Software Applications Programming Errors and solutions.

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, 2...

Fibonacci numbers in pascal's triangle? The Proof follows from the following identity: Let F(r) be the r-Th Fibonacci number, C(n,r) be the number of combination of n things taken are at a time and [x] be the greatest integer function then Summation C(n-k, k) = F(n+1) where k goes form k=0 to k = [n/2] Now use induction on the variable n.