# 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 word addend appears to mean the same thing as summand, such that the two words may be used interchangeably.

The word term is frequently seen for summand, but term also has other meanings.

If it is important to avoid ambiguity then it is recommended that summand is used.

The term augend can sometimes be seen for (specifically) the first of a pair of summands, so in the context of $a + b = c$:

$a$ is the augend
$b$ is the addend
$c$ is the sum.

In the context of summations, 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.