In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. For example: Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... 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. ...

4 • (2 + 3) = (4 • 2) + (4 • 3)

In the left-hand side of the above equation, the 4 multiplies the sum of 2 and 3; on the right-hand side, it multiplies the 2 and the 3 individually, with the results added afterwards. Because these give the same final answer (20), we say that multiplication by 4 distributes over addition of 2 and 3. Since we could have put any real numbers in place of 4, 2, and 3 above, and still have obtained a true equation, we say that multiplication of real numbers distributes over addition of real numbers In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, multiplication is an elementary arithmetic operation. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. ...

Definition

Given a set S and two binary operations • and + on S, we say that In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...

• • 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 both left- and right-distributive.

Notice that when • is commutative, then the three above conditions are logically equivalent. In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In logic, statements p and q are logically equivalent if they have the same logical content. ...

Examples

1. Multiplication of numbers is distributive over addition of numbers, for a broad class of different kinds of numbers ranging from natural numbers to complex numbers and cardinal numbers.
2. Multiplication of ordinal numbers, in contrast, is only left-distributive, not right-distributive.
3. Matrix multiplication is distributive over matrix addition, even though it's not commutative.
4. The union of sets is distributive over intersection, and intersection is distributive over union. Also, intersection is distributive over the symmetric difference.
5. Logical disjunction ("or") is distributive over logical conjunction ("and"), and conjunction is distributive over disjunction. Also, conjunction is distributive over exclusive disjunction ("xor").
6. For real numbers (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)).
7. For integers, the greatest common divisor is distributive over the least common multiple, 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)).
8. 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).

A number is an abstract idea used in counting and measuring. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ... Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... This article gives an overview of the various ways to perform matrix multiplication. ... The operations on matrices differ from similar operations of scalar algebra in several respects. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... 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, the symmetric difference of two sets is the set of elements which are in one of either set, but not in both. ... OR logic gate. ... AND Logic Gate In logic and mathematics, logical conjunction (usual symbol and) is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false. ... Exclusive disjunction, also known as exclusive or and symbolized by XOR or EOR, is a logical operation on two operands that results in a logical value of true if and only if one of the operands, but not both, has a value of true. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... The integers are commonly denoted by the above symbol. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ... In arithmetic and number theory the least common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b. ...

Distributivity and rounding

In practice, the distributive property of multiplication (and division) over addition is lost around the limits of arithmetic precision. For example, the identity ⅓+⅓+⅓ = (1+1+1)/3 fails if conducted in decimal arithmetic; however many significant digits are used, the calculation will take the form 0.33333+0.33333+0.33333 = 0.99999 ≠ 1. Even where fractional numbers are representable exactly, errors will be introduced if rounding too far; for example, buying two books each priced at £14.99 before VAT 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. The precision of a value describes the number of digits that are used to express that value. ... Algorism comprises all of the rules of performing arithmetic computations using a decimal system for representing numbers in which numbers written using ten symbols having the values 0 through 9 are combined using a place-value system (positional notation), where each symbol has ten times the weight of the one... Significant figures (also called significant digits and abbreviated sig figs or sig digs, respectively) is a method of expressing errors in measurements. ... vat can be a type of barrel used for storage. ... Rounding is the process of reducing the number of significant digits in a number. ...

Distributivity in rings

Distributivity is most commonly found in rings and distributive lattices. In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. ...

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 v. If either of these operations (say ^) distributes over the other (v), then v must also distribute over ^, and the lattice is called distributive. See also the article on distributivity (order theory). The name lattice is suggested by the form of the Hasse diagram depicting it. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ... 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) 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. In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ... In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R; that is, R consists of idempotent elements. ... In mathematics and computer science, Boolean algebras, or Boolean lattices, are algebraic structures which capture the essence of the logical operations AND, OR and NOT as well as the corresponding set theoretic operations intersection, union and complement. ...

Rings and distributive lattices are both special kinds of rigs, certain generalisations of rings. Those numbers in example 1 that don't form rings at least form rigs. Near-rigs are a further generalisation of rigs that are left-distributive but not right-distributive; example 2 is a near-rig. In abstract algebra, a semiring is an algebraic structure, similar to a ring, but without additive inverses. ...

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 theory 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 implication operator of Heyting algebras. Details of the according definitions and their relations are given in the article distributivity (order theory). This also includes the notion of a completely distributive lattice. Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... 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. ... In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. ... 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. ... 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 intervals. In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

In category theory, if (S, μ, η) and (S', μ', η') are monads 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. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In category theory, a monad or triple is a type of functor, together with two associated natural transformations. ... Look up category in Wiktionary, the free dictionary. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... 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

Look up distributivity in Wiktionary, the free dictionary.

• Proof for children of distributive property with integer as multiplier, with an animation
• Solver showing an animation for user supplied examples of distributive property
• A demonstration of the Distributive Law for integer arithmetic (from cut-the-knot)