Geometric Proof of the Irrationality of Square-Roots for Select Integers

Abstract

This paper presents geometric proofs for the irrationality of square roots of select integers, extending classical approaches. Building on known geometric methods for proving the irrationality of sqrt(2), the authors explore whether similar techniques can be applied to other non-square integers. They begin by reviewing well-known results, such as Euclid's proof for the irrationality of sqrt(2), and discuss subsequent geometric extensions for sqrt(3), sqrt(5), and sqrt(6). The authors then introduce new geometric constructions, particularly using hexagons, to prove the irrationality of sqrt(6). Furthermore, the paper investigates the limitations and challenges of extending these geometric methods to triangular numbers. Through detailed geometric reasoning, the authors successfully generalize the approach to several square-free numbers and identify cases where the method breaks down. The paper concludes by inviting further exploration of geometric irrationality proofs for other integers, proposing potential avenues for future work.

Figures (7)

• Figure 2: Geometric proof of the irrationality of $\sqrt{6}$
• Figure 3: Geometric proof of the irrationality of $\sqrt{6}$: equilateral triangles
• Figure 4: The proof of the irrationality of $\sqrt{n(n+1)/2}$ when $n$ is 2, 3, 4, and 5.
• Figure 5: Proof of congruence for all doubly-covered and triply-covered triangles: triangles on the outside
• Figure 6: Proof of congruence for all doubly-covered and triply-covered triangles: first two rows