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Concept
Distributive property
Summary
In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality
is always true in elementary algebra.
For example, in elementary arithmetic, one has
Therefore, one would say that multiplication distributes over addition.
This basic property of numbers is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers, polynomials, matrices, rings, and fields. It is also encountered in Boolean algebra and mathematical logic, where each of the logical and (denoted ) and the logical or (denoted ) distributes over the other.
Given a set and two binary operators and on
the operation is over (or with respect to) if, given any elements of
the operation is over if, given any elements of
and the operation is over if it is left- and right-distributive.
When is commutative, the three conditions above are logically equivalent.
The operators used for examples in this section are those of the usual addition and multiplication
If the operation denoted is not commutative, there is a distinction between left-distributivity and right-distributivity:
In either case, the distributive property can be described in words as:
To multiply a sum (or difference) by a factor, each summand (or minuend and subtrahend) is multiplied by this factor and the resulting products are added (or subtracted).
If the operation outside the parentheses (in this case, the multiplication) is commutative, then left-distributivity implies right-distributivity and vice versa, and one talks simply of .
One example of an operation that is "only" right-distributive is division, which is not commutative:
In this case, left-distributivity does not apply:
The distributive laws are among the axioms for rings (like the ring of integers) and fields (like the field of rational numbers). Here multiplication is distributive over addition, but addition is not distributive over multiplication.
Official source
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