The irrational nature of exponential and trigonometric functions of rational numbers, and their inverses

It is well-known that if x is non-zero and rational, then exp(x) and tan x are irrational (see, for example, the Mathworld article ‘Irrational Number’). From the former result, exp(1) = e is irrational, and from the latter, since tan pi/4 = 1 is not irrational, pi/4 and hence pi is not rational.

Perhaps one day someone will find a general method for determining, from the coefficients in a function’s Taylor series, whether the function possesses the property of mapping rational numbers almost exclusively to irrational numbers, but that day seems a long way off.

The purpose of this series of posts is to revisit Hermite’s, Niven’s and Lambert’s examinations of the exponential function, sin, cos and tan, and their inverses.

Above all, I will try to make the motivation of the various proofs clear.  The underlying methods are actually quite simple.  Hermite’s method, extended by Niven, uses repeated integration by parts of a product of the relevant function and a carefully selected polynomial.  Lambert’s method uses an iterative method to generate a sequence of rational fraction approximations to tanh x and tan x, from which the results on irrationality can be deduced fairly simply.

I will start (in the next post) with a couple of simple, general theorems, then I will explore Hermite’s and Niven’s methods, and finally Lambert’s method.


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