Addition of Numbers is not Distributive over Multiplication

Theorem

Addition of numbers is not distributive over multiplication.

That is, for numbers $a$, $b$ and $c$ it is not necessarily the case that $a + \paren {b \times c} = \paren {a + b} \times \paren {a + c}$.

Proof

Proof by Counterexample:

\(\ds 2 + \paren {3 \times 6}\)

\(=\)

\(\ds 2 + 18\)

\(\ds \)

\(=\)

\(\ds 20\)

\(\ds \paren {2 + 3} \times \paren {2 + 6}\)

\(=\)

\(\ds 6 + 8\)

\(\ds \)

\(=\)

\(\ds 48\)

\(\ds \)

\(\ne\)

\(\ds 2 + \paren {3 \times 6}\)

$\blacksquare$

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): distributive law
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): distributive
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): distributive