Editing Distributive property (section)

=== 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}}