Prove that 3+2?5 is irrational
Answers
Given: 3 + 2√5
To prove: 3 + 2√5 is an irrational number.
Proof:
Let us assume that 3 + 2√5 is a rational number.
So it can be written in the form a/b
3 + 2√5 = a/b
Here a and b are coprime numbers and b ≠ 0
Solving 3 + 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 of 3 + 2√5 is a rational number is incorrect.
3 + 2√5 is an irrational number
Hence proved
Step-by-step explanation:
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