Dedekind cut
Zeno of Elea (490 c. 430 BC) produced four paradoxes that seem to show the impossibility of change. The Pythagorean school of mathematics originally insisted that only natural and rational numbers exist. The discovery of the irrationality of , the ratio of the diagonal of a square to its side (around 5th century BC), was a shock to them which they only reluctantly accepted. The discrepancy between rationals and reals was finally resolved by Eudoxus of Cnidus (408–355 BC), a student of Plato, who reduced the comparison of two irrational ratios to comparisons of multiples of the magnitudes involved. His method anticipated that of the Dedekind cut in the modern definition of real numbers by Richard Dedekind (1831–1916).