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# Imaginary Numbers Explained with Formula

## What is an imaginary number anyway?

Imaginary numbers are based around mathematical number i
$\text {i is defined to be } \sqrt{-1}$
• $i^2 = \sqrt{-1} \times \sqrt{-1} = -1$
• $i^3 = i^2 \times i = -1 \times i = - i$
• $i^4 = i^2 \times i^2 = -1 \times -1 = 1$

Imaginary numbers also look like $\sqrt{-3}$ or $\sqrt{-5}$

We can use the rules for simplifying square roots to rewrite these kinds of imaginary numbers.
• $\sqrt{-5} = \sqrt{-1 \times 5} = \sqrt{-1} \times \sqrt{5} = i\sqrt{5}$
• $\sqrt{-3} = \sqrt{-1 \times 3} = \sqrt{-1} \times \sqrt{3} = i\sqrt{3}$
• $\sqrt{-4} = \sqrt{-1 \times 4} = \sqrt{-1} \times \sqrt{4} = i\sqrt{4}$
General Formula
$\sqrt{-a} = \sqrt{-1 \times a} = \sqrt{-1} \times \sqrt{a} = i\sqrt{5}$
A Word Of Caution:
You are probably familiar with the following rule from your work with radicals:
However, this rule only works if a > 0 and b > 0. In other words, the product of two radicals does not equal the radical of their products if you are dealing with imaginary numbers.
How to simplify i to any given power:
• Since i4 is the number 1 we can apply the following formula to reduce i to any power
ik = ir
Where r = the remainder of k÷4
Examples
• i9
• Step 1. 9 ÷ 4 has a remainder of 1
• Step 2. i9 = i1
• i22
• Step 1. 22÷ 4 has a remainder of 2
• Step 2. i22 = i2
• i103
• Step 1. 103 ÷ 4 has a remainder of 3
• Step 2. i103 = i3 = -i

### Imaginary Number Calculator and Solver

Calculate the Square Root of:

Simplify

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Imaginary Numbers
Practice Problems
Simplify the imaginary numbers below:

Problem 1)

Problem 2)