(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). |