Real Numbers and Imaginary Numbers

Real numbers are what we get when we combine all the irrational and rational numbers. These numbers are "real" because they're useful for measuring things in the real world such as money, distance, temperature, and Weight Watcher points.

Absolute values and negative signs work the same way for real numbers as they do for integers. We can still use the number line to think about these concepts, but now we can take partial steps. Quick, short little shuffles of the feet, if you will. Fractions, decimals, weird square roots, and even π are all happily having a picnic under the "real number" umbrella.

But here's a question: what number times itself equals -1?

Unfortunately, there's no real number that, multiplied by itself, gives -1 or any other negative integer, for that matter. Any positive real number multiplied by itself is positive, and any negative real number multiplied by itself is also positive: (-1)(-1) = 1, and so on. So if we want a number that creates a negative when multiplied by itself, we actually have to make up a new number—just pull one out of thin air. Whoever said there was no place for imagination in mathematics?

That's where the famous imaginary number i comes in. When you multiply i by itself you get -1.

i2 = -1

In other words, the square root of -1 is i. Now let's hop onto the back of our imaginary pet Hippogriff and fly on over to some exercises.