Question

Prove that root 2 is a irrational Number​

Answer 1

Answer:

Let √2 be a rational number..

Therefore, √2= p/q  [ p and q are in their least terms i.e., HCF of ( p,q )= 1 and q ≠ 0..

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 H.C.F. of (p,q)= 1

Therefore, Our supposition is wrong

Hence √2 is not a rational number i.e., irrational number.

Step-by-step explanation:

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