Empty product

In arithmetic, the empty product, or nullary product, is the result of multiplying no numbers. Its numerical value is one, just as the empty sum -- the sum of no numbers -- is zero. This fact is useful in discrete mathematics, algebra, the study of power series, and computer programs. Two often-seen instances are a⁰ = 1 (any number raised to the zeroth power is one) and 0! = 1 (the factorial of zero is one).

Some examples of the use of the empty product in mathematics may be found at the following pages: binomial theorem, factorial, fundamental theorem of arithmetic, birthday paradox, Stirling number, König's theorem, binomial type, difference operator, Pochhammer symbol, product (category theory), proof that e is irrational, prime factor, binomial series.

More generally, given an operation of multiplication on some collection of objects, the empty product is the result of multiplying no objects together. It is generally defined to be the identity element with respect to the given operation, if such exists. For example, the empty direct product of (isomorphism classes of) groups is (the isomorphism class of) the trivial group, since every group is isomorphic to its direct product with the trivial group.

Proof

The sum of two logarithms is equal to the logarithm of the product of their operands, i.e.:

<math>\log_b n + \log_b m = \log_b nm<math>

and

<math>b^{\log_b n + \log_b m} = nm<math>

Noting these properties, we may define multiplication across all elements of a set as e to the power of the sum of all natural logarithms of the set's elements:

<math>\prod_i x_i = e^{\sum_i \ln x_i}<math>

Because the empty sum is defined to be zero, the right-hand side of this equation evaluates to <math>e^0<math> for the empty set, and therefore the empty product must equal one.

0 raised to the 0th power

Some accounts say that any non-zero number raised to the 0th power is 1. This point is somewhat context-dependent. If f(x) and g(x) both approach 0 from above as x approaches some number, then f(x)^g(x) may approach some value other than one, or fail to converge. In that sense, 0⁰ is an indeterminate form. A case in which the limit is not 1 (but 1/2 instead) is f(x) := 2^-1/x and g(x) := x, as x approaches 0 from above. However, if the plane curve along which the ordered pair (f(x), g(x)) moves through the positive quadrant towards (0,0) is bounded away from tangency to either of the two coordinate axes, then the limit is necessarily one. Thus it may be said that in a sense, the limit is almost always 1. Furthermore, if the functions f and g are analytic at the point that the variable approaches, then the value will converge to 1, unless f is constant.

However, for other purposes, such as those of combinatorics, set theory, the binomial theorem, and power series, one should take 0⁰ = 1. From the combinatorial point of view, the number n^m is the size of the set of functions from a set of size m into a set of size n. If both sets are empty (size 0), then there is just one such mapping: the empty function. From the power-series point of view, identities such as

<math> e^{0} = \sum_{n=0}^{\infty} \frac{0^n}{n!} = \frac{0^0}{0!} + \frac{0^1}{1!} + \frac{0^2}{2!} + \frac{0^3}{3!} + \cdots \! <math>

are not valid unless 0⁰, which appears in the numerator of the first term of such a series, is 1. A striking instance is the fact that the Poisson distribution with expectation 0 concentrates probability 1 at 0; that does not agree with the usual formula for the probability mass function of the Poisson distribution unless 0⁰ = 1.

A consistent point of view incorporating all of these aspects is to accept that 0⁰ = 1 in all situations, but the function h(x,y) := x^y is not continuous. Then 0⁰ is still an indeterminate form, because we don't know the value of the limit of f(x)^g(x) (in the example above), but that's a statement about limits, not about the value of 0⁰, which is still 1. (More nuanced approaches are possible, but this view is simple and will always work.)

Nullary intersection

For similar reasons, the intersection of an empty set of subsets of a set X is conventionally equal to X.