# 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 $a + b = c$:

In the context of **summations**, the **summand** is also known as the **set of summands**.

## Also see

## 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**.