Prove that 3+2√5 is irrational
Answers
Step-by-step explanation:
Prove 3+2
5
is irrational.
→ let take that 3+2
5
is rational number
→ so, we can write this answer as
⇒3+2
5
=
b
a
Here a & b use two coprime number and b
=0.
⇒2
5
=
b
a
−3
⇒2
5
=
b
a−3b
∴
5
=
2b
a−3b
Here a and b are integer so
2b
a−3b
is a rational number so
5
should be rational number but
5
is a irrational number so it is contradict
- Hence 3+2
5
is irrational.
Given:3 + 2√5
To prove:3 + 2√5 is an irrational number.
Proof:
Letus assume that 3 + 2√5 is a rational number.
Soit can be written in the form a/b
3 + 2√5 = a/b
Here a and b are coprime numbers and b ≠ 0
Solving3 + 2√5 = a/b we get,
=>2√5 = a/b – 3
=>2√5 = (a-3b)/b
=>√5 = (a-3b)/2b
This shows (a-3b)/2b is a rational number. But we know that But √5 is an irrational number.
so it contradictsour assumption.
Our assumption of3 + 2√5 is a rational number is incorrect.
3 + 2√5 is an irrational number
Hence proved