prove √5 as irrational
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Let us assume opposite,
√5 is rational.
According to the question :
Form = p / q , where q is not equal to 0.
√5 = p / q
squaring both sides,
=> 5 = p^2 / q^2
=> 5q^2 = p^2
- p is divisible by 5.
- p = 5k , k is a positive integer.
squaring both sides,
=> 5q^2 = 5 × 5k^2
=> 5q^2 = 25k^2
substituting 5q^2 = p^2
=> 5q^2 = 25k^2
=> q^2 = 5k^2
- q is divisible by 5.
- p and q have common factor 5.
- p and q are co - primes.
=> So, √5 is an irrational number.
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