Define what it means for two magnitudes to be commensurable

. Define what it means for two magnitudes to be commensurable and incommensurable. Do incommensurable magnitudes exist? Do incommensurable numbers exist (according to Euclid’s definition of number)? Describe the effect that the discovery of incommensurable magnitudes had on the development of mathematics.

Solution

When the ratio of lengths of two line segments is irrational, the line segments are also described as being incommensurable, meaning they share no measure in common.ex: length of diagonal of square and length of its sides. If two magnitudes are commensurable, then their ratio is a rational number, whereas the ratio of incommensurable magnitudes is irrational. Incommensurability in a way gave way to discovery of pythagerus theorem because greeks started to represent incommensurable square roots using geometric figures only after the discovery of incommensurability.