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Sequence Multiplication (∏) ○◂｜Definition｜1st｜20260604202716-00-⌔

Multiplication - Wikipedia#Capital_pi_notation

Capital pi notation

The product of a sequence of factors can be written with the product symbol $\prod$, which derives from the capital letter Π (pi) in the Greek alphabet (much like the same way the summation symbol $\sum$ is derived from the Greek letter Σ (sigma)).¹² The meaning of this notation is given by

• ✤ $\prod _{i=1}^4(i+1)=(1+1)(2+1)(3+1)(4+1),$

which results in

• ✤ $\prod _{i=1}^4(i+1)=120.$

In such a notation, the variable i represents a varying integer, called the multiplication index, that runs from the lower value 1 indicated in the subscript to the upper value 4 given by the superscript. The product is obtained by multiplying together all factors obtained by substituting the multiplication index for an integer between the lower and the upper values (the bounds included) in the expression that follows the product operator.

More generally, the notation is defined as

• ✤ $\prod _{i=m}^n{x}_i=x_m\cdot x_{m+1}\cdot x_{m+2}\cdot \cdots \cdot x_{n-1}\cdot x_n,$

where m and n are integers or expressions that evaluate to integers. In the case where m = n, the value of the product is the same as that of the single factor x${}_m$; if m > n, the product is an empty product whose value is 1—regardless of the expression for the factors.

Printed 2026-06-28.

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Footnotes

1. Weisstein, Eric W. “Product”. mathworld.wolfram.com. Retrieved 2020-08-16. ↩

2. “Summation and Product Notation”. math.illinoisstate.edu. Retrieved 2020-08-16. ↩

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