Question

prove that underoot 2 is an irrational number

Answer 1

Step-by-step explanation:

do long division for root 2

it would never end

thus it is a irrational number..

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Answer 2

Lets assume that √2 is a rational number

⇒ √2 = p/q where a, b are co primes.

On squaring both sides, we get 

                   p²= 2q² ...(1)

Clearly, 2 is a factor of 2q²

⇒ 2 is a factor of p²  [since, 2q²=p²]

⇒ 2 is a factor of p

Let p =2 m for all m ( where  m is a positive integer)

Squaring both sides, we get 

p²= 4 m²   ...(2)

From (1) and (2), we get 

2q² = 4m²      

⇒q²= 2m²

Clearly, 2 is a factor of 2m²

⇒2 is a factor of q²  [since, q² = 2m²]

⇒ 2 is a factor of q 

Thus, we see that both p and q have common factor 2 which is a contradiction that p amd q are co primes.

Therefore, our assumption is wrong.

Hence √2 is an irrational number.

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