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Number Systems

In this part of BookofProofs, we will be studying the development of different number systems using the axiomatic method. It will turn out that natural numbers $\mathbb N$ play the main role in this development.

Beginning with the axiomatic foundation of natural numbers, we will try to solve certain equations and find out that it is necessary to introduce extended number systems, going beyond the natural numbers. These number systems will consecutively be the systems of integers $\mathbb Z$, of rational numbers $\mathbb Q$, of real numbers $\mathbb R$, of complex numbers $\mathbb C$, and of quaternions. Occasionally, it will be shown that it is possible to derive the same (i.e. isomorphic) number systems using different sets of axioms, but the focus will be set on only one axiomatic system, taken as an example.

Moreover, by studying this part, you will find the foundations of basic arithmetics. If you learned at elementary school facts like

  • it is not possible to divide by $0$,
  • each number has a negative corresponding number,
  • the product of two negative numbers is positive,
  • adding $0$ does not change a sum, etc.

and always asked yourself why? but never got a satisfying answer, then you are invited to dive into the stuff to follow and find the answers you have been looking for.

For the more experienced readers, it will be interesting 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 discrete, countable
Integers $\mathbb Z$ $\ldots,-2,-1,0,1,2,\ldots$ integral domain strict total and Archimedean distcrete, countable
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 discrete, dense, countable, metric space, incomplete (not all Cauchy sequences converge)
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 dense, uncountable, metric space, complete (all Cauchy sequences do converge)
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)
Algebraic Numbers $\mathbb A$ all solutions of polynomials over $\mathbb Q[X]$, $\mathbb R[X]$, or any other field field, every polynomial in $\mathbb A[X]$ can be factored into linear factors no ordering possible countability and completeness depending on the countability of the underlying field, metric space

| | | | Contributors: bookofproofs | References: [979]

1.Natural Numbers

2.Integers

3.Rational Numbers

4.Irrational Numbers

5.Real Numbers

6.Complex Numbers


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Bibliography (further reading)

[979] Reinhardt F., Soeder H.: “dtv-Atlas zur Mathematik”, Deutsche Taschenbuch Verlag, 1994, 10

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