A real number that cannot be expressed as the ratio of two integers, meaning it has a non-terminating, non-repeating decimal expansion. Examples include √2, π, and e.
From Latin 'irrationalis' meaning 'without reason,' literally 'not rational.' The term reflects the ancient Greeks' shock at discovering numbers that couldn't be expressed as ratios, which seemed to defy logical reasoning about the nature of number.
Irrational numbers are infinitely mysterious - their decimal expansions go on forever without any pattern! What's mind-blowing is that there are actually more irrational numbers than rational ones, making the 'unreasonable' numbers far more common in the mathematical universe.
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