Proof the above Show that for any real number x x pi irrati

Proof the above

Show that for any real number x, x + pi irrational or x - pi is irrational.

Solution

Assume pi + x and pi - x are both rational and show that this leads to a contradiction.

pi + x = a/b and pi - x = c/d for some integers a, b, c and d where b and d are not zero.

(pi + x) + (pi - x) = a/b + c/d
pi = (ad + bc)/(bd)
ad+bc is an integer and bd is an integer not equal to zero which means that pi must be rational. This is a contradiction as it is known that pi is irrational. So our supposition is incorrect.

Therefore one or both of pi+x and pi-x must be irrational.