Placeholder Image

Subtitles section Play video

  • What is proof?

  • And why is it so important in mathematics?

  • Proofs provide a solid foundation for mathematicians, logicians, statisticians, economists, architects, engineers, and many others to build and test their theories on.

  • And they're just plain awesome!

  • Let me start at the beginning.

  • I'll introduce you to a fellow named Euclid, as in, "Here's looking at you, Clid."

  • He lived in Greece about 2,300 years ago, and he's considered by many to be the father of geometry.

  • So, if you've been wondering where to send your geometry fan mail, Euclid of Alexandria is the guy to thank for proofs.

  • Euclid is not really known for inventing or discovering a lot of mathematics, but he revolutionized the way in which it is written, presented, and thought about.

  • Euclid set out to formalize mathematics by establishing the rules of the game.

  • These rules of the game are called axioms.

  • Once you have the rules, Euclid says you have to use them to prove what you think is true.

  • If you can't, then your theorem or idea might be false.

  • And if your theorem is false, then any theorems that come after it and use it might be false, too.

  • Like how one misplaced beam can bring down the whole house.

  • So, that's all that proofs are:

  • Using well-established rules to prove beyond a doubt that some theorem is true.

  • Then, you use those theorems like blocks to build mathematics.

  • Let's check out an example.

  • Say I want to prove that these two triangles are the same size and shape.

  • In other words, they are congruent.

  • Well, one way to do that is to write a proof that shows that all three sides of one triangle are congruent to all three sides of the other triangle.

  • So, how do we prove it?

  • First, I'll write down what we know.

  • We know that point M is the midpoint of AB.

  • We also know that sides AC and BC are already congruent.

  • Now, let's see, what does the midpoint tell us?

  • Luckily, I know the definition of midpoint.

  • It is basically the point in the middle.

  • What this means is that AM and BM are the same length, since M is the exact middle of AB.

  • In other words, the bottom side of each of our triangles are congruent.

  • I'll put that as step two.

  • Great! So far, I have two pairs of sides that are congruent.

  • The last one is easy.

  • The third side of the left triangle is CM, and the third side of the right triangle is, well, also CM.

  • They share the same side; of course it's congruent to itself!

  • This is called the reflexive propertyeverything is congruent to itself.

  • I'll put this as step three.

  • Ta-da! You've just proven that all three sides of the left triangle are congruent to all three sides of the right triangle.

  • Plus, the two triangles are congruent because of the side-side-side congruence theorem for triangles.

  • When finished with a proof, I like to do what Euclid did.

  • He marked the end of a proof with the letters QED.

  • It's Latin for "quod erat demonstrandum", which translates literally to "what was to be proven".

  • But I just think of it as "look what I just did!"

  • I can hear what you're thinking,

  • "Why should I study proofs?"

  • One reason is that they could allow you to win any argument.

  • Abraham Lincoln, one of our nation's greatest leaders of all time used to keep a copy of Euclid's "Elements" on his bedside table to keep his mind in shape.

  • Another reason is, you can make a million dollars.

  • You heard me, one million dollars.

  • That's the price that the Clay Mathematics Institute in Massachusetts is willing to pay anyone who proves one of the many unproven theories that it calls "the millennium problems".

  • A couple of these have been solved in the '90s and 2000s.

  • But beyond money and arguments, proofs are everywhere.

  • They underly architecture, art, computer programming, and internet security.

  • If no one understood or could generate a proof, we could not advance these essential parts of our world.

  • Finally, we all know that the proof is in the pudding.

  • And pudding is delicious. QED.

What is proof?

Subtitles and vocabulary

Click the word to look it up Click the word to find further inforamtion about it

B1 TED-Ed congruent euclid triangle proof theorem

【TED-Ed】How to prove a mathematical theory - Scott Kennedy

  • 7850 160
    阿多賓 posted on 2022/11/23
Video vocabulary