# Explanation: Comparison Between the Number Systems

(related to Branch: Number Systems and Arithmetics)

It is instructive to compare the algebraic, ordering and topologic properties of the different number systems. The following table summarizes the main results of this part of BookofProofs with respect to these questions:

Number system Symbol Examples Algebraic Properties Order Properties Topological Properties
Natural numbers $\mathbb N$ $0,1,2,\ldots$ semi-ring strict total and Archimedean
Integers $\mathbb Z$ $\ldots,-2,-1,0,1,2,\ldots$ integral domain strict total and Archimedean
Rational Numbers $\mathbb Z$ $\ldots,-\frac 12,-\frac 11,0,1,\frac 32,\frac 52, \ldots$ field, not every polynomial in $\mathbb Q[X]$ can be factored into linear factors strict total and Archimedean
Real Numbers $\mathbb R$ all $\mathbb Q$, but also $\ldots,-\sqrt{5},\sqrt{2},\pi,e, \ldots$ field, not every polynomial in $\mathbb R[X]$ can be factored into linear factors strict total and Archimedean
Complex Numbers $\mathbb C$ all numbers of the form $x+iy$, with $x,y\in\mathbb R$ and $i$ being the imaginary unit field, every polynomial in $\mathbb C[X]$ can be factored into linear factors no ordering possible uncountable, metric space, complete (all Cauchy sequences do converge).

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### References

#### Bibliography

1. Reinhardt F., Soeder H.: "dtv-Atlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition