# Thread: Simple Irrational Number Proof

1. ## Simple Irrational Number Proof

Prove the statement: If x is irrational and y is irrational, the x + y is irrational.

Contrapositive: If x + y is rational, then x is rational or y is rational.

My best guess at an approach is to prove the contrapositive. This is how I would start:

Proof:
Suppose x + y is rational. Then, by defn., x + y = p/q for some intergers p,q where q ≠ 0...

I'm not sure what to do from here. How do I deduce that x is rational or y is rational?

2. ## Re: Simple Irrational Number Proof

What you are trying to prove is FALSE:

$\sqrt{2} + (-\sqrt{2}) = 0$

and 0 is certainly rational.

3. ## Re: Simple Irrational Number Proof

Wow, duh! I think I'm going to throw up. Thanks..

4. ## Re: Simple Irrational Number Proof

About the only "non-trivial" assertion we have along these lines is:

If $x$ is rational, and $y$ irrational, then $x+y$ is irrational.

If $x$ is rational and $y$ is irrational, but $x+y$ is rational, we then have $y = (x+y) - x$ is rational (since the rationals form an abelian group under addition), contradiction.

5. ## Re: Simple Irrational Number Proof

Yea, I have to prove that one too. But I was told to prove the contrapositive here. That is, assume x + y is rational and x is rational and then duduce that y is rational. A little less intuitive but I suppose it still gets the job done.

6. ## Re: Simple Irrational Number Proof

It's the same proof:

If $x+y$ is rational, and $x$ is rational, then $-x$ is rational

(for if $x = \dfrac{p}{q}$, then $-x = \dfrac{-p}{q}$ and $-p$ is an integer if $p$ is).

Then, surely, $y = (x+y) + (-x)$ is rational, because the sum of any two rational numbers is rational:

let $r_1,r_2$ be rational, so:

$r_1 = \dfrac{a}{b}, a,b,\in \Bbb Z, b \neq 0$

$r_2 = \dfrac{c}{d}, c,d,\in \Bbb Z, d \neq 0$

then $r_1 + r_2 = \dfrac{ad + bc}{bd}$ and clearly $ad + bc$ is also an integer, and so is $bd$ and furthermore, $bd \neq 0$.

In general, any proof by contradiction (of an implication) can be turned into a "straight proof" of the contrapositive, and vice-versa.