Uma demonstração da irracionalidade do número de Euler

Euler's number, denoted by e, stands as one of the most emblematic constants in mathematics. It is an irrational number that serves as the base of natural logarithms, boasting significant historical and theoretical aspects. The present work aims to demonstrate the irrationality of any non-zero rational power of the Euler number. To construct this demonstration, a sequence of real numbers, derived from a definite integral, is used and through the technique of integration by parts and by the use of the second principle of induction, a special characterization is presented allowing to deduce the main result. As an immediate consequence, we obtain the irrationality of Euler's number and also that the natural logarithm of every positive rational number other than one is irrational.