Number theory/Signed Articles/Elementary diophantine approximations: Difference between revisions

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imported>Wlodzimierz Holsztynski
(→‎Notation: non-negative integers)
imported>Wlodzimierz Holsztynski
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* <math>1\ </math>&nbsp; is the only positive divisor of <math>\ 1.</math>
* <math>1\ </math>&nbsp; is the only positive divisor of <math>\ 1.</math>
* Every integer is divisible by <math>\ 1</math>&nbsp; (and by&nbsp; <math>\ -1</math>).
* Every integer is divisible by <math>\ 1</math>&nbsp; (and by&nbsp; <math>\ -1</math>).
&nbsp;
* <math>a|b > 0\ \Rightarrow a\le b</math>
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* <math>a|a\ </math>
* <math>a|a\ </math>

Revision as of 21:18, 12 January 2008

The theory of diophantine approximations is a chapter of number theory, which in turn is a part of mathematics. It studies the approximations of real numbers by rational numbers. This article presents an elementary introduction to diophantine approximations, as well as an introduction to number theory via diophantine approximations.

Introduction

In the everyday life our civilization applies mostly (finite) decimal fractions   Decimal fractions are used both as certain values, e.g. $5.85, and as approximations of the real numbers, e.g.   However, the field of all rational numbers is much richer than the ring of the decimal fractions (or of the binary fractions   which are used in the computer science). For instance, the famous approximation   has denominator 113 much smaller than 105 but it provides a better approximation than the decimal one, which has five digits after the decimal point.

How well can real numbers (all of them or the special ones) be approximated by rational numbers? A typical Diophantine approximation result states:

Theorem  Let   be an arbitrary real number. Then

  •   is rational if and only if there exists a real number C > 0 such that

for arbitrary integers   such that   and

  •   is irrational if and only if there exist infinitely many pairs of integers   such that   and

Notation

  •   —   "equivalent by definition" (i.e. "if and only if");
  •   —   "equals by definition";
  •   —   "there exists";
  •   —   "for all";
  •   —   "  is an element of set ";

 

  •  —  the semiring of the natural numbers;
  •  —  the semiring of the non-negative integers;
  •  —  the ring of integers;
  •  —  the field of rational numbers;
  •  —  the field of real numbers;

 

  •   —   "  divides ";
  •  —  the greatest common divisor of integers   and

 

Divisibility

Definition  Integer   is divisible by integer    

Symbolically:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b|a\ }   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftarrow:\Rightarrow\ }   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists_{c\in\mathbb{Z}}\ a=b\cdot c}


When   is divisible by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ b}   then we also say that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ b}   is a divisor of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a,}   or that   divides Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ a.}

  • The only integer divisible by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 0}   is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 0.}
  •   is divisible by every integer.
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\ }   is the only positive divisor of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 1.}
  • Every integer is divisible by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 1}   (and by  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ -1} ).

 

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a|b > 0\ \Rightarrow a\le b}

 

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a|a\ }
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a|b\ \and\ b|c)\ \Rightarrow\ a|c}

Remark  The above three properties show that the relation of divisibility is a partial order in the set of natural number  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{N},}   and also in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 1}   is its minimal, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ 0}   is its maximal element.