  Book III continues the discussion of ratios; Khayyam sets himself the task of demonstrating the seemingly innocuous proposition A/C = (A/B) (B/C), a fact which is used in the Elements but never proved. During this process he sets an arbitrary fixed magnitude to serve as a unit, to which he relates all other magnitudes of the same kind. This allows Khayyam to incorporate both numbers and geometric magnitudes within the same system. Thus Khayyam thinks of irrational magnitudes as numbers themselves, which effectively defines the set of “real numbers” that we take for granted today. This step was one of the most significant changes of conception to occur between ancient Greek and modern mathematics.