Empty product

In mathematics, an empty product, or nullary product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity 1 (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.

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: ${\displaystyle P_{1}=a_{1}}$ and ${\displaystyle P_{0}=1}$. In other words, a "product" ${\displaystyle P_{1}}$ with only one factor evaluates to that factor, while a "product" ${\displaystyle P_{0}}$ 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.

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