# Sum Of Products notation

"Sum Of Products" notation

Is this notation correct?
(Assuming the equation is valid?)

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## Answers and Replies

0rthodontist
Science Advisor
No, the lower limit on the product is off by 1.

You mean like this? (attached)
I just caught that error myself, yeh. :grumpy:

But my main question was, is the "Sum Of Products" notation correct (even if my math was wrong).

I don't have much experience writing in math notation so I just wonder if I have it typeset correctly. Does it need extra parentheses or anything?

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LOL because how did you know it was the lower limit on the product that's off by 1, rather than the factorial denominator on the left? In fact the number I am describing is correct in the sum of factorials, and yes the sum of products was the one that was off. But couldn't it have been the other way around? How did you know? Just lucky? Is the expression to the left always the one that's "correct" and if there's an error it must be on the right side of the equals sign?

I'm still impressed you analyzed it so quick.

But I still wonder about my "spelling" ... does the upper limit in the product need to be in parentheses, for example? or maybe the entire product expression?

I can't find a clear guide to mathematical notation. The best I found was a meager list of symbols. I even see sums being written with their limits to the right rather than above & below. Cripes! Do mathematicians even care about such concerns? I know, in the English language, there are precise rules for where commas go, how spaces get used (or not) with dashes & ellipses, yada yada. Are mathies just more laid back?

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Zurtex
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Your notation is fine because it is clear what you mean.

It's all about clarity. For example, this isn't fine:

$$\sum_{k=0}^m \prod_{i=0}^n ki + n + \frac{3}{2} + j$$

That could mean in many many different things, in such a case you need to put parenthesis to explain clearly what you mean, for example:

$$\sum_{k=0}^m \left( \prod_{i=0}^n \left(ki + n + \frac{3}{2}\right) \right) + j$$

töff said:
I can't find a clear guide to mathematical notation. The best I found was a meager list of symbols. I even see sums being written with their limits to the right rather than above & below. Cripes! Do mathematicians even care about such concerns? I know, in the English language, there are precise rules for where commas go, how spaces get used (or not) with dashes & ellipses, yada yada. Are mathies just more laid back?
The way an expression is written can depend on context. This is no different from your example of how spelling and grammar works in English. Two grammatically correct sentences can say the same thing in very different ways. Sums usually have their limits written above and below the summation sign, but space considerations might force the limits to the right. If the context is clear, sometimes the entire set to be summed over is written under the summation sign, with nothing above. There are rules for when you can do this, just like using "I" vs. "me" in English.

Aside from what Orthodontist pointed out, your "spelling" looks fine. The upper limits don't need parentheses, though you can put them there if you want to. The product expression doesn't need parentheses around it, either, since it's clear what's being summed without the parentheses. Again, you can put parentheses around it if you want to, but it's redundant.

Clarity, yes, I understand there must not be ambiguity in formulae. So I'm glad you say my notation is unambiguous. Thank you! Nothing like a little validation, even from a complete stranger! In language, ambiguity is also a concern, but punctuation/spelling/grammar are concerns separate from ambiguity ... so now I guess I amend my question to: "Is there an official standard of mathematical notation?" (That's what I should have asked in the first place.)

Sadly, in the world of writing, there are several competing standards across several subdomains. The Chicago Manual Of Style does not coincide everywhere with the Associated Press Style And Libel Manual, neither of which jibes with Turabian or Strunk. Journalists do not agree with fiction-writers, who do not agree with academicians, who don't agree with educators, who don't agree with linguists. And that's just for the English language! and let's not even get into British and American spelling, or even variant spellings in American such as "grey" and "gray." Yikes! (But I digress [as usual].) end/rant

So. Can anybody point me to one (or more) official references for mathematical notation? Do they exist?

Actually, something like
$$\sum_{k=0}^m \prod_{i=0}^n ki + n + \frac{3}{2} + j$$
has a definite meaning, but it probably doesn't mean what is intended. As written, its meaning is the same as
$$\sum_{k=0}^m \left( \prod_{i=0}^n ki \right) + n + \frac{3}{2} + j$$.
Clarity is usually what counts the most, though, so the latter expression is definitely preferable.

That would be the "order of operations" rule, right? which is basically a PUNCTUATION rule ... where to put your parentheses.

A comprehensive (:tongue2:) collection of such rules is what I seek.

Nimz said:
... the entire set to be summed over is written under the summation sign, with nothing above. There are rules for when you can do this, just like using "I" vs. "me" in English
Yep, those rules are what I am looking for.

(Putting it up in LATEX just for giggles ...)

$$\sum_{P=1}^I \frac{\left(P+S-1\right)!}{\left(S-1\right)!} = \sum_{P=1}^I \prod_{i=S}^{P+S-1} i$$

Heh. Looks like Latex requires some "parens" (actually curly braces, which don't render) in the product's upper limit:

\prod_{i=S}^{P+S-1} i

yields ... $$\prod_{i=S}^{P+S-1} i$$

... otherwise the order of operations confuses it ...

