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Question

# Prove that positive or negative irrational number is always an irrational number

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Solution

## let suppose x is a number.We can prove x irrational by contradiction By definition of rational, we have x = a/b for some integers a and b with b ≠ 0.​​​​​​​​​​​​​​​​​​​​​ x = (a/b) But a and b are integers [since a and b are integers] and b ≠ 0 [by zero product property.] Thus, x is a ratio of the two integers a and b with b ≠ 0. Hence, by definition of ration x is rational, which is a contradiction. for negative Irrational number Assume, to the contrary, that For every irrational number x such that −x is rational. By definition of rational, we have −x = a/b for some integers a and b with b ≠ 0. ( By zero product property ) Multiply both sides by −1, gives x = −(a/b) = −a/b But −a and b are integers [since a and b are integers] and b ≠ 0 [by zero product property.] Thus, x is a ratio of the two integers −a and b with b ≠ 0. Hence, by definition of ration x is rational, which is a contradiction. This contradiction shows that the supposition is false and so the given statement is true. This completes the proof.

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