Prove that a sum of rational number & Irrational number is irrational

2 + root2 remains as 2 + root 2 only
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Suppose not. [We take the negation of the theorem and suppose it to be true.] Suppose ∃ a rational number x and an irrational number y such that (x − y) is rational. [We must derive a contradiction.] By definition of rational, we have

                                 x = a/b        for some integers a and b with b ≠ 0.

and                     x − y = c/d        for some integers c and d with d ≠ 0.

By substitution, we have

                                                x  −  y  =  c/d

                                             a/b  −  y  =  c/d

                                                        y  =  a/b  −  c/d

                                                            = (ad − bc)/bd

But (ad − bc) are integers [because a, b, c, d are all integers and products and differences of integers are integers], and bd ≠ 0 [by zero product property]. Therefore, by definition of rational, y is rational. This contradicts the supposition that y is rational. [Hence, the supposition is false and the theorem is true.]

And this completes the proof.

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Sum of 5,√6 is irrational
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