prove that √5 is an irrational number
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Let us suppose that √5 is rational. Then there exist two positive integers a and B such that
√5 = a/b
Where a and B are co primes
Squaring on both side gives us
5=a^2/b^2
5b^2 = a^2
It means 5 is a factor of a^2 and a as well
5c = a. (as 5 is a factor of a)
Squaring on both sides gives us
25c^2 = a^2
25c^2 = 5b^2. ( As proved above)
b^2 = 5c^2
It means 5 is also a factor of B.
Hence it is a contradiction as a and b were co primes.
Hence our supposition is wrong and √5 is irrational.
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