# “Sigma Notation” vs. “Summation Notation”

I was having a discussion with a colleague a few weeks ago, and she opted to use the term "Summation Notation" whereas I had a preference for "Sigma Notation". We didn't have the time to tangentially discuss the differences, so I put it on the back burner. After revisiting it, I've done some cursory research, and there doesn't appear to be a preferred or agreed upon name. To me, "Summation Notation" is cumbersome sounding, so I've always gone with the former, but this could just be because I learned it as such.

Is there a general consensus amongst mathematicians which term to use?

• When tutoring, I've found that some students were taught it as "Sigma notation" and had never heard the word "summation". But I've also found that some students did not know what the greek letter Sigma looked like, and only knew "sum(mation) notation with that big E". I think the best term to use is whatever is used in most other courses at the college/university. – Mark S. Jan 5 '17 at 2:19

Probably there is no consensus... as usual.

But there is a small argument in favor of calling it "summation notation", even if the character is literally an upper-case sigma, since the name "summation..." suggests its function, as opposed to merely its appearance. For that matter, any instance of it is literally a sum (even if infinite), so it makes sense to refer to "the sum" rather than the symbols or characters (or font!) that denote the sum.

• I would take the opposite view. I would think the "sigma notation" is a way of representing the summation operation, and refers to specifically to that notation as opposed to some other way of representing the same operation. Therefore it's appropriate that the name be based on the symbol rather than the meaning. – Henry Towsner Jan 4 '17 at 22:45
• @HenryTowsner: an interesting counterpoint, which surprises me a little, but not toooo much. I do suppose that answers to this question depend on language-philosophy opinions, syntax versus semantics, and so on. – paul garrett Jan 4 '17 at 23:30
• $1 + \frac{1}2 + \frac{1}4 + \frac{1}8 + \cdots$ is also a summation notation, but it isn't sigma notation. – user253751 Jan 5 '17 at 0:47
• @immibis, I think this helps to elucidate my biggest issue with the use of "summation notation" – Andrew Sanfratello Jan 5 '17 at 6:48
• @AndrewSanfratello, when the description is substantially more complicated, a "global" name can be significantly advantageous, yes. Even so, we do distinguish Riemann's zeta from Weierstrass' zeta by extra modifiers. – paul garrett Jan 5 '17 at 13:54

I do not think that you could speak of a consensus there. People will probably use the term they prefer. In fact, Columbia University, for example, uses both terms here (my emphasis):

Often mathematical formulae require the addition of many variables Summation or sigma notation is a convenient and simple form of shorthand used to give a concise expression for a sum of the values of a variable.

However, if we have a look at some numbers, the term "summation notation" appears to be more frequent. A full text search of arXiv for "summation notation" returns 200 entries while the same search for "sigma notation" returns only a number of 109 articles. Other search enginges return similar results. For example, Project Euclid lists 36 results for "summation notation" but only 23 for "sigma notation". Last but not least Google Scholar finds 4750 results containing "Summation notation" while only returning 1310 results containing "Sigma notation".

Thus, "Summation notation" is the term that is used more frequently.

I'd argue for not using either of the two, but if I have to chose I'd much prefer summation notation.

The symbol $\sum$ is the summation symbol. While it is derived from a capital sigma, usually one does not even literally use a capital sigma to typeset it. Observe $\sum_{i=1}^n i^2$ versus $\Sigma_{i=1}^n i^2$

One can then denote a sum using the summation symbol. One might say:

Express the sum of the first 230 odd natural numbers using the summation symbol.

Rewrite $1 -2 + 3 -4 + \dots +99-100$ using the summation symbol.

In this lecture we will recall how to use the summation symbol to express sums in a convenient form.

And if one says some formula like $\sum_{i=1}^{20}$ I would argue for "sum from one to twenty" and not "sigma from one to twenty."

I do not really see the need for a more dedicated term. But perhaps I am missing something.

If one does it like this, one then has in analogy $\prod$ the product symbol, derived from a capital pi $\Pi$. If one has used summation notation, one can call this product notation. Yet, if one has called it sigma notation one would call it pi notation? I find this not optimal.

There are various related symbols and notations, and I find it of considerable advantage to insist on the meaning rather than the appearance or even etymology of the symbol when referring to it.

Finally, while it is a arguably a bit farther away, consider the fact that the integral symbol is also derived from a letter, namely a long s, but we do not talk about the long s notation.