What in the world is an imaginary numbers

What is the square root of a negative number? Did you know that no real number multiplied by itself will ever produce a negative number? Finding the square root of 4 is simple enough: either 2 or -2 multiplied by itself gives 4. However, there is no simple answer for the square root of -4.

So, what do you do when a discriminant is negative and you have to take its square root? This is where imaginary numbers come into play. Essentially, mathematicians have decided that the square root of -1 should be represented by the letter i. So, $i = \sqrt{-1}$, or you can write it this way: $-1^{.5}$ or you can simply say: $i^2 = -1$.

What you should know about the number i:

1) i is not a variable.
2) i is not found on the real number line.
3) i is not a real number.

Example A:

Simplify $(4i)^2$

Steps:

1) Multiply 4i times 4i. This will produce $16(i^2)$.

2) Multiply 16 times -1 because $i^2$ equals -1.

The answer is: -16.

Example B:

Simplify $\sqrt{-80}$.

Steps:

1) Factor the existing expression into the product of two or more radicands, keeping in mind that one of them has to be a perfect square. How about $\sqrt{-1}*\sqrt{16}*\sqrt{5}$? Yes, this will produce $\sqrt{-80}$. Don't forget the -1 part!

2) Simplify square roots where needed. For example, $\sqrt{16}$ becomes 4, and $\sqrt{-1}$ simply becomes the number i.

3) Put it all together this way: $4i\sqrt{5}$ or 4i times the square root of 5.

NOTE: You cannot reduce $\sqrt{5}$ anymore because it is already in lowest terms.

Here's another lesson on imaginary numbers if you would like another view point.

Provided by Mr. Feliz