IMI
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.
z + f ==
z + t = t = -f
z=-f
R=
Statement
HU () () () (#)
z = bm-on
H=
bm- an
br
-
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?