Editing Distributive property (section)

== Examples ==

=== Real numbers ===

In the following examples, the use of the distributive law on the set of real numbers <math>\R</math> is illustrated. When multiplication is mentioned in elementary mathematics, it usually refers to this kind of multiplication. From the point of view of algebra, the real numbers form a [[field (mathematics)|field]], which ensures the validity of the distributive law.

{{glossary}}
{{term|First example (mental and written multiplication)}}{{defn|During mental arithmetic, distributivity is often used unconsciously:
<math display="block">6 \cdot 16 = 6 \cdot (10 + 6) = 6\cdot 10 + 6 \cdot 6 = 60 + 36 = 96</math>

Thus, to calculate <math>6 \cdot 16</math> in one's head, one first multiplies <math>6 \cdot 10</math> and <math>6 \cdot 6</math> and add the intermediate results. Written multiplication is also based on the distributive law.
}}
{{term|Second example (with variables)}}{{defn|
<math display="block">3 a^2 b \cdot (4 a - 5 b) = 3 a^2 b \cdot 4a - 3 a^2 b \cdot 5 b = 12 a^3 b - 15 a^2 b^2</math>
}}
{{term|Third example (with two sums)}}{{defn|
<math display="block">\begin{align}
(a + b) \cdot (a - b) & = a \cdot (a - b) + b \cdot (a - b) = a^2 - ab + ba - b^2 = a^2 - b^2 \\
                      & = (a + b) \cdot a - (a + b) \cdot b = a^2 + ba - ab - b^2 = a^2 - b^2 \\
\end{align}</math>

Here the distributive law was applied twice, and it does not matter which bracket is first multiplied out.
}}
{{term|Fourth example}}{{defn|Here the distributive law is applied the other way around compared to the previous examples. Consider
<math display="block">12 a^3 b^2 - 30 a^4 b c + 18 a^2 b^3 c^2 \,.</math>

Since the factor <math>6 a^2 b</math> occurs in all summands, it can be factored out. That is, due to the distributive law one obtains
<math display="block">12 a^3 b^2 - 30 a^4 b c + 18 a^2 b^3 c^2 = 6 a^2 b \left(2 a b - 5 a^2 c + 3 b^2 c^2\right).</math>
}}
{{glossary end}}

=== Matrices ===

The distributive law is valid for [[matrix multiplication]]. More precisely,
<math display="block">(A + B) \cdot C = A \cdot C + B \cdot C</math>
for all  <math>l \times m</math>-matrices <math>A, B</math> and <math>m \times n</math>-matrices <math>C,</math> as well as
<math display="block">A \cdot (B + C) = A \cdot B + A \cdot C</math>
for all <math>l \times m</math>-matrices <math>A</math> and <math>m \times n</math>-matrices <math>B, C.</math> 
Because the commutative property does not hold for matrix multiplication, the second law does not follow from the first law. In this case, they are two different laws.

=== Other examples ===

* [[Ordinal arithmetic#Multiplication|Multiplication]] of [[ordinal number]]s, in contrast, is only left-distributive, not right-distributive.
* The [[cross product]] is left- and right-distributive over [[vector addition]], though not commutative.
* The [[Union (set theory)|union]] of sets is distributive over [[Intersection (set theory)|intersection]], and intersection is distributive over union.
* [[Logical disjunction]] ("or") is distributive over [[logical conjunction]] ("and"), and vice versa.
* For [[real number]]s (and for any [[totally ordered set]]), the [[maximum]] operation is distributive over the [[minimum]] operation, and vice versa: <math display="block">\max(a, \min(b, c)) = \min(\max(a, b), \max(a, c)) \quad \text{ and } \quad \min(a, \max(b, c)) = \max(\min(a, b), \min(a, c)).</math>
* For [[integer]]s, the [[greatest common divisor]] is distributive over the [[least common multiple]], and vice versa: <math display="block">\gcd(a, \operatorname{lcm}(b, c)) = \operatorname{lcm}(\gcd(a, b), \gcd(a, c)) \quad \text{ and } \quad \operatorname{lcm}(a, \gcd(b, c)) = \gcd(\operatorname{lcm}(a, b), \operatorname{lcm}(a, c)).</math>
* For real numbers, addition distributes over the maximum operation, and also over the minimum operation: <math display="block">a + \max(b, c) = \max(a + b, a + c) \quad \text{ and } \quad a + \min(b, c) = \min(a + b, a + c).</math>
* For [[Binomial (polynomial)|binomial]] multiplication, distribution is sometimes referred to as the [[FOIL Method]]<ref>Kim Steward (2011) [http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut28_multpoly.htm Multiplying Polynomials] from Virtual Math Lab at [[West Texas A&M University]]</ref> (First terms <math>a c,</math> Outer <math>a d,</math> Inner <math>b c,</math> and Last <math>b d</math>) such as: <math>(a + b) \cdot (c + d) = a c + a d + b c + b d.</math>
* In all [[semirings]], including the [[complex number]]s, the [[quaternion]]s, [[polynomial]]s, and [[matrix (mathematics)|matrices]],  multiplication distributes over addition: <math>u (v + w) = u v + u w, (u + v)w = u w + v w.</math>
* In all [[Algebra over a field|algebras over a field]], including the [[octonion]]s and other [[non-associative algebra]]s, multiplication distributes over addition.