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=t, where a and b are integers and bz0. 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 mand n are integers and n *0. The
process for rewriting the sum for x is shown.
Statement
*- (1) (2) (2) (4)
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?