Question

Prove that route P + route q is irrational where, p, q are primes.

Answer 1

Let us assume that P + √q is a rational number.

And a rational number can be written in the form of a / b where a & b are co - prime numbers .

p + √q = a/b

√q = ( a/b ) - p

√q = ( a - bp )/ b

Since ,√q is an irrational number and
( a - bp ) /b is a rational number.

And an irrational number can't be equal to a rational number.

So , Our assumption is wrong

p + √q is an irrational number.

Answer 2

Let as assume that p + √q is a rational no.

then,

p + √q = a/b

√q = a/b - p

√q = ( a - bp )/b

Since √q is irrational and ( a - bp )/b is a rational number...and irrational number and rational number can't be equal so our assumption is wrong.

p + √q is an irrational number.