Below is a proof showing that the sum of a rational number and an irrational number is an irrational number.
Let a be a rational number and b be an irrational number.
Assume that a + b = x and that x is rational.
Then b = x – a = x + (–a).
x + (–a) is rational because _______________________.
However, it was stated that b is an irrational number. This is a contradiction.
Therefore, the assumption that x is rational in the equation a + b = x must be incorrect, and x should be an irrational number.
In conclusion, the sum of a rational number and an irrational number is irrational.