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• 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?

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B1 TED-Ed congruent euclid triangle proof theorem

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

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阿多賓 posted on 2022/11/23
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