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Concept
Empty product
Summary
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (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. When numbers are implied, the empty product becomes one.
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.
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 convention . In other words, a "product" with no factors at all evaluates to 1.
Allowing a "product" with zero factors reduces the number of cases to be considered in many mathematical formulas. Such a "product" is a natural starting point 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.
The notion of an empty product is useful for the same reason that the number zero and the empty set are useful: while they seem to represent quite uninteresting notions, their existence allows for a much shorter mathematical presentation of many subjects.
For example, the empty products 0! = 1 (the factorial of zero) and x0 = 1 shorten Taylor series notation (see zero to the power of zero for a discussion of when x = 0). Likewise, if M is an n × n matrix, then M0 is the n × n identity matrix, reflecting the fact that applying a linear map zero times has the same effect as applying the identity map.
As another example, the fundamental theorem of arithmetic says that every positive integer greater than 1 can be written uniquely as a product of primes. However, if we do not allow products with only 0 or 1 factors, then the theorem (and its proof) become longer.
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