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Question

If p, q are prime positive integers, prove that p+q is an irrational number.
 


Solution

Let √p+√q is rational number
A rational number can be written in the form of a/b ( where a and b are co-prime ie they have no common factor other than 1.)
√p+√q=a/b
√p=a/b-√q
√p=(a-b√q)/b

p, q are integers and we supposed that √q is rational then (a-b√q) /b is a rational number.
So, √p is also a rational number. 
But this contradicts the fact that √p is an irrational number. 
so, our supposition is false
√p+√q is an irrational number

Mathematics
RD Sharma
Standard X

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