Definition:Summation/Propositional Function

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 $S$.

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

Then we can write the summation as:

$\ds \sum_{\map R j} a_j = \text{ The sum of all $a_j$ such that $\map R j$ 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 $\ds \sum_{j \mathop = 1}^n$ is merely another way of writing $\ds \sum_{1 \mathop \le j \mathop \le n}$.

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

Iverson's Convention

Let $\ds \sum_{\map R j} a_j$ be the summation over all $a_j$ such that $j$ satisfies $R$.

This can also be expressed:

$\ds \sum_{j \mathop \in \Z} a_j \sqbrk {\map R j}$

where $\sqbrk {\map R j}$ 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 $\ds \sum_{i \mathop = 1}^{i \mathop = \infty}$ 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.

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: $(2)$