How did Book V help Euclid handle irrational numbers?

Book V of Euclid’s Elements helps handle irrational numbers by providing a general theory of proportions that does not require magnitudes to be commensurable (measurable by a common unit).12

Before this development, Greek mathematics faced an impasse because the discovery of incommensurable quantities—now known as irrational numbers—meant that ratios could not always be expressed as simple relationships between integers. Attributed to the mathematician Eudoxus of Cnidus, the theory in Book V bypassed this limitation by defining ratios in a way that accounted for all magnitudes, whether commensurable or incommensurable.341

By establishing this rigorous, general framework, Euclid provided a foundation for geometry that remained valid even when dealing with quantities like the square root of 2, which cannot be expressed as a ratio of integers. This approach is considered a significant precursor to 19th-century definitions of real numbers based on rational numbers.314