ABSTRACT. APythagoreantripleisanorderedtripleofintegers(a,b,c)̸=(0,0,0)such 222 that a + b = c . It is well known that the set P of all Pythagorean triples has an intrinsic structure of commutative monoid with respect to a suitable binary operation, (P, ⋆). In this article, we will introduce the “commensurability” relation R among Pythagorean triples, and we will see that it induces a group quotient, P/R, which is isomorphic with the direct product of infinite (countable) copies of C∞ , the infinite cyclic group, and a cyclic group of order 4. As an application, we will see that the acute angles of Pythagorean triangles are irrational when measured in degrees.
ON THE ALGEBRAIC STRUCTURE OF PYTHAGOREAN TRIPLES
Vincenzi Giovanni
2024-01-01
Abstract
ABSTRACT. APythagoreantripleisanorderedtripleofintegers(a,b,c)̸=(0,0,0)such 222 that a + b = c . It is well known that the set P of all Pythagorean triples has an intrinsic structure of commutative monoid with respect to a suitable binary operation, (P, ⋆). In this article, we will introduce the “commensurability” relation R among Pythagorean triples, and we will see that it induces a group quotient, P/R, which is isomorphic with the direct product of infinite (countable) copies of C∞ , the infinite cyclic group, and a cyclic group of order 4. As an application, we will see that the acute angles of Pythagorean triangles are irrational when measured in degrees.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
