Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Empty product
Summary
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question), just as the empty sum—the result of adding no numbers—is by convention zero, or the additive identity. When numbers are implied, the empty product becomes one.
The term empty product is most often used in the above sense when discussing arithmetic operations. However, the term is sometimes employed when discussing set-theoretic intersections, categorical products, and products in computer programming.
Let a1, a2, a3, ... be a sequence of numbers, and let
be the product of the first m elements of the sequence. Then
for all m = 1, 2, ... provided that we use the convention . In other words, a "product" with no factors at all evaluates to 1.
Allowing a "product" with zero factors reduces the number of cases to be considered in many mathematical formulas. Such a "product" is a natural starting point in induction proofs, as well as in algorithms. For these reasons, the "empty product is one" convention is common practice in mathematics and computer programming.
The notion of an empty product is useful for the same reason that the number zero and the empty set are useful: while they seem to represent quite uninteresting notions, their existence allows for a much shorter mathematical presentation of many subjects.
For example, the empty products 0! = 1 (the factorial of zero) and x0 = 1 shorten Taylor series notation (see zero to the power of zero for a discussion of when x = 0). Likewise, if M is an n × n matrix, then M0 is the n × n identity matrix, reflecting the fact that applying a linear map zero times has the same effect as applying the identity map.
As another example, the fundamental theorem of arithmetic says that every positive integer greater than 1 can be written uniquely as a product of primes. However, if we do not allow products with only 0 or 1 factors, then the theorem (and its proof) become longer.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related courses (4)
CS-101: Advanced information, computation, communication I
Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics a
MATH-113: Algebraic structures
Le but de ce cours est d'introduire et d'étudier les notions de base de l'algèbre abstraite.
CH-335: Asymmetric synthesis and retrosynthesis
La première partie du cours décrit les méthodes classiques de synthèse asymétrique. La seconde partie du cours traite des stratégies de rétrosynthèse basées sur l'approche par disconnection.
Related concepts (16)
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B. In terms of set-builder notation, that is A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value).
Falling and rising factorials
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, rising sequential product, or upper factorial) is defined as The value of each is taken to be 1 (an empty product) when These symbols are collectively called factorial powers. The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation (x)_n , where n is a non-negative integer.
Product type
In programming languages and type theory, a product of types is another, compounded, type in a structure. The "operands" of the product are types, and the structure of a product type is determined by the fixed order of the operands in the product. An instance of a product type retains the fixed order, but otherwise may contain all possible instances of its primitive data types. The expression of an instance of a product type will be a tuple, and is called a "tuple type" of expression.