Distributivity
Encyclopedia
In mathematics
, and in particular in abstract algebra
, distributivity is a property of binary operation
s that generalizes the distributive law from elementary algebra
.
For example:
In the left-hand side of the first equation, the 2 multiplies the sum of 1 and 3; on the right-hand side, it multiplies the 1 and the 3 individually, with the products added afterwards.
Because these give the same final answer (8), we say that multiplication by 2 distributes over addition of 1 and 3.
Since we could have put any real number
s in place of 2, 1, and 3 above, and still have obtained a true equation, we say that multiplication
of real numbers distributes over addition
of real numbers.
Notice that when · is commutative, then the three above conditions are logically equivalent
.
of 17.5%, in two separate transactions will actually save £0.01, over buying them together: £14.99×1.175 = £17.61 to the nearest £0.01, giving a total expenditure of £35.22, but £29.98×1.175 = £35.23. Methods such as banker's rounding may help in some cases, as may increasing the precision used, but ultimately some calculation errors are inevitable.
Janice Vian, Ph.D. (talk) 23:10, 16 November 2011 (UTC)Janice Vian, Ph.D. (talk) 23:17, 16 November 2011 (UTC)
s.
A ring has two binary operations (commonly called "+" and "*"), and one of the requirements of a ring is that * must distribute over +.
Most kinds of numbers (example 1) and matrices (example 3) form rings.
A lattice
is another kind of algebraic structure
with two binary operations, ∧ and ∨.
If either of these operations (say ∧) distributes over the other (∨), then ∨ must also distribute over ∧, and the lattice is called distributive. See also the article on distributivity (order theory)
.
Examples 4 and 5 are Boolean algebras, which can be interpreted either as a special kind of ring (a Boolean ring
) or a special kind of distributive lattice (a Boolean lattice). Each interpretation is responsible for different distributive laws in the Boolean algebra. Examples 6 and 7 are distributive lattices which are not Boolean algebras.
Rings and distributive lattices are both special kinds of rigs, certain generalizations of rings.
Those numbers in example 1 that don't form rings at least form rigs.
Near-rigs are a further generalization of rigs that are left-distributive but not right-distributive; example 2 is a near-rig.
one finds numerous important variants of distributivity, some of which include infinitary operations, such as the infinite distributive law; others being defined in the presence of only one binary operation, such as the according definitions and their relations are given in the article distributivity (order theory)
. This also includes the notion of a completely distributive lattice
.
In the presence of an ordering relation, one can also weaken the above equalities by replacing = by either ≤ or ≥. Naturally, this will lead to meaningful concepts only in some situations. An application of this principle is the notion of sub-distributivity as explained in the article on interval arithmetic
.
In category theory
, if (S, μ, η) and (S', μ', η') are monad
s on a category
C, a distributive law S.S' → S'.S is a natural transformation
λ : S.S' → S'.S such that (S' , λ) is a lax map of monads S → S and (S, λ) is a colax map of monads S' → S' . This is exactly the data needed to define a monad structure on S'.S: the multiplication map is S'μ.μ'S².S'λS and the unit map is η'S.η. See: distributive law between monads
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, and in particular in abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, distributivity is a property of binary operation
Binary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
s that generalizes the distributive law from elementary algebra
Elementary algebra
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. It is typically taught in secondary school under the term algebra. The major difference between algebra and...
.
For example:
- 2 × (1 + 3) = (2 × 1) + (2 × 3).
In the left-hand side of the first equation, the 2 multiplies the sum of 1 and 3; on the right-hand side, it multiplies the 1 and the 3 individually, with the products added afterwards.
Because these give the same final answer (8), we say that multiplication by 2 distributes over addition of 1 and 3.
Since we could have put any real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s in place of 2, 1, and 3 above, and still have obtained a true equation, we say that multiplication
Multiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
of real numbers distributes over addition
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
of real numbers.
Definition
Given a set S and two binary operators · and + on S, we say that the operation ·- is left-distributive over + if, given any elements x, y, and z of S,
-
- x · (y + z) = (x · y) + (x · z);
- is right-distributive over + if, given any elements x, y, and z of S:
- (y + z) · x = (y · x) + (z · x);
- is distributive over + if it is left- and right-distributive.
- x · (y + z) = (x · y) + (x · z);
Notice that when · is commutative, then the three above conditions are logically equivalent
Logical equivalence
In logic, statements p and q are logically equivalent if they have the same logical content.Syntactically, p and q are equivalent if each can be proved from the other...
.
Examples
- Multiplication of numberNumberA number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s is distributive over addition of numbers, for a broad class of different kinds of numbers ranging from natural numberNatural numberIn mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s to complex numberComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s and cardinal numberCardinal numberIn mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
s. - Multiplication of ordinal numberOrdinal numberIn set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
s, in contrast, is only left-distributive, not right-distributive. - The cross productCross productIn mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
is left- and right-distributive over vector addition, though not commutative. - MatrixMatrix (mathematics)In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
multiplicationMatrix multiplicationIn mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
is distributive over matrix additionMatrix additionIn mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations which could also be considered as a kind of addition for matrices, the direct sum and the Kronecker sum....
, though also not commutative. - The unionUnion (set theory)In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
of sets is distributive over intersectionIntersection (set theory)In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
, and intersection is distributive over union. - Logical disjunctionLogical disjunctionIn logic and mathematics, a two-place logical connective or, is a logical disjunction, also known as inclusive disjunction or alternation, that results in true whenever one or more of its operands are true. E.g. in this context, "A or B" is true if A is true, or if B is true, or if both A and B are...
