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. 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.
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Nullary arithmetic product
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).
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
- <math>{2\cdot 3\over 2 \cdot 5},</math>
one may say that 2 divided by 2 is 1, so that we have
- <math>{1 \cdot 3 \over 1 \cdot 5},</math>
but the result is equivalent to what one gets be simply deleting the "2" from the list of factors:
- <math>{3 \over 5}.</math>
If all factors of the numerator or the denominator cancel (as would 2 and 3 in the following example), the remaining value is 1:
- <math>{2\cdot 3 \over 2 \cdot 3 \cdot 5}={\not2\cdot\not3 \over\not2 \cdot\not3 \cdot 5}=\frac15~.</math>
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.
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 zero numbers as 1.
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.:
- <math>\log_b n + \log_b m = \log_b nm</math>
and
- <math>b^{\log_b n + \log_b m} = nm</math>
and more generally
- <math>\prod_i x_i = e^{\sum_i \ln x_i}</math>
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 <math>e^0</math> 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 nm 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) := xy 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.
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.
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.
Quote
"Some textbooks leave the quantity 00 undefined, because the functions x0 and 0x have different limiting values when x decreases to 0. But this is a mistake. We must define x0 = 1 for all x, if the binomial theorem is to be valid when x = 0, y = 0, and/or x = −y. The theorem is too important to be arbitrarily restricted! By contrast, the function 0x is quite unimportant." –– Concrete Mathematics, by Ronald Graham, Donald Knuth, and Oren Patashnik, Addison-Wesley, IBSN 0-21-14236-8, page 162 in the first edition, the chapter on binomial coefficients.