I know that this seems a childish question but i have not been able to find a proper convention that cover all math operations. Obviously the basic is that exponential takes precedence over multiplication (and division) that takes precedence over sum (and difference). A particular case can be the sign "$-$" that indicates both the binary operation of subtraction and the unary operation of negation (sign reversal), since the unary operators should take precedence over the binary operators.

My question is focused about operators like summatory, productory, factorial, gradient, etc...

For example, how the following expression should be evaluated? A or B? $$\sum k!$$ A) $\left(\sum k\right)!$

B) $\sum \left(k!\right)$

  • 3
    $\begingroup$ The answer is (B). But that's really just multiplication over addition, the usual rule. $\endgroup$ – Gregory Grant Feb 3 '16 at 11:25
  • $\begingroup$ Please, be more explicit. $\endgroup$ – gvgramazio Feb 3 '16 at 11:27
  • $\begingroup$ If you give that answer only because the summatory operator gives a sequence of sums while the factorial operator gives a sequence of products please consider $\prod k!$. $\endgroup$ – gvgramazio Feb 3 '16 at 11:36
  • $\begingroup$ Factorial takes precedent over sums and products in expressions like that. But I don't think there are firm rules for all combinations. I don't think a complete set of rules is written anywhere, if it is it's not universally recognized - instead it's more like a social unspoken agreement because either it's clear from context or the writer should assess when it's not clear enough and add parentheses. $\endgroup$ – Gregory Grant Feb 3 '16 at 12:21

I do not think there is a universal convention of notations so sometimes a reader has to understand the context. Here are some pointers that might help you.

The general convention is that we read from left to right and the precedence of functions increases.

So if we are looking at $\prod k!$ the answer is $\prod (K!)$. While factorial is written on the right it is still a function and the expression is just a composition of functions (I recall some old books write factorial on the left !).

Similarly it is not so much as binary / unary operation what matters is their order of precedence.

The most common exception to that is encountered in elementary arithmetic where multiplication precedes addition and again that is just an incarnation of the fact that a field is also a vector space over itself.

Sometimes (like computer code) authors write unclear expressions because (a) the expression is unambiguous you just have to check it or (b) the expression is wrong.


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