  This paper looks at how ancient mathematicians (and especially the
Pythagorean school) were faced by problems/paradoxes associated with the
infinite which led them to juggle two systems of numbers: the discrete
whole/rationals which were handled arithmetically and the continuous magnitude
quantities which were handled geometrically. We look at how approximations and
mixed numbers (whole numbers with fractions) helped develop the arithmetization
of geometry and the development of mathematical analysis and real numbers.
