The integers

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 10 July 2011

The integers

The positive integers is the set $${\mathbb{Z}}_{>0}=\{1,1+1,1+1+1,1+1+1+1,\dots \}$$ with operation given by concatenation so that, for example, $$(1+1+1)+(1+1+1+1)=1+1+1+1+1+1+1.$$

The positive integers are often written as $${\mathbb{Z}}_{>0}=\{1,2,3,\dots \}.$$

The nonnegative integers is the set $${\mathbb{Z}}_{\ge 0}=\{=\{0,1,2,3,\dots \}$$ with addition given by the addition in ${\mathbb{Z}}_{>0}$ and $$\text{if}x\in {\mathbb{Z}}_{\ge 0}\text{then}0+x=x\text{and}\text{and}x+0=x.$$

The integers is the set $$\mathbb{Z}=\{\dots ,-3,-2,-1,0,1,23,\dots \}$$ with addition given by the addition in ${\mathbb{Z}}_{\ge 0}$ and if $$x\in $ {\mathbb{Z}}_{>0}$\text{then}(-x)+x=0\text{and}x+(-x)=0$$ and the conditions $$\text{if}x,y,z\in \mathbb{Z}\text{then}(x+y)+z=x+(y+z)\text{,}\text{and}$$ $$\text{if}x,y\in \mathbb{Z}\text{then}x+y=y+x.$$

Let $k\in \mathbb{Z}$. Then there exists a unique function $m_k:\mathbb{Z}\to \mathbb{Z}$ such that

(a) if $x,y\in \mathbb{Z}$ then $m_k(x+y)=m_k\left(x\right)+m_k\left(y\right)$,   and

(b) $m_k\left(1\right)=k$.

The multiplication operation on $\mathbb{Z}$ is $$\begin{array}{rrcl}& \mathbb{Z}\times \mathbb{Z}& ⟶& \mathbb{Z}\\ & (k,l)& ⟼& m_k\left(l\right).\end{array}$$

Define a relation $\le$ on $\mathbb{Z}$ by $$x\le y\text{if there exists}n\in {\mathbb{Z}}_{\ge 0}\text{such that}x+n=y$$.

The set $\mathbb{Z}$ with the operations of addition and multiplication and the total order $\le$ is an ordered ring.

Notes and References

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.

References

[BouEns] N. Bourbaki, Theorie des Ensembles, Chapter III Sections 4 and 5, Masson, Paris, 1990. MR?????

[BouAlg] N. Bourbaki, Algebra I, Chapter I, Section 2 Nos. 5-8, Springer-Verlag, Berlin 1989. MR?????

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