Editing Distributive property (section)

== Meaning ==

The operators used for examples in this section are those of the usual [[addition]] <math>\,+\,</math> and [[multiplication]] <math>\,\cdot.\,</math>

If the operation denoted <math>\cdot</math> is not commutative, there is a distinction between left-distributivity and right-distributivity:

<math display="block">a \cdot \left( b \pm c \right) = a \cdot b \pm a \cdot c \qquad \text{ (left-distributive) }</math>
<math display="block">(a \pm b) \cdot c = a \cdot c \pm b \cdot c \qquad \text{ (right-distributive) }.</math>

In either case, the distributive property can be described in words as:

To multiply a [[Summation|sum]] (or [[Difference (mathematics)|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 {{em|distributivity}}.

One example of an operation that is "only" right-distributive is division, which is not commutative:
<math display="block">(a \pm b) \div c = a \div c \pm b \div c.</math> 
In this case, left-distributivity does not apply:
<math display="block">a \div(b \pm c) \neq a \div b \pm a \div c</math>

The distributive laws are among the axioms for [[Ring (mathematics)|rings]] (like the ring of [[integer]]s) and [[Field (mathematics)|fields]] (like the field of [[rational number]]s). Here multiplication is distributive over addition, but addition is not distributive over multiplication. Examples of structures with two operations that are each distributive over the other are [[Boolean algebras]] such as the [[algebra of sets]] or the [[switching algebra]].

Multiplying sums can be put into words as follows: When a sum is multiplied by a sum, multiply each summand of a sum with each summand of the other sum (keeping track of signs) then add up all of the resulting products.