Eudoxus's definition of equal ratios corresponds exactly to the modern theory of irrationals. |
His commentary to Euclid is of interest because of its discussion of unordered irrationals. |
Lastly, the Pythagoreans discovered the existence of incommensurable lines, or of irrationals. |
Whether such quirks in the irregularity of irrationals have any implications for number theory remains an open question for mathematicians. |