"Plato denied explicitly the existence of fractional numbers: the numerical unit had no parts and could not be divided. Of course, for practical purposes fractions were commonly required. The use of what we call rational numbers therefore infiltrated almost imperceptibly into theoretical mathematics. It would be hard to say exactly when rational numbers were recognized as numbers, since this requires making a careful distinction between the ratio 1:2 (which had a perfectly good pedigree in Eudoxus' theory of proportion) and the number ½. ...It would be quite a long time after this period before irrational numbers were tolerated, and until this step was taken there was no prospect for describing geometrical problems in arithmetical terms."