July 24, 2011

Properties of Irrational numbers

Properties of Irrational numbers

If x be an irrational number then there are infinitely many relatively prime integers p and q such that
| p/q - x | < 1/q^2
If x is a rational number then there are finitely many solutions to the above equation. This Theorem was first discovered by Dirichlet

Further let {y} be the fractional part of y. x is an irrational number then the sequence {nx} n = 1,2,3, ... is not periodic. If x
is a rational then this sequence is periodic.

A rational number can be represented only by a a finite continued fraction where as an irrational number can be represented by an infinite continued fraction. Each convergent of the continued fraction representation of a irrational number is a better rational approximation of the given irrational number, than any previous convergent.
Eg: The continued fraction representation of Pi is
[3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ..]

hence the successive rational approximation of Pi are 3, 22/7, 355/113
Similarly for an algebraic number number a, Roth proved there are finite number of rational numbers p/q such that absolute value of [p/q -a] < 1/q^?(2+e) where e is any arbitrarily small positive real number.Klaus Roth received Fields medal for this remarkable theorem.