The ancient Greeks wanted to believe that the universe could be described in its entirety using only whole numbers and the ratios between them β€” fractions, or what we now call rational numbers. But this aspiration was undermined when they considered a square with sides of length 1, only to find that the length of its diagonal couldn’t possibly be written as a fraction.
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The first proof of this (there would be several) is commonly attributed to Pythagoras, a 6th-century BCE philosopher, even though none of his writings survive and little is known about him. Nevertheless, β€œit was the first crisis in what we call the foundations of mathematics,” said John Bell, a professor emeritus at Western University in London, Ontario.
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That crisis would not be resolved for a long time. Though the ancient Greeks could establish what $latex \sqrt{2}$ was not, they didn’t have a language for explaining what it was.

https://www.quantamagazine.org/how-the-square-root-of-2-became-a-number-20240621/
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