("or") is distributive over logical conjunctionLogical conjunctionIn logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....
("and"), and conjunction is distributive over disjunction. - For real numberReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s (or for any totally ordered set), the maximum operation is distributive over the minimum operation, and vice versa: max(a,min(b,c)) = min(max(a,b),max(a,c)) and min(a,max(b,c)) = max(min(a,b),min(a,c)). - For integerIntegerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s, the greatest common divisorGreatest common divisorIn mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
is distributive over the least common multipleLeast common multipleIn arithmetic and number theory, the least common multiple of two integers a and b, usually denoted by LCM, is the smallest positive integer that is a multiple of both a and b...
, and vice versa: gcd(a,lcm(b,c)) = lcm(gcd(a,b),gcd(a,c)) and lcm(a,gcd(b,c)) = gcd(lcm(a,b),LCM(a,c)). - For real numbers, addition distributes over the maximum operation, and also over the minimum operation: a + max(b,c) = max(a+b,a+c) and a + min(b,c) = min(a+b,a+c).
Distributivity and rounding
In practice, the distributive property of multiplication (and division) over addition may appear to be compromized or lost because of the limitations of arithmetic precision. For example, the identity ⅓ + ⅓ + ⅓ = (1+1+1)/3 appears to fail if the addition is conducted in decimal arithmetic; however if many significant digits are used, the calculation will result in a closer approximation to the correct results. For example, if the arithmetical calculation takes the form: 0.33333+0.33333+0.33333 = 0.99999 ≠ 1, this result is a closer approximation than if fewer significant digits had been used. Even when fractional numbers can be represented exactly in arithmetical form, errors will be introduced if those arithmetical values are rounded or truncated. For example, buying two books, each priced at £14.99 before a taxVat
Vat or VAT may refer to:* A type of container such as a barrel, storage tank, or tub, often constructed of welded sheet stainless steel, and used for holding, storing, and processing liquids such as milk, wine, and beer...
of 17.5%, in two separate transactions will actually save £0.01, over buying them together: £14.99×1.175 = £17.61 to the nearest £0.01, giving a total expenditure of £35.22, but £29.98×1.175 = £35.23. Methods such as banker's rounding may help in some cases, as may increasing the precision used, but ultimately some calculation errors are inevitable.
Janice Vian, Ph.D. (talk) 23:10, 16 November 2011 (UTC)Janice Vian, Ph.D. (talk) 23:17, 16 November 2011 (UTC)
Distributivity in rings
Distributivity is most commonly found in rings and distributive latticeDistributive lattice
In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection...
s.
A ring has two binary operations (commonly called "+" and "*"), and one of the requirements of a ring is that * must distribute over +.
Most kinds of numbers (example 1) and matrices (example 3) form rings.
A lattice
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
is another kind of algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
with two binary operations, ∧ and ∨.
If either of these operations (say ∧) distributes over the other (∨), then ∨ must also distribute over ∧, and the lattice is called distributive. See also the article on distributivity (order theory)
Distributivity (order theory)
In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima...
.
Examples 4 and 5 are Boolean algebras, which can be interpreted either as a special kind of ring (a Boolean ring
Boolean ring
In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R; that is, R consists only of idempotent elements....
) or a special kind of distributive lattice (a Boolean lattice). Each interpretation is responsible for different distributive laws in the Boolean algebra. Examples 6 and 7 are distributive lattices which are not Boolean algebras.
Rings and distributive lattices are both special kinds of rigs, certain generalizations of rings.
Those numbers in example 1 that don't form rings at least form rigs.
Near-rigs are a further generalization of rigs that are left-distributive but not right-distributive; example 2 is a near-rig.
Generalizations of distributivity
In several mathematical areas, generalized distributivity laws are considered. This may involve the weakening of the above conditions or the extension to infinitary operations. Especially in order theoryOrder theory
Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
one finds numerous important variants of distributivity, some of which include infinitary operations, such as the infinite distributive law; others being defined in the presence of only one binary operation, such as the according definitions and their relations are given in the article distributivity (order theory)
Distributivity (order theory)
In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima...
. This also includes the notion of a completely distributive lattice
Completely distributive lattice
In the mathematical area of order theory, a completely distributive lattice is a complete lattice in which arbitrary joins distribute over arbitrary meets....
.
In the presence of an ordering relation, one can also weaken the above equalities by replacing = by either ≤ or ≥. Naturally, this will lead to meaningful concepts only in some situations. An application of this principle is the notion of sub-distributivity as explained in the article on interval arithmetic
Interval arithmetic
Interval arithmetic, interval mathematics, interval analysis, or interval computation, is a method developed by mathematicians since the 1950s and 1960s as an approach to putting bounds on rounding errors and measurement errors in mathematical computation and thus developing numerical methods that...
.
In category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, if (S, μ, η) and (S', μ', η') are monad
Monad (category theory)
In category theory, a branch of mathematics, a monad, Kleisli triple, or triple is an functor, together with two natural transformations...
s on a category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
C, a distributive law S.S' → S'.S is a natural transformation
Natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
λ : S.S' → S'.S such that (S' , λ) is a lax map of monads S → S and (S, λ) is a colax map of monads S' → S' . This is exactly the data needed to define a monad structure on S'.S: the multiplication map is S'μ.μ'S².S'λS and the unit map is η'S.η. See: distributive law between monads
Distributive law between monads
In category theory, an abstract branch of mathematics, distributive laws between monads are a way to express abstractly that two algebraic structures distribute one over the other one....
.
External links
- A demonstration of the Distributive Law for integer arithmetic (from cut-the-knotCut-the-knotCut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
)