nLab
integer

Context

Algebra

Arithmetic

Contents

Idea

An integer is a number that is a natural number or the negative? of one.

The ring \mathbb{Z} of all integers may defined as the free group on one generator or as the initial ring.

In keeping with a historical point of view in which integers are whole numbers with a sign attached, one may write

={0}{n,n|n,n>0}={,3,2,1,0,1,2,3,}. \mathbb{Z} = \{0\} \cup \{n, -n | n \in \mathbb{N}, n \gt 0\} = \{ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \} \,.

From an nPOV, one may consider this as follows: \mathbb{Z} is a filtered colimit of sets

1+()1+()1+()\mathbb{N} \stackrel{1 + (-)}{\to} \mathbb{N} \stackrel{1 + (-)}{\to} \mathbb{N} \stackrel{1 + (-)}{\to} \ldots

whereby n-n \in \mathbb{Z} is represented by the element 00 in the n thn^{th} copy of \mathbb{N} appearing in this diagram (starting the count at the 0 th0^{th} copy). The resulting induced map to the colimit

× m:(m,n)nm\mathbb{N} \times \mathbb{N} \cong \sum_{m \in \mathbb{N}} \mathbb{N} \to \mathbb{Z}: (m, n) \mapsto n-m

imparts a monoid (in fact a group) structure on \mathbb{Z} descended from the monoid structure on ×\mathbb{N} \times \mathbb{N}; compare double-entry bookkeeping in medieval mathematics (partita doppia).

As a group, \mathbb{Z} is abelian and is the Grothendieck group of the monoid (or semigroup) \mathbb{N} of natural numbers.

The monoid of natural numbers is naturally even a rig – in fact the initial rig – and this multiplicative structure extends to \mathbb{Z} to make it a ring – in fact the initial ring.

Terminology

The underlying sets \mathbb{Z} and \mathbb{N} are isomorphic. Some subcultures of mathematics (and not only set theorists) use the term ‘integer’ synonymously for a natural number. Computer scientists distinguish between ‘unsigned integers’ (natural numbers) and ‘signed integers’ (integers as described here). Translations can also cause confusion with the term ‘whole number’.

In number theory, one generalises integers to algebraic integers, an instance of the red herring principle. Accordingly, some number theorists will call the integers ‘rational integers’ to clarify; \mathbb{Z} is the ring of integers in the number field \mathbb{Q} of rational numbers. (Compare, for example, Gaussian integers and Gaussian numbers.)

The symbol ‘\mathbb{Z}’ derives from the German word ‘Zahlen’, which is a generic word for ‘numbers’. (Compare Dedekind's use of that word in the title of his famous book on the foundations of real numbers.)

References

A formalization in terms of homotopy type theory, using a unary notation, is in

(A different common formalization of integers in type theory is in a binary notation, as in the Coq standard library. Binary notation is exponentially more efficient for performing computations, but the unary notation was convenient for calculating π 1(S 1)\pi_1(S^1).)

Revised on May 24, 2017 10:26:45 by Urs Schreiber (131.220.184.222)