More Fun with Pythagoras

Abstract

Students may be introduced to a set of formulae that defines all primitive Pythagorean triples: For all positive integers i and j where i is an uneven number and j is an even number and i and j have no common factors, h = i2 + ij + j2/2, e = ij + j2/2, u = i2 + ij where h is the hypotenuse, e is the even-numbered leg and u is the uneven-numbered leg of the Pythagorean triangle. A collection of well-defined subsets of the universal set of triangles described by i and j form sequences of triangles that approach in proportion a triangle which has one side defined by an irrational number while the other two are positive integers. This also creates sequences of rational numbers that approach an irrational limit. Finding and defining these series of triangles and numbers require the use of good algebra, giving relevance to the student’s learning of factorising and manipulating algebraic expressions.