Now that the fall term of school is over, we can all relax for the holidays. Here's a gift to everybody out there, students and teachers and especially parents, who might still be befuddled about some of the more puzzling things that came up in math class. This is why the teacher said what she said.

1) You can't divide by 0.

Why not? Well, because if you try to, no matter what answer you write down, it won't make sense. Take 6 divided by 0. What should that equal? A lot of people guess 0. But that doesn't work. If 6 divided by 0 were equal to 0, that would mean that 0x0 would have to equal 6 (just as 6 divided by 2 equals 3 means that 2x3 equals 6). The trouble here is that 6 divided by 0 cannot equal *any* number, because any number times 0 always gives 0, not 6. This is why division by 0 is verboten.

2) 1 is not a prime number

This seems so unfair. 1's got everything going for it. It's divisible only by 1 and itself, so it does everything a prime number is supposed to do, yet it's still not allowed in the club. What disqualifies it?

The trouble is that if 1 were allowed to be a prime number, it would ruin a beautiful fact that mathematicians love, called the unique factorization theorem, which says that any whole number can be factored in primes in only one way. For example, 30 equals 2x3x5, and there's no other way to write it as a product of prime numbers. But if we allowed 1 in, it would mean we could also write 30 as 1x2x3x5 and 1x1x2x3x5 and so on. The agony would never stop; you could have any number of 1's in the product, and all would be considered different prime factorizations. Goodbye, unique factorization theorem. That's why 1 has to be turned away at the door.

This little story reveals something about how math is actually done: sometimes we adjust the definitions to make the theorems come out the way we want, to ensure that our theories are as pretty as possible.

3) Why is a negative times a negative equal to a positive?

Imagine a movie of someone walking. Think of two steps forward as being like the number 2, and two steps backward as being like -2. If you *rewind* a movie of someone walking two steps forward, it looks like they're walking two steps backward. In this sense, -1x2 equals -2; rewinding a positive makes it negative. Finally, picture what happens if you rewind a movie of someone walking two steps *backward*. Amazingly you'll see them walking two steps forward! That's the intuition for why -1 times -2 should equal 2. The same kind of argument works with any number of steps, and any multiple of times you replay the movie in forward or reverse.

So have a safe and happy holiday season, don't drink and drive, and above all, do not divide by zero.

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