## Posts tagged ‘counterexample’

### Blog: Proofs from the Book

I told a friend that Guillermo Bautista had started an interesting new blog called *Proofs from the Book*. “Why would he do that?” my friend asked. “Proofs are so boring!”

I replied in the only way I knew how. “Well, that’s a given.”

Of course, I was making a joke. But lots of people are like my friend and think that proofs are boring. They don’t see the beauty in proofs, probably because they’ve never been exposed to the beauty. That’s why I’m so excited that Guillermo started this blog. His proofs allows students to glimpse the beauty and elegance of mathematical theorems discussed in school mathematics, whether it’s proving that the square root of 3 is irrational, providing multiple proofs for the sum of the first *n* positive integers, or having a little fun and “proving” that 2 = 1.

Two of my favorite proofs follow.

Theorem.Every positive integer is interesting.

Proof.Assume that there is an uninteresting positive integer. Then there must be a smallest uninteresting positive integer. But being the smallest uninteresting positive integer is interesting by itself. Contradiction!

Theorem.A cat has nine tails.

Proof.No cat has eight tails. Since one cat has one more tail than no cat, it must have nine tails.

An excellent proof relies on mathematical insight. The most exciting moment of my mathematical life occurred while I was walking my dog. Though I had found an answer to the Three Points on a Square problem, I had no proof that it was correct, other than thousands of examples generated by Excel. With no pencil, no paper, and no agenda — just some time to think — an elegant proof came to me as I was picking up feces. (I have no idea what that says about me.)

This is my favorite part of mathematics. I can literally spend hours reworking equations, drawing figures, and thinking about a problem, and I’ll make no progress. Then later, when I least expect it, when I’m freed from the confines of pencil and paper, the solution gently alights in my mind like a butterfly coming to rest on a marigold.

My hope is that everyone has a chance to see as much beauty in mathematics as I have seen, and *Proofs from the Book* is a place where you can take a peek.

The following jokes, taken from *Math Jokes 4 Mathy Folks*, provide proof that math can be funny. Sort of.

Did you hear about the one-line proof of Fermat’s Last Theorem?

It’s the same as Andrew Wiles’ proof, but it’s written on a really long strip of paper.

At a conference, a mathematician proves a theorem.

Someone in the audience interrupts him. “But, sir, that proof must be wrong. I’ve found a counterexample.”

The speaker replies, “I don’t care — I have another proof for it.”

What’s the difference between an argument and a proof?

An argument will convince a reasonable man, but a proof is needed to convince an unreasonable one.

A meek man appeared in a court room, and the judge was incredulous when he read the charges against the man. “Sir,” said the judge, “you’re a well educated man. How did you end up here?”

“I’m a mathematical logician, dealing in the nature of proof.”

“Yes, go on,” said the judge.

“Well, I was at the library, and I found the books I wanted and went to take them out. The librarian told me I had to fill out a form to get a library card, so I filled out the forms and got back in line.”

“And?” said the judge.

“And the librarian asked, ‘Can you prove you’re from New York City?’ So I stabbed her.”