# *Proof* Sum of Rational and Irrational Numbers

## Homework Statement

Prove by contradiction: If a and b are rational numbers and b != 0, and r is an irrational number, then a+br is irrational.

In addition, I am to use only properties of integers, the definitions of rational and irrational numbers, and algebra.

You guys should also know that I am new to proofs, so if I'm breaking convention in any blatant way, please lemme know.

## The Attempt at a Solution

First, I figure I need to find the negation of this statement:

Negation: There exists rational numbers a and b, b!=0, and irrational number r, such that a+br is rational.

I'm pretty sure this negation is correct, but I've been wrong before

Anyway, my proof would start like this:

Proof: To prove that for any rational numbers a,b,b!=0, and irrational number r, a+br is irrational, let's suppose not. Suppose there exists rational numbers a,b,b!=0 and irrational number r such that a+br is rational.
Let c,d,e,f be integers, by definition of rational, a+br can be rewritten as:

$$\frac{c}{d}$$ + $$\frac{e}{f}$$*r is rational.

$$\frac{e}{f}$$*r = -$$\frac{c}{d}$$

r = -$$\frac{cf}{de}$$

$$\frac{cf}{de}$$ can be written as a quotient of integers, so it is rational, therefore, our negation is false, therefore, our theorem is true?

I think I'm on the right track, but I know its messy and I might have made some logic mistakes...Also, my professor said that if we didn't have to explicitly say why b!=0, we probably weren't doing it right, and I never really did, so I have no idea...any help would be greatly appreciated.

edit-I guess if b was 0, that would make the quantity a+(br) 0, which leaves you with a, which is just a rational number

Mark44
Mentor
In a proof by contradiction, you assume that the original hypothesis is true and that the conclusion is false.

Your original hypothesis is "a and b are rational numbers and b != 0, and r is an irrational number". Your original conclusion is "a + br is irrational".

The negative of your original conclusion is that a + br is not irrational, or in other words that a + br is rational.

Now start by assuming original hypothesis is true and that a + br is rational (the negated original conclusion). You should arrive at a contradiction, from which you can conclude that when the original hypothesis is true, the original conclusion is true, as well.