Numbers that can be written as the quotient of two integers are called rational. Even young children readily understand what the fractions 1/2, 1/3 or 3/4 represent when cutting a cake. One of geometry's most famous surprises is that the length of the diagonal of a square with side length 1, namely is not a rational number. This has been known since antiquity, and it is often said that the proof of this result was due to Hippasus of Metapontum, who thereby supposedly triggered a foundational crisis among the Pythagoreans and was drowned as a result!
If ABCD is a square with side length 1, then the diagonal BD has length √ 2.
One thing is certain: ancient Greek scholars explored questions of this kind, and more specifically the fact that two lengths can be incommensurable, meaning that their ratio is irrational.
A long road to acceptance
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