---
tag: KAEDCQ8RM6
kind: concept
name: Ordinal arithmetic
msc: ["03E10"]
sources:
  - url: https://en.wikipedia.org/wiki/Ordinal_arithmetic
same_as:
  wikidata: Q1142215
  wikipedia: Ordinal arithmetic
---

## Summary

Ordinal arithmetic is the addition, multiplication and exponentiation of [[ordinal numbers|0YSCSX5YZC]].
The operations extend those of the natural numbers to transfinite ordinals, and they are [[defined|2C7MBG93AZ]] by transfinite recursion or by order types of well-ordered sets.
They are not commutative, and they are the language of ordinal notations, such as the [[Cantor normal form|6GD30FGYDZ]].

## Explanations

### L1 human

Think of an ordinal as a queue of positions.
Adding $\alpha + \beta$ means placing the queue $\beta$ after the queue $\alpha$.
Multiplying $\alpha \cdot \beta$ means taking $\beta$ copies of $\alpha$, one after the other.
Order matters: $1 + \omega$ puts one extra point before an infinite queue, which gives again a queue of type $\omega$.
But $\omega + 1$ puts a point after all the natural numbers, which is a longer queue.
So $1 + \omega = \omega$ while $\omega + 1 > \omega$.