Real Line – An Incomplete Number System

Abstract

The algebraic equation a3 + b3 = c3 where a, b & c are real numbers greater than zero, has infinite solutions. ‘R’ denotes the set of real numbers as well as the ‘Real Line’. The real numbers are treated as if those are points on the ‘Real Line’ and the points on the ‘Real Line’ as if those are real numbers. Hence a, b & c will have corresponding points on the Real Line and can be represented by three straight lines denoting three sides of a triangle, say ΔABC, on a two dimensional plane. Consequently the algebraic equation will yield the trigonometric equation Sin3A + Sin3B = Sin3C. Trigonometric properties of such a triangle i.e. DABC, imply that ÐC will be independent of (a & b) if c is constant; but solutions of the algebraic equation show that ÐC is dependent on (a & b) when c is constant, leading to a contradiction. Hence, all real numbers of the form ∛x cannot be considered as if those are points on the ‘Real Line’; and there are no points on the ‘Real Line’ corresponding to the irrational numbers of the form ∛x. This leads to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the ‘Real Line’.