Q33 of 131 Page 1

Prove that √3 - √2 is irrational.

Let √3 - √2 be a rational number, say r

Then √3 - √2 =   r 

On squaring both sides we have
    (√3 - √2)2 =   r 2
  3 - 2√6 + 2 =   r 2 
        5 - 2√6 =  r 2
           - 2√6 =  r 2 - 5
               √6 = -(  r 2 - 5)/2

Now -(  r 2 - 5)/2 is a rational number and √ 6 is an irrational number. 
Since a rational number cannot be equal to an irrational number. Our assumption that
√3 - √2 is rational is wrong. 

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