Empty Product A Conceptual Rationale A More Technical

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 paradoxThe birthday paradox states that if there are 23 people in a room then there is a slightly more than 50:50 chance that at least two of them will have the same birthday. For 60 or more people, the probability is greater than 99%. This is not a paradox in t, Stirling numberIn combinatorics, unsigned Stirling numbers of the first kind s ''n ''k (with a lower-case s ) count the number of permutations of n elements with k disjoint cycles. Stirling numbers of the first kind (without the qualifying adjective unsigned are the coe, König's theoremThere is also a proposition in graph theory called Konig's lemma. Set theory In set theory, Konig's theorem states that if I is a set and m and n are cardinal numbers for every i in I and : then : The sum here is the disjoint union of the sets n and the p, binomial typeDefinition In mathematics, a polynomial sequence, i. a sequence of polynomials indexed by { 0, 1, 2, 3,. in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities : Many such sequenc, difference operatorIn mathematics, a difference operator maps a function f ''x to another function f ''x + a − f ''x + b . The forward difference operator : occurs frequently in the calculus of finite differences, where it plays a role formally similar to that of the, Pochhammer symbolIn mathematics, the Pochhammer symbol : is used in the theory of special functions to represent the rising factorial or "upper factorial" : and, confusingly, is used in combinatorics to represent the falling factorial or "lower factorial" : The empty prod, product (category theory)In category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the product of topological spaces. Essentially, the product of a family of objects is the "most gen, proof that e is irrationalIn mathematics, the series expansion : of the number e can be used to prove that e is irrational. Suppose e a ''b for some positive integers a and b''. Consider the number : We will show that x is a positive integer less than 1, and this contradiction wil, 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.

1 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 84 is displayed, 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.

Then the starting value after pressing "CLEAR" has to be 1. Therefore it makes sense to define the product of an empty set of numbers as 1.

2 A more technical justification

The definition of an empty product can be based on that of the empty sum:

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

and

and more generally

i.e., multiplication across all elements of a set is e to the power of the sum of all natural logarithms of the set's elements.

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.

3 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

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.)

4 Nullary intersection

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

5 External link

sci.math FAQ: What is 0⁰?

Arithmetic

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