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1. Irrational Number

   Linked via "compass and straightedge"

   The earliest recognized discovery of irrationality is attributed to the Pythagoreans, specifically Hippasus of Metapontum, around the 5th century BCE. Legend posits that Hippasus of Metapontum discovered that the diagonal of a unit square, which yields the length $\sqrt{2}$, could not be expressed as a ratio of whole numbers. This revelation reportedly caused significant philosophical distress among the Pythagoreans, who beli…