Now examine the sum of a rational number, y, and an irrational number, x. The rational number y can be written as y =
, where a and b are integers and b ≠ 0. Leave the irrational number x as x because it can’t be written as the ratio of two integers.
Let’s look at a proof by contradiction. In other words, we’re trying to show that x + y is equal to a rational number instead of an irrational number. Let the sum equal
, where m and n are integers and n ≠ 0. The process for rewriting the sum for x is shown.
Statement Reason
substitution
subtraction property of equality
Create common denominators.
Simplify.
Based on what we established about the classification of x and using the closure of integers, what does the equation tell you about the type of number x must be for the sum to be rational? What conclusion can you now make about the result of adding a rational and an irrational number?