# Definition:Summation/Propositional Function

## Definition

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

Let $\left({a_1, a_2, \ldots, a_n}\right) \in S^n$ be an ordered $n$-tuple in $S$.

Let $R \left({j}\right)$ be a propositional function of $j$.

Then we can write the **summation** as:

- $\displaystyle \sum_{R \left({j}\right)} a_j = \text{ The sum of all $a_j$ such that $R \left({j}\right)$ holds}$.

If more than one propositional function is written under the summation sign, they must *all* hold.

Such an operation on an ordered tuple is known as a **summation**.

Note that the definition by inequality form $1 \le j \le n$ is a special case of such a propositional function.

Also note that the definition by index form $\displaystyle \sum_{j \mathop = 1}^n$ is merely another way of writing $\displaystyle \sum_{1 \mathop \le j \mathop \le n}$.

Hence *all* instances of a **summation** can be expressed in terms of a propositional function.

### Iverson's Convention

Let $\displaystyle \sum_{R \left({j}\right)} a_j$ be the summation over all $a_j$ such that $j$ satisfies $R$.

This can also be expressed:

- $\displaystyle \sum_{j \mathop \in \Z} a_j \left[{R \left({j}\right)}\right]$

where $\left[{R \left({j}\right)}\right]$ is Iverson's convention.

## Summand

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

## Notation

The sign $\sum$ is called **the summation sign** and sometimes referred to as **sigma** (as that is its name in Greek).

## Also see

- Results about
**summations**can be found here.

## Historical Note

The notation $\sum$ for a summation was famously introduced by Joseph Fourier in $1820$:

*Le signe $\displaystyle \sum_{i \mathop = 1}^{i \mathop = \infty}$ indique que l'on doit donner au nombre entier $i$ toutes les valeurs $1, 2, 3, \ldots$, et prendre la somme des termes.*

- (
*The sign $\displaystyle \sum_{i \mathop = 1}^{i \mathop = \infty}$ indicates that one must give to the whole number $i$ all the values $1, 2, 3, \ldots$, and take the sum of the terms.*)

- -- 1820:
*Refroidissement séculaire du globe terrestre*(*Bulletin des Sciences par la Société Philomathique de Paris***Vol. 3, 7**: pp. 58 – 70)

- -- 1820:

However, some sources suggest that it was in fact first introduced by Euler.

## Sources

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: $(2)$