Use proof by Contradiction to prove that the sum of an irrational number and a rational number is irrational.

Chesley 2020-11-09 Answered
Use proof by Contradiction to prove that the sum of an irrational number and a rational number is irrational.
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Usamah Prosser
Answered 2020-11-10 Author has 86 answers

A real number r is rational if and only if there exists two integer a and b such that r=a/b ( b is not equal to zero) Let us assume that x is an irrational number and y is a rational number . Suppose for the sake of contradiction that x+y is not irrational ya/b since s is irrational , x cannot be written as the ratio of two integers Since x+y in not irrational, x+y is rational . By the definition of rational, there exists integers c and d not equal to zero such that x+y=c/d
since y=a/b
x+a/b=c/d
x=c/da/b
x=(bcad)/bd Since a,b,c,d are integers bc - ad and bd are also integers. Moreover bd is zero as b and d are both non zero. However , this then implies that x is a rational number (as it is written as the ratio of tow integers). which is in contradiction with the fact that with the fact that x is an irrational number. Thus our supposition that x+y is not irrational is false. Which means that x+y is irrational .

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