Is the sum of an irrational number and rational number always irrational?

Yes.

It can be proved by contradiction.

Let:
a - a rational number
b - an irrational number
c - the sum of a and b



Let assume that c is a rational number. Then a and c can be expressed as fractions with integer numerator and denominator:
      where



Since are all integers, then the products and the difference are integers as well. It means that the number is a rational number, but this on the other hand contradicts the earlier assumption that is an irrational number. Therefore must be an irrational number.

Just turn the whole gym into pie

Google it that might work