proove that sum of rational and irrational is always an irrational

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A rational number can always be represented by a fraction p/q where p and q are integers.

An irrational number cannot be so represented. Let √n represent an irrational number where n is positive and not a perfect square. The sum of rational and irrational is p/q+√n=(p+q√n)/q. The numerator is irrational since q is an integer and q√n is a multiple of an irrational number, which is also irrational, and the irrational cannot be expressed as the quotient of two integers. So p/q+√n can be represented by √m, an irrational number.

 

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