Golden ratio
By irrationality of √ 5

Another short proof – perhaps more commonly known – of the irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication. If φ = 1 2 (1 + 5) {\displaystyle \varphi ={\tfrac {1}{2}}(1+{\sqrt {5}})} is rational, then 2 φ − 1 = 5 {\displaystyle 2\varphi -1={\sqrt {5}}} is also rational, which is a contradiction if it is already known that the square roots of all non- square natural numbers are irrational.
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