Let $\alpha , \beta$ and $c$ be positive numbers less than 1 , with $c$ rational and $\alpha , \beta$ irrational.\\
(A) The number $\alpha + \beta$ must be irrational.\\
(B) The infinite sum $\sum _ { i = 0 } ^ { \infty } \alpha c ^ { i } = \alpha + \alpha c + \alpha c ^ { 2 } + \cdots$ must be irrational.\\
(C) The value of the integral $\int _ { 0 } ^ { \pi } ( \beta \cos x + c ) d x$ must be irrational.