Question

prove that the sum of a rational number and an irrational number is always irrational

Answer 1

Solution:- Assume x is an irrational number, and the sum of x and a rational a/b is a rational c/d, where a,b,c and d are integers (b,d≠0).Then x+a/b=c/d By subtraction, x=c/d–a/b, and x= cb-ad/bd. Since integers are closed under multiplication and subtraction, cb,ad and bd and cb-ad are integers making cb-ad/bd a rational number by definition.This is a contradiction to the given fact that x is an irrational number. The assumption is wrong.The sum of a rational number and an irrational number is an irrational number.
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