Part D
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 n0. The process for rewriting the sum for
x is shown.
Statement
A T
=− (†) (;) – (4) (†)
substitution
Reason
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?