ON THE THEORY OF DIOPHANTINE APPROXIMATIONS AND CONTINUED FRACTIONS EXPANSION

Hailiza Kamarul Haili

ICMSA
Vol 1, No 1 (2005): ICMSA 2005

Approximation of irrational numbers by the rationals has been studiedfor decades and one of the methods to approximate the irrationals is byusrng the diophantine approximation and the convergents of itscontinued fractions expansion Many results pertaining to this area ofresearch has been developed and documented. In 1963, Niven [Niv] inhis monograph had shown a sketch proof of several results related tothis problem. Now, in this paper we attempt to write a morecomprehensive and well sfuctured proofs. Given a real number 0,how closely can it be approximated by rational numbers?.Keywords: Diophantine approximatioq continued fractions, irrationalnumber, and rational number.