05. The Incommensurables – Geometrical Proof

The text discusses the discovery of incommensurable magnitudes, a fundamental concept in mathematics. This discovery, made by the Pythagoreans, demonstrated that not all line segments can be measured using a common unit of length. The text uses the example of a square's diagonal and its side to illustrate this concept. The process of repeatedly trying to find a common unit of length between the diagonal and side of the square reveals that this task is impossible, as the process continues indefinitely. This led to the creation of irrational numbers, a new category of numbers that cannot be expressed as fractions of integers. This discovery had a profound impact on the development of mathematics, expanding the understanding of numbers and their relationship to geometry.