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Math and science::Analysis::Tao::05. The real numbers

The irrationality of \( \sqrt(2) \)

Consider a rational number fully reduced, \( \frac{p}{q}\) such that \(\frac{p^2}{q^2} = 2\). Investigate whether \(p\) or \(q\) are odd or even numbers leads to a contradiction which shows that \(2\) cannot be rational.

  • \(p\) and \(q\) can't both be even, as [...].
  • \(p\) and \(q\) must both be odd, as [...].
  • \(p\) is even as [...].

These points lead to a contradiction.