Empty product

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). Another commonplace instance is that when one cancels (in this case) 2 and 3 from both the numerator and the denominator in a fraction such as

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, multiset.

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.

A conceptual rationale

Imagine a calculator that can only multiply.
It has an "ENTER" key and a "CLEAR" key.
One would wish that, for example, if one presses "CLEAR", 7, 3, 4, then the display reads 84, because 7 × 3 × 4 = 84. More precisely, we specify:

• A number is displayed just after pressing "CLEAR";
• When a number is displayed and one enters another number, the product is displayed;
• Pressing "CLEAR" and entering a number results in the display of that number.

A more technical justification

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

Using this property as definition, and extending this to the empty product, the right-hand side of this equation evaluates to for the empty set, because the empty sum is defined to be zero, 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 functionss 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
:
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 do not know the value of the limit of f(x)^g(x) (in the example above), but that is 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. See nullary intersection for more information.

Complex numbers

In computer programming

Most programming languages do not permit the direct expression of the empty product, because multiplication is taken to be a binary operator. (A programmer may, of course, implement it.) Lisp languages are an exception, where fully parenthesized prefix notation and variadic functions give rise to a natural notation for nullary functions.

(* 2 2) ; evaluates to 4 (* 2) ; evaluates to 2 (*) ; evaluates to 1

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