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Why the Square Root of Two is an Irrational Number


The square root of two is irrational because it cannot be expressed as an irreducible fraction of integers.  It was very important to the ancient Pythagoreans that all numbers be expressible in such a way.  According to legend, a follower of Pythagoras, a man named Hippasus, discovered no such ratio exists for the square root of two.  Since the basic tenet of the Pythagoreans was "all is numbers," this absolutely horrified them because this meant their system of expressing all numbers is insufficient.  They tried to keep it a secret and failed and this eventually led to the collapse of Pythagoras' school.  Here is why the square root of two is irrational.



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