Zur Irrationalit\"at in der Schule

TL;DR

This paper demonstrates elementary calculus-based proofs of the irrationality of numbers like e and pi, and offers geometric variants to illustrate these analytical statements, making the concepts accessible at various educational levels.

Contribution

It introduces simple calculus methods to prove irrationality of e and pi and provides geometric variants to enhance understanding across different levels.

Findings

Elementary proofs of irrationality for e and pi using calculus

Geometric variants for analytical statements to aid visualization

Creates variability in educational complexity

Abstract

Irrational numbers are introduced usually already introduced in lower secondary level schools. But typically, maybe with the exception of $\sqrt{2}$, no mathematical proof of irrationality is provided. In particular it is not proven that famous Euler's number $e$ as well as the number $\pi$ are irrational. In this article we want to show how this can be done with very elementary methods from calculus. In addition, we offer geometrical variants for many of the analytical statements, which in particular create variability in the level of requirements. ----- Irrationale Zahlen werden in der Schule bereits in der Sekundarstufe I eingef\"uhrt. Allerdings wird typischerweise, mit Ausnahme vielleicht f\"ur $\sqrt{2}$, kein mathematischer Beweis zur Irrationalit\"at gef\"uhrt. Insbesondere wird nicht bewiesen, dass die ber\"uhmte Eulersche Zahl $e$ sowie die Kreiszahl $\pi$ irrationale Zahlen sind. In diesem Artikel wollen wir aufzeigen, wie dies mit recht elementaren Methoden der Analysis m\"oglich ist. Dar\"uber hinaus bieten wir f\"ur viele der analytischen Aussagen geometrische Varianten zur Veranschaulichung, die insbesondere Variabilit\"at im Anspruchsniveau schaffen.