Question

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Prove that √3 – √2 and √3 + √5 are irrational.​

Answer 1

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 qp form where p,q are integers and q isn't equal to zero.

3+5=qp

3=qp−5

squaring on both sides,

 

3=q2p2−2.5(qp)+5

⇒q(25p)=5−3+(q2p2) 

⇒q(25p)=q22q2−p2

⇒

Answer 2

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 √5 is an irrational number.

So, it contradicts our assumption. Our assumption of 3 + 2√5 is a rational number is incorrect.

3 + 2√5 is an irrational number

Hence proved