# Imaginary Numbers – HSN.CN.A.1

A lot of students get hung up when dealing with imaginary numbers. They can be used in electrical engineering and in Integrated Math 2 complex numbers, or numbers formed by real and imaginary numbers, can be used to represent the “solutions” of a parabola that never crosses the x-axis. That being said they don’t have many simplistic uses. Nonetheless imaginary numbers are an important concept to learn.

So, what exactly are imaginary numbers you ask? Well mathematicians needed something to represent what happens when you try to take a square root of a negative number. The reason why they call it imaginary is that you can’t really take the square root of a negative number. Why you ask?

Well when you take a square root you are looking for a number when multiplied by itself will equal the number of the square root you are taking. So if you take the square root of 4, you will get 2 because 2 x 2 = 4. If you take the square root of 9, you will get 3 because 3 x 3 = 9. So what happens when you try to take the square root of -1? Well, you might be tempted to think that it is just -1, but not so fast! -1 x -1 = +1. In fact any number multiplied by itself, whether it is positive or negative, will equal a positive number. (Remember a negative x a negative = a positive).

So that’s where the i comes in. That represents the square root of -1. It’s imaginary. O.k. so what does i2 equal? Well, remember when you multiply a square root by itself you get the number under the square root sign. So i2 equals -1, a real number! Whoa!

What about i3? Well in that case you multiple i by i by i or -1 times i. So you get –i. You can keep going i4 is -1 times -1 which equals +1, while i5 is back to plain old i and on and on and on …