Commutativity of addition

Statement

For all natural numbers $a$ and $b$, $a + b = b + a$. Here $+$ is the addition of natural numbers.

Remarks

The proof from the recursive definition of addition uses the principle of mathematical induction two times. An operation $\ast$ on a set is commutative when $x \ast y = y \ast x$ for all $x$ and $y$. Multiplication of natural numbers is also commutative. Subtraction of integers is not commutative: $3 - 1 \neq 1 - 3$.

Explanations

L1 human

The order of two numbers in a sum does not change the result: $2 + 5$ and $5 + 2$ are both $7$. Think of two piles of stones: you count the same total when you count the second pile first.