Empty product

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In mathematics, an empty product, or nullary product, is the result of multiplying no numbers. Its numerical value is 1, the multiplicative identity, just as the empty sum — the sum of no numbers — is zero, or the additive identity.^[1]^[2]^[3]

The empty product is used in discrete mathematics, algebra, the study of power series, and computer programs.

The term "empty product" is most often used in the above sense when discussing arithmetic operations. However, the term is sometimes employed when discussing the value of 0⁰, set-theoretic intersections, categorical products, and products in computer programming; these are discussed below.

1 Nullary arithmetic product
2 0 raised to the 0th power
3 Nullary intersection
4 Nullary Cartesian product
- 4.1 Nullary Cartesian product of functions
5 Nullary categorical product
6 In computer programming
7 References
8 See also
9 External links

[edit] Nullary arithmetic product

[edit] Frequent examples

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).

[edit] A motivation

The idea that the empty product is 1 can be motivated by considering cancellation from the numerator and the denominator of a fraction. When one cancels the factor 2 from

${2\cdot 3\over 2 \cdot 5},$

one may say that 2 divided by 2 is 1, so that we have

${1 \cdot 3 \over 1 \cdot 5},$

but the result is equivalent to what one gets by simply deleting the "2" from the list of factors:

${3 \over 5}.$

If all factors of the numerator or the denominator cancel (as would 2 and 3 in the following example), the remaining value is 1:

${2\cdot 3 \over 2 \cdot 3 \cdot 5}={\not2\cdot\not3 \over\not2 \cdot\not3 \cdot 5}=\frac15~.$

This deletion of all factors is equivalent to dividing by all factors. The numerator becomes here a "product of no numbers", i.e. equal to 1. (Also see 1 (number).)

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, proof that e is irrational, prime factor, binomial series, multiset.

[edit] Conceptual justification

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 "ENTER", 3 "ENTER", 4 "ENTER", 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.

Then the starting value after pressing "CLEAR" has to be 1. After one has pressed "clear" and done nothing else, the number of factors one has entered is zero. Therefore it makes sense to define the product of zero numbers as 1.

[edit] 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.:

log b n + log b m = log b n m

and

$b^{\log_b n + \log_b m} = nm$

and more generally

$\prod_i x_i = e^{\sum_i \ln x_i}$

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 $e 0$ for the empty set, because the empty sum is defined to be zero, and therefore the empty product must equal one.

[edit] 0 raised to the 0th power

Main article: Exponentiation#Zero_to_the_zero_power

From the set-theoretic and combinatorial point of view, the number nm is the size of the set of functions from a set of size m into a set of size n. If n is zero, then in general there are no such functions, because there are no elements in the latter set to map those of the former set into; however if both sets are empty (size 0), then there is exactly one such mapping: the empty function. (This justifies the convention that $0 m = 0$ except when m is zero.) The situation where m and n have no elements can be interpreted as taking a product of 0 copies of the 0 element set; the fact that this empty product equals 1 can be used to support the convention that $00 = 1$ . For more information on the value of $00$ , see the article on exponentiation.

[edit] 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.

[edit] Nullary Cartesian product

Consider the general definition of the Cartesian product:

$\prod_{i \in I} X_i = \{ f : I \to \bigcup_{i \in I} X_i\ |\ (\forall i)(f(i) \in X_i)\}.$

If I is empty, the only satisfying f is the empty function:

$\prod \empty = \{ f_\empty: \empty \to \empty \}.$

Under the perhaps more familiar n-tuple interpretation,

$\prod \empty = \{ ( ) \},$

that is, the singleton set containing the empty tuple. Note that in both representations the empty product has cardinality 1.

[edit] Nullary Cartesian product of functions

The empty Cartesian product of functions is again the empty function.

[edit] Nullary categorical product

In any category, the product of an empty family is a terminal object of that category. In the category of sets, for example, this is a singleton set, while in the category of groups, this is a trivial group with one element.

Dually, the coproduct of an empty family is an initial object. Nullary categorical products or coproducts may not exist in a given category; e.g. in the category of fields, neither exists.

[edit] In computer programming

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

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

Many programming languages with infix multiplication also offer a generalized multiplication function, often called "product", which can be applied to a list of numbers. Such functions return 1 when applied to an empty list.

[edit] References

^ Jaroslav Nešetřil, Jiří Matoušek (1998). Invitation to Discrete Mathematics. Oxford University Press, 12. ISBN 0-198-50207-9.
^ A.E. Ingham and R C Vaughan (1990). The Distribution of Prime Numbers. Cambridge University Press, 1. ISBN 0-521-39789-8.
^ Serge Lang (2002). Algebra. Springer-Verlag, 9. ISBN 0-387-95385-X.