Question

So that the reciprocal of rational number root 2 is irrational

Answer 1

Answer:

1. Let     $\mathbf{X=\frac{1}{\sqrt{2}}}$  is a rational number.

2.  $\mathbf{X=\frac{1}{\sqrt{2}}=\frac{p}{q}}$  where p and q is a prime number and no common factor  between p and q except 1.

3. $\mathbf{\frac{1}{\sqrt{2}}=\frac{p}{q}}$

   On squaring both side ,we get

   $\mathbf{\frac{1}{2}=\frac{p^{2}}{q^{2}}}$

   $\mathbf{\frac{q^{2}}{2}=p^{2}}$       ...1)

   It means q is divisible by 2

   So we can write

   q =2 m        ...2)

3. So we can write from equation 1) and equation 2)

   $\mathbf{\frac{(2m)^{2}}{2}=p^{2}}$

 

   $\mathbf{\frac{4\times m^{2}}{2}=p^{2}}$

   $\mathbf{2\times m^{2}=p^{2}}$       ...3)

4. Equation 3) can be written as

    $\mathbf{m^{2}=\frac{p^{2}}{2}}$           ...4)

 

   From above equation 4), it is clear that p is divisible by 2 because m is a   integer.

  So we can write

   p =2 n               ...5)

5. From equation 2) and equation 5), it is clear that p and q is common factor, which is 2.

6. This make our assumption wrong that p and q is no common factor.

7. So if p and q have common factor, it make our assumption that x is a rational number.

8. Means X is a irrational number.

9. So $\mathbf{\frac{1}{\sqrt{2}}}$  is a irrational number.