A proof by contradiction is one in which we assume the opposite of what we want to prove is true, and then explain how that assumption leads to a contradiction.
Here, we want to prove that the sum of an irrational number and a rational number is irrational, so we're going to assume the sum is rational and see why that can't possibly be true.
The question sets up this equation as a starting point:
[tex]x+\frac{a}{b} =\frac{m}{n}[/tex]
and we're told that a, b, m, and n are all integers. We can now subtract [tex]\frac{a}{b}[/tex] from both sides to get the equation in the form
[tex]x=\frac{m}{n}-\frac{a}{b}[/tex]
And here's where we run into our contradiction: [tex]\frac{a}{b}-\frac{m}{n}[/tex], being the difference of two rational numbers, gives us a rational number as a result, but we also stated at the beginning that x was an irrational number. If we accept that, our equation now makes the absurd claim that
irrational number = rational number
which is like saying that peanut butter = jelly, black = white, or up = down; it just doesn't make sense!
Since our reasoning was airtight, the only thing that could've led us down this tragic path was our assumption that the sum of an irrational and a rational number was rational, so we know now that the sum of a rational number and an irrational number must be irrational.
