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Question

Prove that 3 is an irrational number.


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Solution

Let us suppose that 3 is a rational number.

Then there are positive integers a and b such that 3=ab, where a and b are co-prime, meaning their HCF is 1.

3=ab3=a2b23b2=a23dividesa2[3divides3b2]3dividesa.....................(i)a=3cforsomeintegerca2=9c23b2=9c2[a2=3b2]b2=3c23dividesb2[3divides3c2]3dividesb..............................(ii)

We can see that a and b share at least 3 as a common factor from (i) and (ii).

Because of the fact thata and b are co-prime, however, contradicts this and indicates that our hypothesis is incorrect.

Hence, 3​ is an irrational number.


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