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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.
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:
Since