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Irrational Numbers arising from Certain Differential Equations

Published online by Cambridge University Press:  20 November 2018

M. Ram Murty  and
V. Kumar Murty
M. Ram Murty
Affiliation:
Department of MathematicsCarleton University Ottawa, Ontario
V. Kumar Murty
Affiliation:
Department of MathematicsCarleton University Ottawa, Ontario
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Niven [3] gave a simple proof that π is irrational. Koksma [2] modified Niven's proof to show that er is irrational for every non-zero rational r. Dixon [1] made a similar modification to show that π is not algebraic of degree 2. In this note, we prove a general theorem which gives Niven's and Koksma's results as easy corollaries. A suitable modification in our proof also gives Dixon's result.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 20 , Issue 1 , 01 March 1977 , pp. 117 - 120
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Dixon, J. D., π is not algebraic of degree one or two, Amer. Math. Monthly, 69 (1962), 632.Google Scholar
2. Koksma, J. F., On Niven's proof that π is irrational, Nieuw Archief voor Wiskunde, (2)23 (1949), 39.Google Scholar
3. Niven, I., A simple proof that π is irrational, Bull. Amer. Math. Soc., 53 (1947), 509.Google Scholar