# Definition:Summation/Summand

## Definition

Let $\struct {S, +}$ be an algebraic structure where the operation $+$ is an operation derived from, or arising from, the addition operation on the natural numbers.

Let $\set {a_1, a_2, \ldots, a_n} \subseteq S$ be a set of elements of $S$.

Let $\map R j$ be a propositional function of $j$.

Let:

$\ds \sum_{\map R j} a_j$

be an instance of a summation on $\set {a_1, a_2, \ldots, a_n}$.

The set of elements $\set {a_j \in S: 1 \le j \le n, \map R j}$ is called the summand.

## Infinite Summand

Let an infinite number of values of $j$ satisfy $\map R j = \T$.

The set of elements $\set {a_j \in A: \map R j}$ is called an infinite summand.

## Also known as

The summand is also known as the set of summands.

## Linguistic Note

The extensions -and and -end derive from the Latin gerundive forms which impart the meaning that which must be ... to a word.

Thus the word summand, and its synonym addend, literally mean: that which must be summed (or added).

In natural language, the word addendum is more common than either, and similarly means something which is to be added (usually, by linguistic coincidence, to the end).

The archaic term augend has the same lingustic root as augment, which means to make larger. Hence augend is interpreted as something which is to be made larger by adding an addend.