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Proposition 1 Prove {\sqrt{2}} is irrational.

Here is a proof using a traditional method (See Euclid’s Elements Book X which incorporates Theatetus work on incommensurable numbers. It includes a proof that {\sqrt{2}} is irrational (Proposition 22), and ends with a proof that there are infinitely many distinct irrational numbers (Proposition 115): Read the rest of this entry »

This is a typical proof one may see in a undergraduate introduction to proofs mathematics course.

Proposition 1 Let the constant {R} be defined such that

\displaystyle R:=\frac{1}{F_{1}}+\frac{1}{F_{2}\cdot F_{1}}+\frac{1}{F_{3}\cdot F_{2}\cdot F_{1}}+\frac{1}{F_{4}\cdot F_{3}\cdot F_{2}\cdot F_{1}}\cdots

and {F_{n}} are the Fibonacci numbers where {F_{n+k}>F_{k}} for {n>0}. Then {R} is irrational.

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