Golden Ratio is Irrational from a Regular Pentagon (visual proof)

In this video, we show how to prove that the golden ratio is the length of a diagonal in a regular pentagon with a side length of 1. We do this using similar triangles and angle analysis using the circumscribed circle. As a bonus, we use nearly the same diagram (just scaled appropriately if needed) to prove that the golden ratio is irrational, that is, that it cannot be written as the ratio of two integers.

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This animation is based on an argument in Roger Nelsen’s book Nuggets of Number Theory (I may receive a commission from Amazon, at no cost to you, if you click the link below):

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Here are other videos related to the Golden ratio:
   • Golden Ratio Visual Computation  
   • Golden ratio!  
   • The Golden Ratio rolling in Euler's c...  

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