\prod_{i=S}^P+S-1 i

yields ... $$\prod_{i=S}^P+S-1 i$$

I know, I know, it's obvious why, even to non-programmers.

Still ... interesting, considering the topic! But I digress again.

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Normally I am an above-average web searcher.
But I cannot find any significant guide to mathematical notation.
(I should head over to the university library. I love that place.)

I find it hard to believe there's just NOT one out there! 0rthodontist
Science Advisor
Why do you care so much about this? People use whatever notation is convenient and clear for a given context.

Zurtex
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töff said:
Normally I am an above-average web searcher.
But I cannot find any significant guide to mathematical notation.
(I should head over to the university library. I love that place.)

I find it hard to believe there's just NOT one out there! That's because there isn't a universal standard notation for evertyhing in mathematics. It's about clarity and context, that is all that matters.

For an example of context, if I was in my PDE class and lecturer wrote:

$$a_i x^i$$

I would assume it meant some constant coefficient ai multiplied by some variable x to the power of some natural number i.

However, if I was in my vector calculus class last semester and lecturer wrote:

$$x^v y_v$$

I would assume that it meant that it meant:

$$\sum_{v=0}^n x^v y_v$$

Where xv is the vth x, yv is the vth y and n is the number of dimensions of the space in which we are dealing with.

Nimz said:
Actually, something like
$$\sum_{k=0}^m \prod_{i=0}^n ki + n + \frac{3}{2} + j$$
has a definite meaning, but it probably doesn't mean what is intended. As written, its meaning is the same as
$$\sum_{k=0}^m \left( \prod_{i=0}^n ki \right) + n + \frac{3}{2} + j$$.
Clarity is usually what counts the most, though, so the latter expression is definitely preferable.
I'm sure, but I have seen simmilar notation be used and parenthesis not added and it is just assumed the reader understands what is going on.

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0rthodontist said:
Why do you care so much about this?
Because I'm a writer. I want to write the math properly. I'm sorry if I have become annoying, but I did kinda assume that some other people might also care* about prescribed rules of notation and maybe be able to point me at them.

* [Edit: Sorry if that sounded critical. If you don't care about this topic, that's fine with me! I didn't mean to come off as disparaging. Nobody cares about everything. I'm sure a lot of things that are important and interesting to me will leave you completely flat, and vice versa. My apologies for the provocative wording there.]

Most of you seem concerned only with clarity, which of course is indispensible. But I can write a sentence in English that very clearly means one and only one thing, while it completely disregards the rules of grammar, punctuation, and spelling.

I want to write math BOTH unambiguously AND within the rules and traditions of the notation. Thus I asked for a reference, which apparently is not readily available online, if it exists anywhere.

In (very briefly!) reviewing the history of mathematical notation all the way back to Babylonian clay tablets, I see a lot of arbitrary and proprietary notation systems. If the authors were foresightful enough to create guides to their notation, their work can live on; otherwise it becomes cryptographic , and modern analysts must decipher and translate it. Fortunately the math community at large seems to have settled on a pretty consistent standard nowadays (since about Euler or slightly before). But everyone seems to know the rules, and take them for granted, until some neophyte nimrod like me comes along and asks for a cheat sheet. It would be very surprising to me, and unfortunate to the grandscale history of mathematics, if no clear guide to the modern notation system exists.

Has anyone here ever even seen or heard of one?

(Heck, there is even a side-debate here about whether Zurtex's original example of an expression that "isn't fine" is actually fine or not.)

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matt grime
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There is far too much notation for there to be a single guide, and one smybol has many uses ($\oplus$ can be direct sum or exclusive or, or mean something else entirely if the writer so chooses, such as some binary operation; context makes it clear).

But there are guides as to how one should structure mathematical writing, there is even a book on it 'How to write mathematics' by Steenrod, Halmos et al.

In particular all 'sentences' should be genuine sentences (the flaw that most people make in writing mathematics is presuming it is OK to write a series of unexplained symbols).

A lot of it is 'good etiqutte' and not binding, though: some places will tell you not to use '(resp.)' notation eg 'If A is even (resp. B is odd) then f(A) is odd (resp F(B) is even)'.

I will examine this book. Thank you!

I've found "how to write math into sentences for papers" guidelines on the web. It's not entirely what I was after.

Still, thanks for the referral!

But, stuff like this ...
Nimz said:
... the entire set to be summed over is written under the summation sign, with nothing above. There are rules for when you can do this, just like using "I" vs. "me" in English
Where are those rules recorded?

matt grime
Science Advisor
Homework Helper
Nowhere, necessarily. Why would they be? It is clear what

$$\sum_{n \in \mathbb{N}}x_n$$

means, and if you're not sure you ask. It is common through usage, no one sits down and writes out the rules. In fact I would say that the only 'rule' in this case is the intuitively obvious one: if you can simply describe the set then do so if you wish to. It is by far the most common way of writing sums. Using

$$\sum_{r=1}^n$$

is by far the least useful way of doing it since it is only usable on sets of the form {r,r+1,r+2,..n}.

