Definition:Product Notation (Algebra)/Index

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Definition

Let $\struct {S, \times}$ be an algebraic structure where the operation $\times$ is an operation derived from, or arising from, the multiplication 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 composite is called the product of $\tuple {a_1, a_2, \ldots, a_n}$, and is written:

$\displaystyle \prod_{j \mathop = 1}^n a_j = \paren {a_1 \times a_2 \times \cdots \times a_n}$


Multiplicand

The set of elements $\left\{{a_j \in S: 1 \le j \le n, R \left({j}\right)}\right\}$ is called the multiplicand.


Notation

The sign $\displaystyle \prod$ is called the product sign and is derived from the capital Greek letter $\Pi$, which is $\mathrm P$, the first letter of product.


Also see

  • Results about products can be found here.


Historical Note

The originally investigation into the theory of infinite products was carried out by Leonhard Paul Euler.


Sources