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