# Definition:Summation/Inequality

## 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 $\tuple {a_1, a_2, \ldots, a_n} \in S^n$ be an ordered $n$-tuple in $S$.

The summation of $\tuple {a_1, a_2, \ldots, a_n}$ can be written:

$\ds \sum_{1 \mathop \le j \mathop \le n} a_j = \paren {a_1 + a_2 + \cdots + a_n}$

### Multiple Indices

Let $\ds \sum_{0 \mathop \le j \mathop \le n} a_j$ denote the summation of $\tuple {a_0, a_1, a_2, \ldots, a_n}$.

Summands with multiple indices can be denoted by propositional functions in several variables, for example:

$\ds \sum_{0 \mathop \le i \mathop \le n} \paren {\sum_{0 \mathop \le j \mathop \le n} a_{i j} } = \sum_{0 \mathop \le i, j \mathop \le n} a_{i j}$

$\ds \sum_{0 \mathop \le i \mathop \le n} \paren {\sum_{0 \mathop \le j \mathop \le i} a_{i j} } = \sum_{0 \mathop \le j \mathop \le i \mathop \le n} a_{i j}$

## 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).

## Examples

### From $1$ to $\pi$

Let $n = 3 \cdotp 1 4$.

Then:

$\ds \sum_{1 \mathop \le j \mathop \le n} a_j = a_1 + a_2 + a_3$

## 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 $\ds \sum_{i \mathop = 1}^{i \mathop = \infty}$ 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 $\ds \sum_{i \mathop = 1}^{i \mathop = \infty}$ 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)

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