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Now parentheses "()" is used for both function, e.g. $f(x)$, and for order of operations, e.g. $(3+5)*2$.

Ever in math history be suggested to differentiate? for example square bracket "[]" can be dedicated to order of operation, and "()" only for function?

this is natural as in primary schools we were taught to use (), [], then {} for order of operation, limiting to [] won't cause any confusion.

just a wild thought...

athos
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  • that's what Mathematica does: [] for functions, () for priority of operations. – Ned Jul 30 '14 at 00:43
  • @Ned so was this idea accepted by professionals, on publications? – athos Jul 30 '14 at 00:44
  • Apparently, Isaac Newton used notation similar to $f[x]$ for $\int f(x),\mathrm{d}x$ (like how one would write f'(x)). Personally, I think the notation should make a comeback. Also, using different brackets in nested brackets can make things a lot clearer, e.g. ([(...)]). – Jam Jul 30 '14 at 00:46
  • no, it's a programming language, where the ambiguity that can arise in a term like a(b) can not be tolerated, whereas in human-to-human mathematical communication, the context makes clear which interpretation is intended. – Ned Jul 30 '14 at 00:48
  • You could do this in your own work, but it would be more trouble than it was worth to change widely. – Kevin Carlson Jul 30 '14 at 01:12
  • For a tiny fraction of a second, I thought you wanted to differentiate the brackets themselves. :) – user_of_math Jul 30 '14 at 03:53

1 Answers1

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Elegance of notation, even at the expense of some easy-to-resolve ambiguity is important. To see why everybody is OK with using $f(x)$ and also $(c+d)(a+b)$ even though $c(a+b)$ becomes ambiguous, consider that a mathematician steeped in analysis would write something like $$ f : \Bbb{R} \rightarrow \Bbb{R} | \forall x \in \Bbb{R} f(x) = x^2 $$ where some more sloppy people might just write $$f(x) = x^2$$

This might illustrate why too much worry about precision in notation is good for some purposes but not for all purposes.

At any rate, certainly before the time of Fermat, the notation $f(x)$ was not in use; and it was surely in use by the time of Gauss. Not being a historian, I don't knwo where it originated.

Mark Fischler
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  • $ f : \Bbb{R} \rightarrow \Bbb{R} | \forall x \in \Bbb{R} f(x) = x^2 $ is a bit too much, but replacing $f(x) = x^2$ with $f[x] = x^2$ won't affect the elegance? or as a layman as me feel... – athos Jul 30 '14 at 00:55