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Prove √n-1+√n+1 is rational number and prove also prove√3+√7 as irrational

9 months ago

Let √n−1+√n+1 be a rational number which can be expressed as p/q, p and q are integers and coprime. q is not equal to 0
squaring on both sides we get n-1+n+1+2 √n2−1
2n+2 √n2−1 =p2q2
2(n+√n2−1)=p2q2
2(n+√n2−1)q2=p2
this mean 2 divides p2 and also divides p.
then let p=2k for any integer k
then 2(n+√n2−1)=(2k)2q2
2(n+√n2−1)=4k2q2
q2=2k2/(n+√n2−1)
so 2 divides q2 and also q
p and q have common factors 2 which contradicts the fact that p and q are co-primes which is due to our wrong assumption. so
√n−1+√n+1 is irrational.

9 months ago
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