Question

Prove that V2+1 is an irrational number.
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Answer 1

Required Proof:

Let,

consider in a contradiction that √2 + 1 is a rational number that can be expressed in p/q form where p and q are coprimes.

Then,

⇒ √2 + 1 = p/q

⇒ √2 = p/q - 1

⇒ √2 = p - q/q

If p/q is a rational number, then p - q/q is also a rational number but this contradicts the fact that √2 is an irrational number$.$

Hence,

Our assumption was wrong. √2 + 1 is an irrational number.

Note:

• As a base, we should know the method of proving √2, √3 and √5 irrational numbers.

• Rest all the proofs of complex rational numbers can be proved by contradictions as above.

Answer 2

Answer:

To Proof :-

Prove that √2 + 1 is irrational number

Solution :-

Firstly,

Let's assume √2+1 as a rational number.

Now,

As we know that all rational number can be represented as p/q form

$\sf \: \sqrt{2} + 1 = \dfrac{p}{q}$

$\sf \: \sqrt{2} = \dfrac{p}{q} - 1$

$\sf \: \sqrt{2} = \dfrac{p - q}{q}$

Now,

We are assuming p/q as rational number. So, p-q/q is also a rational number.

But √2 is left. And we know √2 is an irrational number.

Hence we proved that √2 + 1 is an irrational number.

Extra Information :-

Rational number :- Number that can be represented as p/q form is called a rational number.

Irrational number :- Number that cannot represented as p/q form is called irrational number.