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Last week one teacher used the pound sign # for implying that we were done proving something and a different teacher used the same for highlighting a contradiction.

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I'm from the UK, and in my experience "#" is used invariably to mean 'contradiction' (though in practice it is usually a bit larger than the standard hash symbol and more slanted) –  Daniel Freedman Nov 21 '11 at 0:07

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Many symbols in math don't have a single meaning; and in fact this is not quite mathematics, but rather the use of a symbol as a shorthand, among mathematicians, for what would otherwise be an ordinary statement in one's language. One common symbol for "we are done proving this" is the tombstone (e.g. $\blacksquare$ or $\Box$), and I've even seen $\clubsuit$. I've also seen $\Rightarrow\Leftarrow$ for "this is a contradiction". So, the fact that two people use the pound (a.k.a. hash, number) sign $\#$ for these two different things is not a problem. For example, I wouldn't use $\#$ for either of those purposes, but I might use $\# S$ to denote the cardinality of the set $S$.

In general: if you're confused, ask the person using it what they mean. And as GEdgar points out below, when you yourself use such symbols, be sure to explain what they mean at the outset.

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So the moral is: Don't use sign # for either of those, except perhaps in your private notes. Or when you are standing there speaking the full meaning as you write it. –  GEdgar Nov 20 '11 at 21:40
For those who were as confused as I was that the hash sign # is also called the pound sign, here's an explanation. I suspect I'm not the only one who thought the pound sign would mean £. –  t.b. Nov 20 '11 at 21:48
@t.b. I'd heard the names "hash sign" and "number sign" before, but I wasn't aware that in fact these were the dominant ones outside the U.S. I will keep that in mind! :) –  Zev Chonoles Nov 20 '11 at 21:57
"or, try what Paul Sally does! books.google.com/… instead of the box, he uses a picture of himself smoking a cigar." - quoting @Eugene Bulkin from math.stackexchange.com/questions/56606. :) –  Srivatsan Nov 21 '11 at 3:04

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