How to Picture the Meaning of Algebra
How can you picture the meaning of the Pythagorean Theorem? What are the geometric meanings of expressions, equations, and all of algebra? Keep on reading to find out!
As we’ve seen, an equation like the Pythagorean Theorem is really a kind of rule saying that whenever we plug in numbers for the variables in the equation, the combination of values in the expression on the left side of the equals sign must have the same value as the combination in the expression on the right.
At least, that’s the algebraic way of looking at things. And while this algebraic viewpoint is helpful, fantastic, and all-around super important, it’s good to keep in mind that it’s not the only way to think about equations. To show you what I mean, today we’re going to learn how to look at math from another angle. In particular, we’re going to learn how to visualize the meaning of everybody’s favorite theorem—the Pythagorean Theorem—using just a simple picture and some clever thinking.
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A Picture of the Pythagorean Theorem
To get things started, I want you to picture one of those beautiful 3–4–5 triangles that we’ve been talking about lately while uncovering the long lost story of Knot Dude. Remember, the 3–4–5 here means that the two legs of the right triangle are 3 and 4 units long, and that the hypotenuse, c, is 5 units long.
Next, draw perfect squares attached to each side of your triangle. The lengths of the sides of the squares should be the same as the lengths of the side of the triangle they’re attached to. In other words, the side of the triangle with a length of 3 should end up attached to a square whose sides are 3 units long, and so on. Finally, further divide up each of these squares you’ve drawn into a bunch of little 1-by-1 unit squares.
Why in the world did we do this? In particular, what’s up with all of those 1-by-1 unit squares? Believe it or not, those little guys are going to play a key role in helping you picture the meaning of the Pythagorean Theorem. How? Before I tell you, I encourage you to first see if you can figure it out first. Then, when you’re ready, continue on!
The Geometric Meaning of Expressions
Let’s start by taking a closer look at all of those unit squares you’ve drawn. First, think about how many fit inside each larger square. For example, as you can see in your drawing, you can fit 4 unit squares from left-to-right across the square attached to the 4-unit-long leg of the triangle, and you can also fit 4 unit squares from top-to-bottom. So a total of 4×4=16 unit squares will fit inside this larger square. No big surprise!
But now imagine that this larger square doesn’t have a length and width of 4, but instead has a length and width represented by a variable called a. That means that the larger square measures a 1-by-1 unit squares across by a 1-by-1 unit squares high. The number of unit squares that will fit inside the larger square is therefore equal to a x a = a^2. Do you see what that means?
In the case of our 3–4–5 triangle whose sides are represented by the variables a, b, and c, it means that you can think of the a^2 part of the Pythagorean Theorem as the number of 1-by-1 unit boxes that will fit inside a 3-by-3 box, and the b^2 part of the equation as the number of 1-by-1 unit boxes that fit inside a 4-by-4 box. And what about c^2? Yep, you guessed it—it’s exactly the same idea but for the big box drawn along the hypotenuse of the triangle.
The Geometric Meaning of Equations
While it’s all well and good to know how to think about the geometric meaning of a^2, b^2, and c^2 separately, we’re really interested in knowing how we can put them all together to understand the entire Pythagorean Theorem. So, let’s do exactly what the Pythagorean Theorem says—let’s combine the big squares containing a^2 and b^2 small 1-by-1 unit boxes. In other words, let’s take all the 1-by-1 boxes that make up the 3-by-3 and 4-by-4 squares and combine them together into a bigger square.
What does this show us? Well, if you count up all the unit boxes, you’ll find that the big box created by adding a^2 1-by-1 boxes to b^2 1-by-1 boxes—for a total of a^2 + b^2 boxes—is the exact same size as the big box from the hypotenuse of the triangle that contains c^2 1-by-1 unit boxes. So what? Well, for one thing, this shows us why the Pythagorean Theorem works for our 3–4–5 triangle. And it also shows us that we now have another way to think about what the Pythagorean Theorem means. Finally, it’s a reminder that problems in algebra—and life—can usually be viewed in multiple ways, and doing so is often a great way to obtain a better understand of their true meaning.
Pythagorean Triples Puzzle
Before we finish up, I want to fill you in on the answer to last week’s brain teaser: Are there other Pythagorean triples besides the simple ones we found? And the answer is… Yes, there are an infinite number of them! In fact, if you pick any two positive integers, let’s call them x and y (where y > x), and calculate:
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a = y^2 – x^2
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b = 2 x y
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c = x^2 + y^2
then these three numbers will always be Pythagorean triples. Go ahead and try it out with a few numbers and test if the results agree with the Pythagorean Theorem!
Wrap Up
Okay, that’s all the math we have time for today. If you want to learn more about algebra, please check out my book The Math Dude’s Quick and Dirty Guide to Algebra.
Remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too. Finally, please send your math questions my way via Facebook, Twitter, or email at mathdude@quickanddirtytips.comcreate new email.
Until next time, this is Jason Marshall with . Thanks for reading, math fans!
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About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.
