Prove sqrt(2) is irrational

Discussion in 'Calculator Requests' started by math_celebrity, Dec 31, 2022.

  1. math_celebrity

    math_celebrity Administrator Staff Member

    Use proof by contradiction. Assume sqrt(2) is rational.

    This means that sqrt(2) = p/q for some integers p and q, with q <>0.

    We assume p and q are in lowest terms.

    Square both side and we get:
    2 = p^2/q^2
    p^2 = 2q^2

    This means p^2 must be an even number which means p is also even since the square of an odd number is odd.

    So we have p = 2k for some integer k. From this, it follows that:
    2q^2 = p^2 = (2k)^2 = 4k^2
    2q^2 = 4k^2
    q^2 = 2k^2
    q^2 is also even, therefore q must be even.
    So both p and q are even.
    This contradicts are assumption that p and q were in lowest terms.

    So sqrt(2) cannot be rational.

    Last edited: Dec 31, 2022

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