Last updates: 10 July 2011
The positive integers is the set with operation given by concatenation so that, for example,
The positive integers are often written as
The nonnegative integers is the set with addition given by the addition in and
The integers is the set with addition given by the addition in and if and the conditions
Let . Then there exists a unique function such that
The multiplication operation on is
Define a relation on by .
The set with the operations of addition and multiplication and the total order is an ordered ring.
In many curricula the positive integers is one of the first number systems that is taught (usually at the age of 1 or 2). The positive integers is the free monoid without identity on one generator. By the age of 3 it is not unusual to be familiar with the number system of nonnegative integers. The nonnegative integers is the free monoid with identity on one generator. Most primary school curricula include treatment of the number system of integers. The integers is the free group on one generator.