empty product
The empty product of numbers is the borderline case of product, where the number of is empty. The most usual examples are the following.

•
The http://planetmath.org/Introducing0thPowerzeroth power of a nonzero number: ${a}^{0}$

•
The factorial of 0: 0!

•
The http://planetmath.org/FundamentalTheoremOfArithmeticsprime factor presentation^{} of unity, which has no prime factors
The value of the empty sum of numbers is equal to the additive identity number, 0. Similarly, the empty product of numbers is equal to the http://planetmath.org/Unitymultiplicative identity^{} number, 1.
Note. When considering the complex numbers as pairs of real numbers one often identifies the pairs $(x,\mathrm{\hspace{0.17em}0})$ and the reals $x$. In this sense one can think that the Cartesian product^{} $\mathbb{R}\times \{0\}$ is equal to $\mathbb{R}$. This seems to the equation
$$\mathbb{R}\times {\mathbb{R}}^{0}={\mathbb{R}}^{1+0}={\mathbb{R}}^{1}=\mathbb{R},$$ 
although the http://planetmath.org/GeneralAssociativityassociativity of Cartesian product is nowhere stated. Nevertheless, it is sometimes natural to define that the Cartesian product of an empty collection of sets equals to a set with one element; so it may that e.g. ${\mathbb{R}}^{0}=\{0\}.$
One can also consider empty products in categories^{}. It follows directly from the definition that an object in a category is a http://planetmath.org/CategoricalDirectProductproduct of an empty family of objects in the category if and only if it is a terminal object^{} of the category. Sets are a special case of this: in the category of sets the singletons are the terminal objects, so the empty product exists and is a singleton.
Title  empty product 
Canonical name  EmptyProduct 
Date of creation  20130322 14:48:13 
Last modified on  20130322 14:48:13 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  15 
Author  pahio (2872) 
Entry type  Definition 
Classification  msc 00A99 
Related topic  EmptySet 
Related topic  IndeterminateForm 
Related topic  LocallyEuclidean 
Related topic  AnalyticContinuationOfGammaFunction 
Related topic  Introducing0thPower 
Related topic  EmptySum 