Question

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

Answer 1

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.