Distributive Laws/Arithmetic

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Theorem

On all the number systems:

natural numbers $\N$
integers $\Z$
rational numbers $\Q$
real numbers $\R$
complex numbers $\C$

the operation of multiplication is distributive over addition:

$m \paren {n + p} = m n + m p$
$\paren {m + n} p = m p + n p$


Proof

This is demonstrated in these pages:

Natural Number Multiplication Distributes over Addition
Integer Multiplication Distributes over Addition
Rational Multiplication Distributes over Addition
Real Multiplication Distributes over Addition
Complex Multiplication Distributes over Addition

$\blacksquare$


Examples

Arbitrary Example

\(\ds 2 \times \paren {3 + 6}\) \(=\) \(\ds 2 \times 9\)
\(\ds \) \(=\) \(\ds 18\)
\(\ds \) \(=\) \(\ds 6 + 12\)
\(\ds \) \(=\) \(\ds \paren {2 \times 3} + \paren {2 \times 6}\)


Also known as

The Distributive Laws of Arithmetic are collectively also known as the Distributive Property.

As such, it typically refers to the various results contributing towards this.


At elementary-school level, this law is often referred to as (the principle of) multiplying out brackets.


Also see



Beware:


Sources

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