A Study on Irrationality of some Numbers
Investigating the irrationality of certain numbers
by Priyanka*,
- Published in Journal of Advances in Science and Technology, E-ISSN: 2230-9659
Volume 1, Issue No. 1, Feb 2011, Pages 0 - 0 (0)
Published by: Ignited Minds Journals
ABSTRACT
Supposethat and are positiveirrational numbers. In this paper, we give the following criterion that is an irrationalnumber.
KEYWORD
irrationality, numbers, positive, criterion
INTRODUCTION
Suppose that and are positive irrational numbers. In this paper, we give the following criterion that is an irrational number. THEOREM : Suppose that and are positive functions such that . Let k be an arbitary fixed positive number with , and suppose that as . Let and be positive irrational numbers. Suppose that : (1) there exists a number q' = q'() such that for all with . and (II) the inequality has infinitely many solutions . Then is an irrational number. Proof : Let q be a sufficiently large integer to ensure the validity of the later argument. Without any loss of generality, we may assume that . Suppose that is a rational number, so that , where P and Q are positive integers. Now, let be a rational number such that . Since . we find that . where p* = Pp and q* = Qq. By the statement (I), we obtain This implies that (1) and by the assumption, we obtain
(2)
The relation (2) contradicts (1), and the theorem is proved.
EXAMPLES
For , Okano [3] proved that Example 1 : If is a Liouville number, then is an irrational number.
Proof : As we can put , we deduce that is an irrational number. Example 2 : Let m and k be integers with and . If , then is an irrational number. Proof : Let be the nth convergent of . Since for , we have for . Hence, . Accordingly, for all sufficiently large n, then has ifinitely many solutions (qn,qn). Consequently, as we can put we deduce that is an irrational number.
REFERENCES
P. Bundschuh, On simple continued fractions with partial quotients in arithmetic progressions, Lithunian Math. J., 38 (1998),15-26. S. Lang, Introduction to Diophantine Approximations, Addison-Wesley, 1966. T. Okano, A note on the retaional approximations to e, Tokyo J. Math., 15 (1992), 129-33. W.M Schmidt, Diophantine Approximation, Lecture Notes in Math., 785, Springer, 1980.
