Number theory/Signed Articles/Elementary diophantine approximations: Difference between revisions
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'''Definition''' Integer <math>\ a</math> is divisible by integer <math>\ b</math> <math>\Leftarrow:\Rightarrow\ </math> <math>\exists_{c\in\mathbb{Z}}\ a=b\cdot c.</math> | '''Definition''' Integer <math>\ a</math> is divisible by integer <math>\ b</math> <math>\Leftarrow:\Rightarrow\ </math> <math>\exists_{c\in\mathbb{Z}}\ a=b\cdot c.</math> | ||
When <math>\ a</math> is divisible by <math>\ b</math> then we also say that <math>\ b</math> is a divisor of <math>\ a,</math> or that <math>\ b</math> divides <math>\ a.</math> | |||
* The only integer divisible by <math>\ 0</math> is <math>\ 0.</math> | |||
* <math>0\ </math> is divisible by every integer. | |||
* <math>1\ </math> is the only positive divisor of <math>\ 1.</math> | |||
* Every integer is divisible by <math>\ 1</math> (and by <math>\ -1</math>). |
Revision as of 20:32, 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 ring of integers;
- — the field of rational numbers;
- — the field of real numbers;
- — the greatest common divisor of integers and
Divisibility
Definition Integer is divisible by integer
When is divisible by then we also say that is a divisor of or that divides
- The only integer divisible by is
- is divisible by every integer.
- is the only positive divisor of
- Every integer is divisible by (and by ).