Imaginary Numbers Explained with Formula

What is an imaginary number anyway?

And the Answer is...

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

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
  i^k = i^r

  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

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Imaginary Numbers
Practice Problems
Simplify the imaginary numbers below:

Problem 1)

Answer

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Problem 2)

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Problem 3)

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Problem  4) 

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Problem 5)

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Problem 6)

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