Question

prove that under root under root 2 under root 3 and under root 5 irrational numbers

Answer 1

All work in the same principal, so I would take the eg. of √2 only.
So,
Let's assume that √2 is Rational no.
Then, √2 = p/q where p and q are Co-prime.
Now, squaring both sides we have,
p^2/q^2=2
therefore,
p^2 = 2*q^2
Hence p is an even no. ( as the squares of even nos. only are Even)
Now,
p is hence even.
therefore, p = 2* m ( where m can be any integer)
therefore,
2*q^2 = (2*m)^2
q^2=4m^2/2=2m^2
therefore,
q is also an even no. ( as squares of even now. are only even)

so we conclude that,
p and q both are even that is both are divisible by 2.
But we assumed in that p and q are co prime, that is there common factor is just 1.
Hence our Assumption is Wrong and thus we proved that√2 is an Irrational Number.


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