Distributive law

Statement

For all natural numbers $a$, $b$ and $c$, $a \cdot (b + c) = a \cdot b + a \cdot c$. Here $+$ is the addition and $\cdot$ is the multiplication of natural numbers.

Remarks

This is the left distributive law. With commutativity of multiplication, it gives the right law $(b + c) \cdot a = b \cdot a + c \cdot a$. The proof from the recursive definitions uses induction on $c$. The same law holds in the integers, the rationals, the reals and the complex numbers.

Explanations

L1 human

To multiply a number by a sum, you can multiply it by each part and then add. For example, $3 \cdot (10 + 2) = 30 + 6 = 36$, and this is how many people compute $3 \cdot 12$.