Question

Prove that √5 is an irrational number. Hence, show that -3 + 2 √5 is an irrational number.​

Answer 1

Given:

• A Irrational number √5 is given .

To Prove:

• It is irrational and further prove that -3+2√5 is also irrational.

Proof:

On the contarary let us suppose that √5 is a Rational number , So it can be expressed in the form of p/q where p and q are integers and q ≠ 0 .

Also p and q are co - primes , meaning to say that HCF of p and q is 1.

=> √5 = p/q .

=> (√5)² = (p/q)²

=> 5 = p²/q².

=> p² = 5q².

Now this implies that 5 is a factor of p² .So from the Fundamental Theorem of Arithmetic , we can say that 5 is also a factor of p .

=> p = 5k .

where k is some integer.

Putting this value in above equⁿ , we get ,

=> (5k)² = 5q².

=> 25k² = 5q².

=> 5k² = q².

Again , now this implies that 5 is a factor of q² .So from the Fundamental Theorem of Arithmetic , we can say that 5 is also a factor of q .

=> q = 5m.

where m is some integer .

This contradicts our assumption that p and q are co primes .Hence our assumption was wrong , √5 is Irrational number.

Now we have to prove that -3+2√5 is Irrational.

• Firstly √5 is Irrational and 2 is Rational , And products of Rational number and a Irrational number is always a Irrational number. Hence 2√5 is Irrational.
• Now sum of a Rational number and a Irrational number is always a Irrational number .Hence here -3 is Rational and 2√5 is Irrational .So -3+2√5 is Irrational .

Hence proved.