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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. 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 one may say that 2 divided by 2 is 1, so that we have but the result is equivalent to what one gets by simply deleting the "2" from the list of factors: If all factors of the numerator or the denominator cancel (as would 2 and 3 in the following example), the remaining value is 1: 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: 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. 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. 0 raised to the 0th power If y approaches 0 as x approaches 0 from above, then xy may approach any nonnegative value, or fail to converge. If y = log(a)/log(x), then xy = a for all positive x. The function xy is not continuous in (x,y)=(0,0). So 00 cannot be determined simply by continuity. (See indeterminate form). However, if the plane curve along which the point (x,y) moves towards (0,0) is bounded away from tangency to the y-axis, then the limit is one. Thus it may be said that, the limit is almost always 1. Furthermore, if the function y=f(x) or the inverse function x=f−1(y) is a nonzero analytic function with f(0) = 0, then limx→0 xy = 1. (Note that neither y=log(a)/log(x) nor x=a1/y are analytic around x = 0). For purposes such as those of combinatorics, set theory, the binomial theorem, and power series, one should also take 00 = 1. 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 0m is zero except when m is zero.) 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; which agrees with the formula for the probability mass function of the Poisson distribution only if 00 = 1. The consistent point of view incorporating these aspects is to define 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 Cartesian product Consider the general definition of the Cartesian product: If I is empty, the only satisfying f is the empty function: Under the perhaps more familiar n-tuple interpretation, that is, the singleton set containing the empty tuple. Note that in both representations the empty product has cardinality 1. Nullary Cartesian product of functions The empty Cartesian product of functions is again the empty function. 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: ( evaluates to 4 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. See also | |||||||
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