---
tag: 0C32J27WQC
kind: statement
role: theorem
name: Associativity of addition
msc: [03F30]
uses: [0W5DSDFQ4G, 0ASJVEFA9J]
sources:
  - url: https://en.wikipedia.org/wiki/Associative_property
    locator: Definition
  - url: https://en.wikipedia.org/wiki/Addition
    locator: Associativity
---

## Statement

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

## Remarks

The proof from the recursive definition of addition uses [[induction|0ASJVEFA9J]] on $c$.
Because of this law, we can write $a + b + c$ without parentheses.
Multiplication of natural numbers is also associative.

## Explanations

### L1 human

When you add three numbers, you can add the first two first or the last two first.
For example, $(2 + 3) + 4 = 5 + 4 = 9$ and $2 + (3 + 4) = 2 + 7 = 9$.