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Lesson 6: Pythagorean theorem proofs

# Garfield's proof of the Pythagorean theorem

James Garfield's proof of the Pythagorean Theorem. Created by Sal Khan.

## Want to join the conversation?

• what is theta • At , what do you mean by "multiply out" (a+b) square and get a square + 2ab + b square? • I'm reading through the slightly different answers people all gave to the different people who asked the same question- "what is theta?". Well, my question is what is the most commonly used definition for theta? • How is it posable for someone to come up with a theorem that is right? how was he able to make this up suddenly? A squrd x B squrd = C squrd, how can someone come up with something like this? • Theorems aren't right they are just scaffolding which is used to build up more theorems. For this they only need to be consistent, which is different from being right or true or fact or real or any of these other categories. As for how thoughts become theorems, it's a process of de-simplification. Which isn't ever just made up, but is built upon other ideas then articulated. Flying squirrels and ants perform trigonometry, but humans talk about it and put it in language. Humans 40,000 years ago started articulating and thinking and then 4,000 years ago, which is when written history starts, we see they were turning thought into theory, and since then it's just been a matter of building up scaffolding, in different languages. One being algebra which started being used to express this ancient relationship of the subtend to the arms (of a triangle but also of a square and a rectangle as the ancient Chinese, Indians, and Babylonian theories went) after the Middle Ages when Italian and German and others discovered ancient Greek and current Arabic/Hindu math, formed groups and schools of method and adopted symbols and abstractions instead of writing everything out in Latin. It was also around that time, coinciding with the rise of professional mathematicians, when simply 'learning for learnings sake' or 'contributing to the stores of wisdom in God's House', sadly gave way to the modern idea of 'originality' and 'research' which has a completely different ethic about it. But it seems Mr. Garfield here wasn't hindered and took the time to light upon an ancient idea even though it had been looked at by countless eyes before.
• Was James Garfield the most recent to prove the Pythagorean Theorem in a new way? • why do we need to equate the area of trapezoid to the are of the three triangles?? • Mr. Garfield realized that there are two ways of calculating the area, and it is understood that, since we are calculating the area of the same figure, both those methods SHOULD give the same result. So he just tried to calculate using both those ways, and saw what it gave him. This is a common way of proof in mathematics - U know something through method A, and you know it using method B also. And since you know that both these, should be giving the same result, by equating them, you might stumble upon wonderful new things :). That is the magic of mathematics.

Beware that sometimes, you may end up with something obvious like a.b = a.b, but sometimes, you might end up proving Pythagoras theorem or that e^i(pi) + 1= 0. :D
• At how does he get the 2ab by multiplying (a+b) squared out? I don't understand. Shouldn't it just be a squared plus b squared? • Work it out with real numbers to see how it comes out.
Let a = 3
Let b = 5
Plugging in the numbers, we'd have (3 + 5)^2 or 8-squared which is equal to 64.

But if we presume that (a + b)^2 is equal to a-squared + b-squared, and if we plug in the numbers, we'd have 3-squared ( which is 9) plus 5-squared (which is 25) and added together they make only 34. Not even close to 64.

On the other hand, we see that a^2 + 2ab + b^2 when substituted would work out to:
3-squared + 2(3 times 5) + 5-squared, or 9 + 2(15) + 25, which is equal to 9 + 30 + 25 = 64.
• Is Ө just a symbol, or does it have mathematical meaning? • Did anyone understand this? I am very confuzzled. • Why (A+B) squared is equal to A squared + 2AB + B squared? 