals zero we would be left with a purely real number since there would be no imaginary part left.

{eq}a + (0)i = a + 0 = a {/eq}