Operations With i

Powers of i

*i^1=i i^2= -1 i^3=-i i^4=1

To find what i to a different power is, however, you take the number that i is to the power to and divide that number by four. Whatever the remainder is is your new power of i then based on the four powers you know, you can solve from there. So let's say you have i^22... You would divide 15 by 4 and get 5 with a remainder of 2 (5 R 2). Next you would take i^2 and your answer would be -1.

*^ means square the number before the ^ by the number after

Practicality

Imaginary numbers are used everyday and most don't know it. Electricity and circuits are run by imaginary numbers. Everybody uses imaginary numbers when they use electricity, but the people who understand it the most are electricians. In AC (alternating current), the electricity produced is represented through positive and negative imaginary numbers and an electricians job is to balance them out. So basically, electricity runs by the balance of positive and negative imaginary numbers.

History

Imaginary numbers were first discovered by Heron of Alexandria. Heron of Alexandria was a Greek mathematician who lived from AD 10-70. Heron discovered imaginary numbers while trying to find the area of a frustum and is also known for his formula for finding the area of a triangle with its side lengths called Herons Formula. 

 

Although Heron discovered imaginary numbers, Rafael Bombelli was the first to discover the rules of multiplication for them in 1572. Rafael Bombelli was an Italian mathematician who lived from around AD 1526-1572.