|
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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
In mathematics, multiplication is an elementary arithmetic operation. ...
Look up one in Wiktionary, the free dictionary. ...
In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...
In mathematics, the empty sum, or nullary sum, is the result of adding no numbers. ...
Addition is one of the basic operations of arithmetic. ...
0 (zero) is both a number — or, more precisely, a numeral representing a number — and a numerical digit. ...
Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. ...
Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. ...
In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...
Computer programming (often simply programming) is the craft of implementing one or more interrelated abstract algorithms using a particular programming language to produce a concrete computer program. ...
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. Arithmetic or arithmetics (from the Greek word αÏιθμός = number) is the oldest and simplest branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...
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). In mathematics, exponentiation (frequently known colloquially as raising a number to a power) is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. ...
The beginning of the sequence of factorials (sequence A000142 in OEIS) In mathematics, the factorial of a natural number n is the product of all positive integers less than or equal to n. ...
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).) Look up one in Wiktionary, the free dictionary. ...
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. In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...
The beginning of the sequence of factorials (sequence A000142 in OEIS) In mathematics, the factorial of a natural number n is the product of all positive integers less than or equal to n. ...
In mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer greater than 1 is either a prime number or can be written as a product of prime numbers. ...
The birthday paradox states that if there are 23 or more people in a room then there is a chance of more than 50% that at least two of them will have the same birthday. ...
In mathematics, Stirling numbers arise in a variety of combinatorics problems. ...
In set theory, Königs theorem (named after the Hungarian mathematician Julius König) colloquially states that if the axiom of choice holds and if I is a set and mi and ni are cardinal numbers for every i in I, and then The sum here is the cardinality...
Definition In mathematics, a polynomial sequence, i. ...
In mathematics, a difference operator maps a function f(x) to another function f(x + a) − f(x + b). ...
In mathematics, the Pochhammer symbol, introduced by Leo August Pochhammer, 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 To distinguish the two, the notations and are commonly used...
In mathematics, the series expansion of the number e can be used to prove that e is irrational. ...
In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. ...
In mathematics, the binomial series generalizes the purely algebraic binomial theorem; it is the series in which where is the Pochhammer symbol, and in particular because it is the product of no terms at all. ...
In mathematics, a multiset (sometimes also called a bag) differs from a set in that each member has a multiplicity, which is a natural number indicating (loosely speaking) how many times it is a member, or perhaps how many memberships it has in the 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 calculator is a device for performing calculations. ...
- 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.
Technical justification The definition of an empty product can be based on that of the empty sum: In mathematics, the empty sum, or nullary sum, is the result of adding no numbers. ...
The sum of two logarithms is equal to the logarithm of the product of their operands, i.e.: Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...
- logbn + logbm = logbnm
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. The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ...
Using this property as definition, and extending this to the empty product, the right-hand side of this equation evaluates to e0 for the empty set, because the empty sum is defined to be zero, and therefore the empty product must equal one. In mathematics, LHS is informal shorthand for the left-hand side of an equation. ...
In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...
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). In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ...
In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...
In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression whose limit cannot be evaluated by substituting the limits of the subexpressions. ...
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. In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ...
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). In mathematics, an analytic function is a function that is locally given by a convergent power series. ...
For purposes such as those of combinatorics, set theory, the binomial theorem, and power series, one should also take 00 = 1. Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. ...
In abstract mathematics, naive set theory1 was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...
In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...
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.) In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ...
Partial plot of a function f. ...
In the case where the domain of the function is the empty set {}, there is only one function with that domain (given any codomain), the empty function, and any formula can be used to define the empty function, since the formula wont apply to anything and will therefore never...
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. In probability theory and statistics, the Poisson distribution is a discrete probability distribution. ...
In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...
The consistent point of view incorporating these aspects is to define
 However many people still believe it is undefined
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. In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...
In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...
A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ...
In arithmetic, the empty product, or nullary product, is the result of multiplying no numbers. ...
Nullary Cartesian product Consider the general definition of the Cartesian product: In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y: The Cartesian product is named after René Descartes...
 If I is empty, the only satisfying f is the empty function: In the case where the domain of the function is the empty set {}, there is only one function with that domain (given any codomain), the empty function, and any formula can be used to define the empty function, since the formula wont apply to anything and will therefore never...
 Under the perhaps more familiar n-tuple interpretation, In mathematics, a tuple is a finite sequence of objects, that is, a list of a limited number of objects. ...
 that is, the singleton set containing the empty tuple. Note that in both representations the empty product has cardinality 1. In mathematics, a singleton is a set with exactly one element. ...
In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ...
In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ...
Nullary Cartesian product of functions The empty Cartesian product of functions is again the empty function. In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y: The Cartesian product is named after René Descartes...
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. In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ...
In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C...
In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ...
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. ...
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 category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in...
In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ...
In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
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: Infix notation is the common arithmetic and logical formula notation, in which operators are written infix-style between the operands they act on (e. ...
In computer programming, a variadic function is a function of variable arity; that is, one which can take different numbers of arguments. ...
An S-expression (S stands for symbolic) is a convention for representing data or an expression in a computer program in a text form. ...
Lisp is a family of computer programming languages with a long history and a distinctive fully-parenthesized syntax. ...
In mathematics, the arity of a function or an operator is the number of arguments or operands it takes, A function or operator can thus be described as unary, binary, ternary,etc. ...
(* 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. In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...
Concrete Mathematics by Ronald L. Graham, Donald E. Knuth and Oren Patashnik is a textbook that provides its readers with mathematical background that can be especially useful in computer science. ...
Ronald L. Graham (born October 31, 1935) is a mathematician credited by the American Mathematical Society with being one of the principle architects of the rapid development worldwide of discrete mathematics in recent years[1]. He has done important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness. ...
Donald Knuth at a reception for the Open Content Alliance. ...
Oren Patashnik (born 1954) is a computer scientist. ...
"I belive 0 to the power 0 is undefined -Chris Chiang from NE
See also In mathematics, an iterated binary operation is an extension of a binary operation on a set S to a function on finite sequences of elements of S through repeated application. ...
External links This article incorporates material from Empty product on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative, online mathematics encyclopedia. ...
PlanetMath is a free, collaborative, online mathematics encyclopedia. ...
|
Feeds |
Contact