The method of not specifying the index on the symbol explicitly is useful for example when you want to do things like taking a sum or product over primes:

$$\prod_p (1- 1/p^s)$$

is common short hand: we know from experience that this sort of product is over p a prime. If you don't know that then it's a fair bet that you wouldn't be reading a paper where it is assumed you do: papers are written with target audiences in mind, and the notation the author chooses reflects that.

There is also nothing to stop you writing: $\sum x_i$ where i runs over the set {1,3,4,6,7} if that is what is needed. You could even write that as

$$\sum_{i \in \{1,3,4,6,7\}} x_i$$

if you wanted but that would be messy.

When would you use the word autodidact and when the phrase 'self taught'? That would depend on the audience in mind, and how fancy you wanted to appear, but there are no had and fast rules written down are there?

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So all of these "common usages" and "common shorthands" are passed down by word of mouth? or at best written singly in scattered explanatory texts, but not compiled somewhere?

I see a trend emerging.

When would you use the word autodidact and when the phrase 'self taught'? ... there are no had and fast rules written down are there?
That is a matter of rhetoric, which I grant is open to a lot of subjectivity, interpretation, and craftsmanship. However, there certainly are many hard and fast rules written down for grammar, spelling, and punctuation, in the field of language. Mathematics at the medial level (I ain't talking about the acroloci where math verges into philosophy) is a precise science. The notation you use is not open to interpretation. You have some latitude about how you describe & write an expression (my equation of sum-of-factorials = sum-of-products being an example!), but your notation system for writing that expression is as precise as the rules of grammar and punctuation that writers follow despite whatever rhetorical approaches they take.

I know comparing math to language, or mathematicians to linguists & writers, is not fair. But I am still quite surprised that the world of math (so far in my limited investigation) has nothing comparable to a "Manual Of Punctuation" in the world of language.

If you don't know that then it's a fair bet that you wouldn't be reading a paper where it is assumed you do
You'd be surprised who reads what. I mean, how do you think I wound up here? I ain't the only cross-discipline freak out there lurking in places I don't belong. Those freaks who lurk into writing & stylistics will find plenty of resources for understanding syntax, dialect, vocabulary, implication & connotation, capitalization, the Oxford Comma, use of single and double quotes, etc.etc. Those freaks (like me) who lurk into math, apparently, are S.double-O.L. for ready references, and are going to have to come in here and ask for help ... which, I suppose, amounts to word-of-mouth.

I have thoroughly enjoyed this discussion so far. It has been quite illuminating! Thanks to everyone who has participated so far. I hope it will continue.

-- Still OMG @ there being no Mathematical Notation Stylistics Guide. arildno
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An important thing to remember, is that it is damaging for your mental flexibility to be too dependent on unique, fixed symbolic notations for your concepts.

Essentially, such a focus is preoccupied with "unimportant" matters.

What IS important, are the concepts themselves, in particular their definitions.
What notational convention you use in describing your concepts should be a matter of unambiguity (NOT the same as fixedness!), efficiency, and taste.

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matt grime
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töff said:
The notation you use is not open to interpretation.

That is because you get to decide, and state, what it means. If it is useful, it catches on. (Often deciding what notation you want to use is the hardest part of doing it.)

Each area of mathematics has its own usage too. Any introductory calculus text book will define the things you need to know to do calculus and so on

$$\sum$$

usually means sum, and as long as you indicate in whatever way you want to what the index set is you're fine. Put the description above, below, both, after, before, whatever you want, it's all fine as long as you put it in somewhere.

That symbol definitely doesn't mean 'sum' in many papers I read, for instance, but they always state what it does mean.

There can be no exhaustive list because people keep inventing new notation all the time (often, nay always, using existing symbols or bastardizations thereof), but any introductory text will define the symbols it uses.

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There can be no exhaustive list
Yeh but apparently there is no list whatsoever, except the briefest of lists of what each symbol means, which is tantamount to a chart of the alphabet with the sounds each letter makes.

Where, for example, has anybody ever described how upper & lower limits can be specified or omitted/implied in sigma notation?

arildno said:
An important thing to remember, is that it is damaging for your mental flexibility to be too dependent on unique, fixed symbolic notations for your concepts.
True! and the same applies to language, as portrayed wonderfully by George Orwell's "Newspeak" in 1984. That was a double-plus-good book!

arildno
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töff said:
True! and the same applies to language, as portrayed wonderfully by George Orwell's "Newspeak" in 1984. That was a double-plus-good book!
Evidently, you are still having the wrong focus on these matters.