Show that there is no positive integer for which √ + 1− √ −1 is rational
Answers
Answered by
1
Answer:
Suppose there exists a positive integer n for which is a rational number.
Where p and q positive integers and q
=0
⇒
p
q
=
n−1
+
n+1
1
⇒
p
q
=
(n−1)−(n+1)
n−1
−
n+1
=
−2
n−1
−
n+1
⇒
p
2q
=
n+1
−
n−1
(
n−1
+
n+1
)+(
n+1
−
n−1
)=
q
p
+
p
2q
⇒2
n+1
=
pq
p
2
+2q
2
⇒
n+1
=
2pq
p
2
+2q
2
.....(1)
(
n−1
+
n+1
)−(
n+1
−
n−1
)=
q
p
−
p
2q
⇒2
n−1
=
pq
p
2
−2q
2
⇒
n−1
=
2pq
p
2
−2q
2
.....(2)
Similar questions