# “Total” relates to “sum” as X relates to “product”

In algebra, there is (as I see it) a subtle difference between “sum” and “total”: a sum is a term consisting of several summands that are added up,e.g., “5+5”, whereas the total is the result of this summation, i.e., “10”.

Is this distinction made, or is that only my way to read it?

If the distinction is made, is there similarly a separate word for the result of a product?

• You are probably overdoing it. – Hot Licks Dec 9 '15 at 13:17
• I don't think "total" is really a mathematical term. I think if a mathematician wants to make that distinction, s/he would use "sum" and "result". – Ernest Friedman-Hill Dec 9 '15 at 13:35
• @HotLicks You might be right. I'm a terminologist. – Sebastian Dec 9 '15 at 13:38
• @ErnestFriedman-Hill You might be right, total is not extremely common in math. It does exist in Mathematica, though, as a command to compute the total of a list or vector: reference.wolfram.com/language/ref/Total.html (Sum is already blocked there: reference.wolfram.com/language/ref/Sum.html.) – Sebastian Dec 9 '15 at 13:41
• @Raphael It might fit better on the math educators site. Mathematicians don't spend a lot of time speaking about the fine details of finite sums of real numbers. If it came up, I believe Ernest (above) has it right. – Matt Samuel Dec 10 '15 at 3:33

This may not completely answer your question, but I thought it might be helpful to hear a mathematician's perspective.

In modern (pure) mathematics, numbers are very important (they're the building blocks of much of the subject) but sums and products are things that were likely known to prehistoric humans and at this point they are, to say the least, extremely well understood. Thus we have moved on. We also mostly finished with calculus of a single real variable almost two hundred years ago, contrary to the curious belief among some of my previous students that calculus is the cutting edge of modern mathematics.

These days, the result of a sum is usually not a number. It could be a polynomial, an element of a Hilbert space, a vector field on a manifold, a derivation in the cotangent bundle of an algebraic variety, or a linear operator on a graded ring, to name some examples. There is also a common construction called a "free abelian group" which is us giving ourselves permission to add whatever things we want together, with integer coefficients. There would be no particular reason for it, but we could add a squirrel, a refrigerator, and a toaster if we were freeing frisky that day. The result would be

squirrel + refrigerator + toaster

unless we imposed some relations, like

2squirrel -3refrigerator + 79toaster = 0

Then the group would no longer be free, and we'd have things like

4squirrel - 6refrigerator = -158toaster

The point is that for a mathematician it often doesn't make sense to call the result of a sum a "total." For example, I'm sure we all agree that minus one hundred fifty eight toasters is not what you get when you remove six refrigerators from four squirrels. So we don't have to use a different word for this or other more serious examples, we just call it the "sum," or the "result."

That was to answer whether the distinction is made. For the rest of your question, "product" means "result" even when we aren't working with numbers. If you need an analogous word to make a distinction, I'd say call the expression the "multiplication" and the result the "product."

• Thanks for the detailed answer! Your idea to use multiplication and product might be the way to go. Regarding the modern algebra-based argument; I think it is more that the group operation has been simply called “+” because we are accustomed to its looks, and the term “sum” is actually abused there to mean something abstract. There was just no good word and algebraists were not courageous enough to coin one. Of course you know there are also “multiplicative groups” using product syntax. – Sebastian Dec 10 '15 at 7:40
• @Sebastian In certain structures, other words exist or sum/product are not used. For instance, strings with concatenation. – Raphael Dec 10 '15 at 9:29
• @Sebastian I do not believe that using sum and product for groups is in any way an abuse if terminology, including for concatenation of strings. – Matt Samuel Dec 10 '15 at 12:22