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

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SUMMARY

The discussion centers on the proof that sqrt(5) is irrational using proof by contradiction. The proof begins with the assumption that sqrt(5) can be expressed as a fraction a/b, leading to the conclusion that both a and b must be divisible by 5, contradicting the assumption that they are coprime. Additionally, participants explore the irrationality of sqrt(6) through similar reasoning, demonstrating that both sqrt(5) and sqrt(6) are irrational due to their prime factors. The conversation also touches on the general theorem that sqrt(n) is irrational if n is not a perfect square.

PREREQUISITES
  • Understanding of proof by contradiction
  • Familiarity with prime factorization and properties of prime numbers
  • Knowledge of rational and irrational numbers
  • Basic algebraic manipulation and equations
NEXT STEPS
  • Study the proof by contradiction in greater depth
  • Learn about the properties of prime numbers and their implications in number theory
  • Explore the concept of perfect squares and their relation to irrational numbers
  • Investigate other proofs of irrationality for square roots of non-square integers
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Mathematicians, students of number theory, educators teaching proofs and irrational numbers, and anyone interested in the foundations of mathematics.

  • #61
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.
 
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