Is sqrt(5) Rational? Understanding the Proof by Contradiction

Thread starter Klion
Start date Nov 23, 2003
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Irrational

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The discussion centers on the proof that sqrt(5) is irrational using proof by contradiction, where it is assumed that sqrt(5) can be expressed as a fraction a/b with integers a and b having no common factors. The proof shows that this leads to a contradiction, demonstrating that 5 must divide both a and b, which contradicts the assumption that they are coprime. Additionally, the conversation explores similar reasoning for sqrt(6), concluding that it is also irrational by analyzing the implications of its prime factorization. Various participants debate the validity and clarity of different proofs, emphasizing the importance of establishing foundational principles in proving irrationality for square roots of non-square integers. The thread ultimately highlights the complexity and nuances involved in mathematical proofs regarding irrational numbers.

Feb 22, 2011

mathwonk

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here's another explanation of gibz's proof which is actually also my favorite: if sqrt(n) = a/b is in lowest form, then n = a^2/b^2 is also in lowest form, but lowest form is unique and n/1 is a lowest form for n, so b^2 = 1 and a^2 = n. i.e. if sqrt(n) is rational, then sqrt(n) is an integer.

Last edited: Feb 22, 2011