In the Real World
The Pythagorean Theorem is good for one thing: finding distances. So long, ruler, there's a new measuring stick in town. This doesn't mean only finding the distance between two objects, however. We can use the Pythagorean Theorem to find the distance between things in three or more dimensions, and also to find the distance between anything that can be measured with numbers.
Colors, for example. Yep, that's right. Every single color can be identified and measured with numbers. Isn't it starting to feel like everything has a number? What's next? Billiard balls and chemical elements?
Radicals and square roots are important because they appear when we compute areas, which is a fairly practical application. Suppose, one day, that you are renting an apartment. (Yes, you will need to move out of your parents' house eventually.) This new apartment is square and has 400 square feet, which seems like a lot of feet...especially having been confined to that cupboard under the stairs for the past 11 years. You know by taking the square root that this must be a 20 foot by 20 foot room.
Even cooler is the fact that square roots give us some of our examples of irrational numbers. In fact, is the "most" irrational number. We know that statement itself sounds a little irrational, but it's true. It is hard to explain what exactly we mean by that, but it is simply very far away from any rational number. That's got to be a difficult feat to accomplish since rational numbers are everywhere. This number , or rather a version of this number , is used by nature to construct almost everything. It determines how sunflower seeds are packed into the face of the sunflower and how branches are distributed in trees so that the leaves receive an optimal amount of sunlight.
You knew trees had roots, but we bet you didn't know they also had square roots.