Prove that V3 + V7
is irrational.
Answers
Answered by
11
Given:
- 3 + 5√7
To Prove:
- 3 + 5√7 is an irrational number.
Proof:
- Let us assume, to the contrary ,that 3 + 5√7 is rational.
Then, there exists co-primes a and b ( b≠0 ) such that
➽ 3 + 5√7 = a/b
➽ 5√7 = a/b – 3
➽ 5√7 = a – 3b/b
➽ √7 = a – 3b/5b
- Since, a and b are integers , so (a – 3b/5b) is rational.
- This √7 is also rational.
- But this contradicts the fact that √7 is irrational. So, our assumption is wrong.
Hence, (3 + 5√7) is irrational.
Answered by
0
Answer:
We have to prove that 3+
7
is irrational.
Let us assume the opposite, that 3+
7
is rational.
Hence 3+
7
can be written in the form
b
a
where a and b are co-prime and b
=0
Hence 3+
7
=
b
a
⇒
7
=
b
a
−3
⇒
7
=
b
a−3b
where
7
is irrational and
b
a−3b
is rational.
Since,rational
= irrational.
This is a contradiction.
∴ Our assumption is incorrect.
Hence 3+
7
is irrational.
Hence proved.
Similar questions