prove that√3-√5 is an irrational number
Answers
Answer:
To prove that 3+√5 is an irrational number, first assume it to be a rational number. ... As both p and q are integers, so p−3q is also an integer. As q is not equal to 0, p−3qq is a rational number. ⇒√5 = p−3qq is also a rational number.
Answer:
Let us assume it to be a rational number.
Rational numbers are the ones that can be expressed in
q
p
form where p,q are integers and q isn't equal to zero.
3
+ Ruchi 3 - root 5 is irrational number
5
=
q
p
3
=
q
p
−
5
squaring on both sides,
3=
q
2
p
2
−2.
5
(
q
p
)+5
⇒
q
(2
5
p)
=5−3+(
q
2
p
2
)
⇒
q
(2
5
p)
=
q
2
2q
2
−p
2
⇒
5
=
q
2
2q
2
−p
2
.
2p
q
⇒
5
=
2pq
(2q
2
−p
2
)
As p and q are integers RHS is also rational.
As RHS is rational LHS is also rational i.e
5
is rational.
But this contradicts the fact that
5
is irrational.
So our consumption is rong root 3 minus root 5 is irrational number