---
tag: R68D2P175Z
kind: example
name: The natural numbers under addition are not a group
msc: [20A05]
not_instance_of:
  - target: G5S1M9
    note: the element 1 has no inverse
uses: [0MWKJ44V5A, 0BRHKKP333]
sources:
  - url: https://en.wikipedia.org/wiki/Natural_number
    locator: Addition
---

## Example

Let $\mathbb{N} = \{0, 1, 2, \dots\}$ be the [[natural numbers|0MWKJ44V5A]] with addition.
Addition is associative, and $0$ is an identity element.
The element $1$ has no [[inverse|0BRHKKP333]], because $1 + m \ge 1$ for each $m \in \mathbb{N}$, so $1 + m \ne 0$.
Thus $(\mathbb{N}, +)$ is not a group. It is a commutative monoid.

## Remarks

If we add the negative integers, we get the group $(\mathbb{Z}, +)$.