Question

prove that underroot 2 is irrational number​

Answer 1

Answer:

Step-by-step explanation:

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.

HOPE IT HELPS YOU.

Answer 2

Step-by-step explanation:

lets assume that root 2 is rational,

so,

    √2 = p/q

       squaring on both sides

       √2² = (p/q)²

        now solve

hope this helps........

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