# Supernatural number

## Contents

## Definition

A **supernatural number** (also called a **surnatural number**, **generalized natural number**, or **Steinitz number**) is defined as a formal product of the form:

where runs over all the prime numbers, and each is either zero, or a natural number, or .

If none of the s is and all but finitely many of them are zero, then the supernatural number can be identified with the natural number obtained by viewing the formal product as an actual product in the integers. The unique factorization of natural numbers tells us that every natural number has a unique representation as a supernatural number where none of the s is and all but finitely many of them are zero.

## Operations

### For two supernatural numbers

Suppose we have two supernatural numbers:

We can define the following operations:

Operation | Output |
---|---|

product of supernatural numbers | . Here the denotes usual addition if both and are finite. If either is , the sum is . |

lcm of supernatural numbers | . Here, the denotes usual maximum if both and are finite. If either is , the maximum is . |

gcd of supernatural numbers | . Here, the denotes usual minimum if both and are finite. If either is , the minimum is the other number. If both are , the minimum is . |

When both inputs are *natural* numbers, the definitions of these operations agree with the usual definition for natural numbers.

### For many supernatural numbers

The definitions are similar to those for two supernatural numbers. Note that unlike the case of the natural numbers, we *can* take the product, lcm, and gcd of *infinitely* many supernatural numbers and still obtain a supernatural number. In particular, if we start with infinitely many *natural* numbers and perform one or more of these operations, we may end up with a supernatural number that is not a natural number.