In this short, we use a famous argument by Tom Apostol to prove that the square root of two is irrational by infinite descent using a right isosceles triangle. We also go a bit further and show how this proof hints at the number theoretic construction of the convergents of the square root of two, which are the best rational approximations of root 2.

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For an alternate visual proof of this fact, see this video:    • Visual irrationality proofs from the ...  

This animation is based on an argument due to Tom Apostol from issue 9 of the 2000 American Mathematical Monthly: https://doi.org/10.1080/00029890.2000...

To learn more about the convergents argument and the relationship between this proof and convergents, see this wonderful article by Doron Zeilberger:
https://sites.math.rutgers.edu/~zeilb...
and here you can learn more about convergents:
https://en.wikipedia.org/wiki/Continu...

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