Challenge problem: is it possible to find the square root of a negative number? why or why not?

We know that a square root is a number that when multiplied by itself, makes a certain number. That sounds vague, but you probably have some intuition about this concept. You may realize that the square root of 9, for example, is 3, because 3 times 3 is 9. But -3 is also a square root of 9. Indeed, -3 times -3 is 9.

Now this is an interesting property. These two numbers both make positive 9. But what about -9? What number multiplied by itself gives -9?

You may recall that the product of two positive numbers is positive, and that the product of two negative numbers is also positive. The square root, by definition, requires that the number times itself equals its square, so we can't have two numbers of different signs. Thus, you can't "really" have the square root of a negative number.

But some clever mathematicians—Euler and Gauss—didn't accept this dogma. They defined our familiar numbers as "real" numbers, denoted ℝ, and made a new set of numbers. These are called "imaginary" numbers (no joke), whose set of numbers are sometimes denoted ℂ (there's a bit more nuance to this definition). The imaginary numbers allow you to have a square root for a negative number. They are defined such that , or . This has opened up an unbelievable number of pathways in mathematics, but I'm sure you'll learn about it in due time. :)