prove that 3-√5 is irrational given that√5 is irrational
Answers
Answer:
Let us assume that 3 + √5 is a rational number. Here, {(a - 3b) ÷ b} is a rational number. But we know that √5 is a irrational number. ... → a and b both are co-prime numbers and 5 divide both of them.
Answer:
Step-by-step explanation:
ANSWER
To prove :
3
+
√5
is irrational.
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+ √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.
This contradiction arose because of our false assumption.
so,
3
+
5
irrational.