Prove that ?7+3/2 is irrational

Dear Student,

Please find below the solution to the asked query:

Let  732 is a rational number .
Then we we can represent it in form of pq , where p and q are co-prime integers, So
732    =  pq

pq  - ​7     =  32

Now squaring both side ,we get

pq  - ​7  )2322

p2q2 + 7 - 27pq  = 94

 p2q2 + 7 - 94 = 27 pq

p2q2 +  194 = 27 pq

4 p2 + 19 q24 q2 × q2 p = 7  

4 p2 + 19 q28 pq  = 7  

Hence 

7    is a rational number  .             p ,q are integers  4 p2 + 19 q28 pq   is a rational number


But we know that 7   is a irrational number , So that contradict fact that 7   is a irrational number .
So,
Our assumption is incorrect ,

Hence 732   is a irrational number                                (  Hence proved )

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