The distributive law

Division

The left distributive law for division over addition does not hold.

For example:

12 ÷ (3 + 4) ≠ 12 ÷ 3 + 12 ÷ 4

The LHS = \(\dfrac{12}{7}\) and the RHS = 7.

The right distributive law for division over subtraction does hold.

\((a + b) ÷ c = a ÷ c + b ÷ c\)

For example:

\((12 + 9) ÷ 3 = 12 ÷ 3 + 9 ÷ 3\)

We are used to seeing these using the following notation for division:

\(\dfrac{12}{3+4}\neq\dfrac{12}{3}+\dfrac{12}{4}=7\)  and  \(\dfrac{12+9}{3}=\dfrac{12}{3}+\dfrac{9}{3}=7\)