---
tag: 0DFZNHJ0S2
kind: statement
role: theorem
name: Commutativity of addition
msc: [03F30]
uses: [0W5DSDFQ4G, 0ASJVEFA9J]
sources:
  - url: https://en.wikipedia.org/wiki/Commutative_property
    locator: Definition
  - url: https://en.wikipedia.org/wiki/Addition
    locator: Commutativity
---

## Statement

For all natural numbers $a$ and $b$, $a + b = b + a$.
Here $+$ is the [[addition of natural numbers|0W5DSDFQ4G]].

## Remarks

The proof from the recursive definition of addition uses the [[principle of mathematical induction|0ASJVEFA9J]] 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.