Geodesic
Irrational Numbers arising from Certain Differential Equations
Canadian mathematical bulletin, Tome 20 (1977) no. 1, pp. 117-120

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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. 
Murty, M. Ram; Murty, V. Kumar. Irrational Numbers arising from Certain Differential Equations. Canadian mathematical bulletin, Tome 20 (1977) no. 1, pp. 117-120. doi: 10.4153/CMB-1977-021-x
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[1] 1. Dixon, J. D., π is not algebraic of degree one or two, Amer. Math. Monthly, 69 (1962), 632. Google Scholar

[2] 2. Koksma, J. F., On Niven's proof that π is irrational, Nieuw Archief voor Wiskunde, (2)23 (1949), 39. Google Scholar

[3] 3. Niven, I., A simple proof that π is irrational, Bull. Amer. Math. Soc., 53 (1947), 509. Google Scholar

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