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

Prove by contradiction. Your proof should be based only on properties of the integers, simple algebra, and the definition of rational and irrational.

If a and b are rational numbers, b does not equal 0, and r is an irrational number, then a+br is irrational.

2. Relevant equations

rational numbers are equal to the ratio of two other numbers

3. The attempt at a solution

I wrote a proof but am not sure it is correct. Please tell me what I did wrong and show me the way to do it right if this is not correct. My teacher indicated that we need to make use of the fact that b does not equal 0 (from a+br). Did I do that sufficiently as well?:

Proof:Suppose not. That is suppose that there exists rational numbers a and b, b does not equal zero, and irrational number r such that a+br is rational [We must deduce a contradiction].

By definition of rational, a = c/d, b= e/f , a+br = g/h for some integers c,d,e,f,g,and h with h,f,d, and b not equal to 0.

By substitution, a+b(r) = c/d +(r)( e/f) = g/h.

Solving for r gives: r = (fgd-chf) / (ehd)

Now fgd and chf are integers (being products of integers) and ehd does not equal 0 (by zero product property). Thus by definition of rational, r is rational which contradicts the supposition that r is irrational [ Hence the supposition is false and the statement is true].

**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: Discrete Math irrational and rational numbers proof

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