(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

3. 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:

[tex]\frac{c}{d}[/tex] + [tex]\frac{e}{f}[/tex]*r is rational.

[tex]\frac{e}{f}[/tex]*r = -[tex]\frac{c}{d}[/tex]

r = -[tex]\frac{cf}{de}[/tex]

[tex]\frac{cf}{de}[/tex] 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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: *Proof* Sum of Rational and Irrational Numbers

**Physics Forums | Science Articles, Homework Help, Discussion**