1
$\begingroup$

Is $\sum a$ a customary (standard) shorthand for $\sum_{i\in\operatorname{dom}a} a_i$, where $a$ is an indexed family of say integers?

  • 5
    Duplicate of [124354](http://math.stackexchange.com/questions/124354/what-does-sum-mean-without-a-starting-index-and-limit/124355#124355)?2012-03-27
  • 0
    I have never seen such a notation. Only $\sum_1^n a=na$ (with $a$ constant) or e. g. $\sum_{i\in\operatorname{dom}A} a_i=a_1+a_2+\dots a_n$, with $A=\{a_1,a_2,\dots,a_n\}$. But you mean something different, don't you?2012-03-27
  • 1
    http://math.stackexchange.com/questions/124322/a-contradiction-in-notation#comment287236_1243222012-03-27

3 Answers 3

4

You will sometimes see it used that way, but in my view it’s a dismally poor abuse of notation. At the very least the index should appear somewhere in the expression: $\sum_ia_i$ is fine, given a reasonable context, or even $\sum a_i$, but $\sum a$ is at best annoying and at worst confusing, especially since $\sum_{k=1}^na$ has the completely different unambiguous meaning $na$.

Added: It occurs to me belatedly that there is one context in which I would not at all object to the notation $\sum a$: if $a$ is a finite set of real numbers, say, $\sum a$ is perfectly acceptable shorthand for $\sum\{x:x\in a\}$, just as in set theory $\bigcup a$ is unambiguously $\bigcup\{x:x\in a\}$ if $a$ is a set of sets.

  • 0
    IMHO, $\sum a_i$ is far worse than $\sum a$ because it has the variable $i$ turning up in only one place, which is paradox since the point of a variable is that an expression can be reused. — In $\sum a$ it should be obvious that $a$ itself is _not_ a summable expression, so it's clear and unambiguous that the subscript must be provided from the sum. Whereas $a_i$ is an evaluated expression that could indeed be taken as a constant to sum some over with some other index variable.2012-03-27
  • 0
    @leftaroundabout: It is neither obvious nor necessarily true that $a$ in $\sum a$ is not a summable expression; that is precisely one of the sources of ambiguity here. And anyone who can interpret $a_i$ as a constant to be summed over some other index variable can certainly also interpret $a$ as a constant to be summed over some unexpressed index variable. Mathematical notation is not a programming language, even an idealized one.2012-03-28
  • 0
    @leftaroundabout: If you don't like $\Sigma a_i$, you may also dislike the [Einstein summation notation](http://en.wikipedia.org/wiki/Einstein_notation) which drops the sigma altogether and introduces conventions regarding which indices are summed over and which are not.2012-03-28
  • 0
    Of course, when $a$ is e.g. a vector then it's not clear. Even more so as in that case $\sum_i a_i$ is suspicious, because basis-dependent. But for a sequence, not treated as an element of a vector space, I'd consider it perfectly obvious — No, the notation is not a programming language. Alas! — @mhum: actually, I use Einstein convention very often. I can't say I like it too much (the distinction between greek and roman letters in some contexts), but I certainly don't dislike it. In that convention, index variables always turn up two times, so you don't get what I criticised about $\sum a_i$.2012-03-28
2

Yes. IMO there's not much of a problem with it: in a Haskell-ish pseudo-lambda-calculus-notation $$\begin{align} &\Sigma\ ::\ (J\text{ countable}, S\text{ additive})\Rightarrow\ (J \to S) \to S \\ &\Sigma f = \underbrace{f(j_1) + f(j_2) + \ldots}_{\text{all }j_k\in A} \end{align}$$ with the more common general notation just being shorthand $$ \sum_{i\in I}a_i := \Sigma\bigl(\lambda i.\ a_i\,\chi_I(i)\bigr) $$ where $\chi_I(i)=1$ for $i\in I$ and $0$ otherwise.

  • 0
    You are mostly right, but this does not work for such operators as set-union: http://math.stackexchange.com/questions/124322/a-contradiction-in-notation#comment287236_1243222012-03-27
  • 0
    The ambiguity also comes up with summation (see Brian M. Scott's edit), but as a set is never a function it can always be resolved. Was it this you meant, or the characteristic function $\chi$?2012-03-28
1

Often, yes. The $a_i$ need not be integers, and the index set can also be different from integers - it's usually understood from context. The same goes for products, $\prod a_i$.