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Show that for any positive integer n ,(√n 1) (√n-1) is not a rational number?
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Show that for any positive integer n ,(√n 1) (√n-1) is not a rationa...
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.
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.
Community Answer
Show that for any positive integer n ,(√n 1) (√n-1) is not a rationa...
Proof that (√n + 1) / (√n - 1) is not a rational number:
- Assume (√n + 1) / (√n - 1) is rational:
Assume that (√n + 1) / (√n - 1) is a rational number, which means it can be expressed as a fraction of two integers: a/b, where a and b are integers and b is not equal to 0.
- Express the given expression in terms of a and b:
(√n + 1) / (√n - 1) = a/b
This can be rearranged to: √n + 1 = a/b(√n - 1)
- Simplify the expression:
√n + 1 = (a√n - a) / b
√n + 1 = (a√n - a) / b
√n + 1 = (a√n - a) / b
b(√n + 1) = a√n - a
b√n + b = a√n - a
b + a = √n(a - b)
- Contradiction:
This equation shows that √n is a rational number, which is a contradiction since the square root of a non-perfect square is irrational. Therefore, our initial assumption that (√n + 1) / (√n - 1) is rational is false. Hence, it is proved that (√n + 1) / (√n - 1) is not a rational number.
- Assume (√n + 1) / (√n - 1) is rational:
Assume that (√n + 1) / (√n - 1) is a rational number, which means it can be expressed as a fraction of two integers: a/b, where a and b are integers and b is not equal to 0.
- Express the given expression in terms of a and b:
(√n + 1) / (√n - 1) = a/b
This can be rearranged to: √n + 1 = a/b(√n - 1)
- Simplify the expression:
√n + 1 = (a√n - a) / b
√n + 1 = (a√n - a) / b
√n + 1 = (a√n - a) / b
b(√n + 1) = a√n - a
b√n + b = a√n - a
b + a = √n(a - b)
- Contradiction:
This equation shows that √n is a rational number, which is a contradiction since the square root of a non-perfect square is irrational. Therefore, our initial assumption that (√n + 1) / (√n - 1) is rational is false. Hence, it is proved that (√n + 1) / (√n - 1) is not a rational number.
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