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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)2011-11-21

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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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    "or, try what Paul Sally does! http://books.google.com/books?id=0mOzVTs2vmwC&lpg=PP1&dq=paul%20sally&pg=PA6#v=onepage&q&f=false instead of the box, he uses a picture of himself smoking a cigar." - quoting @Eugene Bulkin from http://math.stackexchange.com/questions/56606/. :)2011-11-21