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Prove that root 8 is irrational …

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Prove that root 8 is irrational number

    • Posted by Deepak Paul 4 years, 3 months ago

    CBSE > Class 10 > Mathematics
    • 1 answers

    Preeti Dabral 4 years, 3 months ago

    suppose √8 = a/b with integers a, b
    and gcd(a,b) = 1 (meaning the ratio is simplified)

    then 8 = a²/b²
    and 8b² = a²

    this implies 8 divides a² which also means 8 divides a.

    so there exists a p within the integers such that:
    a = 8p
    and thus,
    √8 = 8p/b
    which implies
    8 = 64p²/b²
    which is:
    1/8 = p²/b²
    or:
    b²/p² = 8
    which implies
    b² = 8p²
    which implies 8 divides b² which means 8 divides b.

    8 divides a, and 8 divides b, which is a contradiction because gcd (a, b) = 1
    therefore, the square root of 8 is irrational.

    1Thank You

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