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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 show up when we compute areas, which is a fairly practical application. Suppose, one day, that you're renting an apartment. (Yes, you'll need to move out of your parents' house eventually.) This new apartment has a square floor plan that covers 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's hard to explain what exactly we mean by that, but it's 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.
We've seen a number of equations over these pages that have no real number solutions. Some of these equations have no solutions because all of mathematics would come crashing down if they did. Scary. Hey, at least that would free up an extra hour of your school day.
An example is the equation x = x + 1.
This can only be true if 0 = 1, which is a contradiction when we're dealing with the usual everyday integers. Although, if you ever have the chance, ask an algebraist about the field of characteristic 1, in which 0 and 1 are the same thing. That'll be one memorable conversation. Or, you know, zero memorable conversations.
A more interesting case is when an equation has no real number solutions, not because it's merely stating a contradiction, but because there is no real number that can possibly fit the job description. For example, the equation x2 = -1 has no real number solutions because there's no real number we can square to give us a negative value.
Remember how we needed to introduce the concept of negative numbers to answer questions like 3 – 5 = ? and then had to do the same thing with rational numbers to answer questions like 2 ÷ 9 = ?
Now we need imaginary numbers to answer questions like
Mathematicians are like people (we could end the sentence right here) who are in possession of a genie in a magic lamp. Every time they need something to make an equation work, they simply rub the lamp and ask for whatever new type of number they want, and poof! their wish is granted. You'd think they would have asked for a billion dollars or to be ruler of the world, but their desires are modest and mainly confined to their work.
The most popular imaginary number, i, can be defined as the square root of -1. If we allow imaginary numbers as solutions to equations, we can solve many more equations than we can with real numbers. This keeps the mathematicians, and their genies, happy. As an example, i is a solution to the equation x2 = -1, an equation which has no real number solutions.
Although i and company are referred to as "imaginary," they're perfectly good numbers. You can't sit down and count out i of something, but i can still be extremely useful in solving complex equations where we'd otherwise be hopelessly lost. If "lost" is how you're feeling right now, then you know exactly how we'd feel without imaginary numbers.
There are three steps to solving a math problem.
If you like drawing pictures, by all means, draw pictures every step of the way; this is where you can let your inner Walt Disney out of the box. We'll draw pictures in every step of this next example.
The length of the hypotenuse of a right triangle is 1 greater than the length of one leg, and 2 greater than the length of the other leg. What are the lengths of the sides of the triangle?
1. Figure out what the problem is asking.
We know there's a right triangle involved, so we'll translate this problem into pictures. Bonus points if you can do this on an Etch-a-Sketch.
Given some information about the relative lengths of the sides, we need to figure out how long each side is. Once we know all of the lengths, we'll be able to make fun of the shortest one.
2. Solve the problem.
We'll name the hypotenuse x, so we don't have equations with the word "hypotenuse" in them. If we redraw the picture, it looks like this:
Since this is a right triangle, let's see if the Pythagorean Theorem tells us anything useful. We have a hunch it might.
(x – 1)2 + (x – 2)2 = x2
Distribute the terms on the left-hand side of the equation:
(x2 – 2x + 1) + (x2 – 4x + 4) = x2
Simplify a bit to get x2 – 6x + 5 = 0.
This factors as (x – 5)(x – 1) = 0, which has roots at x = 5 and x = 1.
3. Check the answer.
We found the values x = 5 and x = 1. These are both solutions to the equation (x – 1)2 + (x – 2)2 = x2.
When x = 5, we find that (5 – 1)2 + (5 – 2)2 = 25, which is indeed 5 squared. When x = 1, on the other hand, we find that (1 – 1)2 + (1 – 2)2 = 1, which is indeed 1 squared. However, before jotting down our solutions and calling it a day, we also need to consider the original triangle and make sure both of these make sense. Neither is a negative number on its own, but what about when we plug them into the information provided to us in the original problem?
Plugging in x = 5 gives us this triangle:
Which is totally hunky-dory. But check out what happens when we plug in x = 1:
This so-called triangle is an impostor. A side of length 0 and a side of length -1? Who does it think it's trying to fool? We weren't born yesterday. If we were, by the way, we would be 0 years old, and we would've been -1 days old two days ago.
Anyway, it's a good thing we checked our answers against the original problem, because only one of our answers works: x = 5.
How do we know that the 4-sided shape in the middle of this picture, with sides of length c, is actually a square? Does it have any credentials or some sort of "square ID" badge?
For starters, the angles of a triangle must add to 180 degrees; that much we know for sure. In a right triangle, the measures of the other two angles must add to 90 degrees:
Any straight line, such as the straight line across the bottom of the big square, is also 180 degrees. Adding the measures of the blue angle and red angle yields 90 degrees, so we have 90 degrees left for the angle between the blue and red line segments; in other words, between the two sides of length c. The figure with 4 sides of length c is a square.
It was nice that our shape passed the initial eyeball test, but we feel even better knowing it can be proven. In fact, we're now on such a proving high that we'll take on the existence of Bigfoot next. Wish us luck.