In mathematics, an empty product, or nullary product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity1 (assuming there is an identity for the multiplication operation in question), just as the empty sum—the result of adding no numbers—is by convention zero, or the additive identity.
The term "empty product" is most often used in the above sense when discussing arithmetic operations. However, the term is sometimes employed when discussing set-theoretic intersections, categorical products, and products in computer programming; these are discussed below.
Nullary arithmetic product
Let a1, a2, a3,... be a sequence of numbers, and let
be the product of the first m elements of the sequence. Then
for all m = 1,2,... provided that we use the following conventions: and . In other words, a "product" with only one factor evaluates to that factor, while a "product" with no factors at all evaluates to 1. Allowing a "product" with only one or zero factors reduces the number of cases to be considered in many mathematical formulas. Such "products" are natural starting points in induction proofs, as well as in algorithms. For these reasons, the "empty product is one convention" is common practice in mathematics and computer programming.