Proofs are hard. So combine that with the fact that what is being proved seems pretty obvious, you get the double whammy of why students hate proofs - they are both hard and pointless.
Do we really need to prove that the sum of the first n odd positive integers is n squared. It wouldn't take to much time to get students to accept that fact without doing a proof.
The square root of two is irrational. "I found the square root of two on my calculator. The decimals don't repeat. Yes. I'll buy it."
It seems that statements that are true for the first ten or so examples are probably going to end up being true. In fact, it's a little difficult to find situations where that doesn't hold true. If I make a statement that a coin I have will always come up heads, you might have your doubts. If I flip it ten time and I get heads each time you probably think that statement is, in fact, true, because it probably is a two headed coin.
I had heard quite a while ago that there was some statement having to do with prime numbers what was true way down the line, but then turned out not to be false at some point. I never knew exactly what that was, though.
Well here it is.
Suppose we want to find out if P is a prime number. Find 2P and then divide it by P. If the remainder is 2, then P is a prime number. If it is something other than 2, P isn't prime.
I was a doubter at first. I did the first several and they all worked out, though. I'll take one example. Let's try the number 17.
217 = 131,072
131,072 divided by 2 is 7,710 remainder 2
Thus 17 is a prime number.
This really works - all the way to 340. Then it doesn't.
2341 is so big I can't tell you what it is. I could get my calculator to go as high as to the 331 power, then it refused to do any more.
Anyway, pretty amazing. It works for 1, 2, 3, 4, 5, ..., 340, correctly determining if the number is prime. Then this method claims 341 is prime. But it isn't, because 11 x 31 is 341.
There you go. Proofs aren't pointless after all. Still hard, but not pointless.
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Now, about last week's post. We worked out a score for the yet to be played Super Bowl of New England 30, Atlanta 27.
And how did that turn out? Pretty darn close I must say. New England 34, Atlanta 28.
I was a doubter at first. I did the first several and they all worked out, though. I'll take one example. Let's try the number 17.
217 = 131,072
131,072 divided by 2 is 7,710 remainder 2
Thus 17 is a prime number.
This really works - all the way to 340. Then it doesn't.
2341 is so big I can't tell you what it is. I could get my calculator to go as high as to the 331 power, then it refused to do any more.
Anyway, pretty amazing. It works for 1, 2, 3, 4, 5, ..., 340, correctly determining if the number is prime. Then this method claims 341 is prime. But it isn't, because 11 x 31 is 341.
There you go. Proofs aren't pointless after all. Still hard, but not pointless.
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Now, about last week's post. We worked out a score for the yet to be played Super Bowl of New England 30, Atlanta 27.
And how did that turn out? Pretty darn close I must say. New England 34, Atlanta 28.
