Question

QNo1 . Prove the under root 2 as irrational number​

Answer 1

Answer:

To prove: √2 is an irrational number.

Proof:  

Let us assume that √2 is a rational number.  

So it can be expressed in the form p/q where p, q are co-prime integers and q≠0  

√2 = p/q  

Here p and q are coprime numbers and q ≠ 0  

Solving  

√2 = p/q  

On squaring both the side we get,  

⇒2 = (p/q)2  

⇒ 2q2 = p2………………………………(1)  

p2/2 = q2  

So 2 divides p and p is a multiple of 2.  

⇒ p = 2 m  

⇒ p² = 4 m² …………………………………(2)  

From equations (1) and (2), we get,  

2q² = 4 m²  

⇒ q² = 2 m²

⇒ q² is a multiple of 2  

⇒ q is a multiple of 2

Hence, p, q have a common factor 2. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number  

√2 is an irrational number.

I hope so that my answer is useful to you. If yes then please mark me the brainliest!!!!

Answer 2

There must always be a simplest rational number and the original assumption that √2 is equal to p/q does not obey this rule. So it can be stated that a contradiction has been reached. ... Hence √2 is not a rational number. Thus, Euclid succeeded in proving that √2 is an Irrational number.