Question

prove √31 is an irrational number​

Answer 1

Answer:

Let √31 be a rational number 

Therefore, √31= 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= 31q²                                                                                    ...(1)

Clearly, √31 is a factor of 31q²

⇒ 31 is a factor of p²                                                                    [since, 31q²=p²]

⇒ 31 is a factor of p

 Let p =31m for all m ( where  m is a positive integer)

Squaring both sides, we get 

            p²= 961m²                                                                                          ...(2)

From (1) and (2), we get 

           31q² = 961m²      ⇒      q²=31m²

Clearly, 31 is a factor of 2m²

⇒       31 is a factor of q²                                                      [since, q² = 31m²]

⇒       31 is a factor of q 

Thus, we see that both p and q have common factor 31 which is a contradiction that H.C.F. of (p,q)= 1

     Therefore, Our supposition is wrong

Hence √31is not a rational number i.e., irrational number.

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