Question

prove that 5√2is an irrational number

Answer 1

this is your answer

I hope you will understand

Answer 2

Assume the contrary that √2 is rational

  √2 =p/q,p and q are co-prime and q≠0

   √2 q=p

squaring

    (√2 q)²=p²

     2 q²=p²

           q²=p²/2

2 divides p²⇒2 divides p also (theorem 1)

        let p=2 m

                q²=(2 m)²/2

                   =4 m²/2

                  2 m²=q²

                     m²=q²/2

2 divides q²⇒2 divides q also (theorem 1)

    ∴ 2 is a common factor for p and q .

This contradicts the fact that p and q are not co-prime,

i.e., our assumption is wrong.

 ∴√2 is irrational.

Assume the contrary that 5√2 is rational .

      5√2=p/q, p and q are co-prime and q≠0

     √2=p/q÷5

    √2=p/5 q                      

Here p/5q is rational since p and q are integers

∴√2is rational

but √2 is irrational (proved above)

i.e., our assumption is wrong.

  ∴5√2 is an irrational number