Why Some Elementary Functions Are Not Rational

Abstract

A classical exercise for college students is to ask them to prove that the sine function is not a polynomial or, more generally, a rational function. This follows from the fact that the sine function has an infinite number of zeros, but this cannot happen to a rational function unless it is identically zero. This, however does not rule out the possibility that the restriction of the sine function to an open interval is rational. One way to prove that a function defined on an open interval is not a polynomial function is to calculate successive derivatives of the function: Since the degree of the derivative of a non-constant polynomial function is smaller than the degree of the polynomial, after some time you will get a constant function. This approach, however, does not work for rational functions. Or does it? In this note we prove, in an elementary way, that the restrictions to an open interval of certain elementary functions are not rational functions, and we do it by using the concept of degree of a rational function. In what follows, the domain of all functions is a fixed nonempty open interval of the real line. When we speak about the exponential function, the sine function, and so on, what we actually mean is the restriction of these functions to this interval.