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 non-zero number:
The factorial of 0: 0!
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 and the reals . In this sense one can think that the Cartesian product is equal to . This seems to the equation
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.
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.
|Date of creation||2013-03-22 14:48:13|
|Last modified on||2013-03-22 14:48:13|
|Last modified by||pahio (2872)|