Does there exist a notation for the number of terms in a sum?

Question

If $S$ were a set comprised of $n$ elements we would say that the cardinality of $S$ is $n$ and denote said cardinality as $|S|=n$. Does there exist any notation alike the mentioned one for the number of terms in a sum?

Answer 1

Mathematical operations deal with properties of mathematical objects. But the number of terms in a sum is a property of the way the sum is written; that is, it is a property of the expression of the sum. In contrast, the number of elements in a set is a property of the set itself, and is independent of the way it is written. For example:

$$\{n\in \Bbb Z \mid n^2\lt 4\}\\ \{-1, 0, 1\}\\ \Bbb Z \cap \left(-\frac\pi2, \frac\pi2\right)\\ $$

are three different ways to write the same set, and regardless of how it is written, the set has exactly 3 elements.

A similar issue arises sometimes when one wants to discuss the numerator of a fraction. One has to be careful to speak only of the numerator of a fraction, which makes sense, and not of the numerator of rational number that might be represented by a fraction, which does not. The rational number $1.25$, for example, could be represented as $\frac54$ or as $\frac{500}{400}$, which have different numerators. One can of course work around the issue. All I claim is that the issue exists.

Here you want to discuss the number of terms in a sum, which is a certain type of mathematical expression. This can be done, but only if you are treating the expression itself as a sort of mathematical object. This kind of thing is done, but usually only in specialized contexts, and I am not aware of any universal standard notation for it.

One context in which we often discuss the number of terms in a sum is the context of polynomials. A polynomial is a special sort of mathematical expression, one which comes up often enough that the expression itself is considered to be a kind of mathematical object. One often talks about the number of terms in a polynomial. The representation issue I mentioned does come up. For example, is the polynomial $x^2 + 1$ the same as the polynomial $x^2 + 0x + 1$? What about $0x^3 + x^2 + 0x + 1$? And how many terms are there? We always agree that these are the same polynomial, and that it has exactly three terms. The jargon for this is that that polynomial has degree 2, which is often written as something like $\deg P = 2$. In some contexts one might want to talk about the number of nonzero terms, so that a polynomial like $x^{53}+x+1$ has three nonzero terms although its degree is 53. I'm not aware of any special notation for this.

In a more specialized context, dealing with more general expressions, here's the kind of thing we might do:

1. We might decide to understand an “expression” as a rooted tree whose leaves are numbers and whose interior nodes are labeled with operations, chosen from some suitable set that includes +, ×, ÷, and so forth. For example, the expression $“2×(9+(5-2))”$ might be identified with this tree:

2. An expression can then be defined to be a “sum” if each of its interior nodes is labeled with either + or -. So the tree of the previous illustration is not a sum, but its right-side subtree is a sum.
3. In this case, the number of terms in the sum can be defined to be the number of leaves in the tree. The right-side subtree here has three leaves, so it is a sum of three terms.
4. Now we can define a notation to represent the number of terms in a sum: If $E$ is a sum expression, then $\operatorname{terms}(E)$ is the number of terms in $E$. Or we could use $|E|$ or $|E|_+$ or whatever. mathematical notation is often fluid and ad hoc.

One could then extend the definition of “expression” to include expressions of the form $\sum_{n\in S} f(n)$ and the like, and the definition of “number of terms” to say that this expression has $|S|$ terms, or $1$, or whatever is desired.

Maybe this was more than you wanted to know, but I hope if so it was at least interesting.