# Thread: proof rational + irrational

1. ## proof rational + irrational

How do I prove a rational +an irrational number = an irrational number?
thanks
Barry

2. Originally Posted by tinhats
How do I prove a rational +an irrational number = an irrational number?
thanks
Barry
It is quite simple - let $\displaystyle n \in \mathbb{R} \setminus \mathbb{Q}, p,q,r,s \in \mathbb{Z}$. Then, if $\displaystyle n+p/q=r/s$ we can quite easily manipulate the equality to get $\displaystyle n=x/y, x,y \in \mathbb{Z}$, a contradiction. I shall, however, leave the manipulation up to you.

3. Hello,
Originally Posted by tinhats
How do I prove a rational +an irrational number = an irrational number?
thanks
Barry
Here is a similar thing : http://www.mathhelpforum.com/math-he...rrational.html .

Put $\displaystyle x=1$, that is to say $\displaystyle x'=x''$ and you're done...

4. here's another example of 2 similar proofs:
http://www.mathhelpforum.com/math-he...bers-even.html

Halfway down the page is a proof that an uneven * even number is an even number.

5. Originally Posted by bmp05
here's another example of 2 similar proofs:
http://www.mathhelpforum.com/math-he...bers-even.html

Halfway down the page is a proof that an uneven * even number is an even number.
Hmm but yours is dealing with even/uneven, while we're talking about rational/irrational here ?

6. Yes, well, I feel that any retort would probably be irrational at this stage.

7. Originally Posted by Moo
Hmm but yours is dealing with even/uneven, while we're talking about rational/irrational here ?
They may not be the same but they are similar.

In general:

Let $\displaystyle G$ be a group and $\displaystyle H$ be a subgroup. Then for any $\displaystyle x,y\in G,$ $\displaystyle x\in H$ and $\displaystyle y\notin H$ $\displaystyle \implies$ $\displaystyle xy\notin H.$

Proof is simple: Suppose $\displaystyle x$ (and therefore $\displaystyle x^{-1})\,\in\,H.$ If $\displaystyle xy\in H,$ then $\displaystyle y=x^{-1}(xy)\in H.$ Therefore by contrapositivity, $\displaystyle y\notin H\ \implies\ xy\notin H.$

In bmp05’s original example, $\displaystyle G$ is the addibive group of the reals and $\displaystyle H$ is the additive group of the rationals; hence rational + irrational = irrational. In the odd/even example, $\displaystyle G$ is the additive group of the integers and $\displaystyle H$ is the subgroup of the even integers – $\displaystyle \therefore$ even + odd = odd.

Further examples:

(i) Even permutation × odd permutation = odd permutation. (Take $\displaystyle G$ to be a symmetric group, $\displaystyle H$ to be the corresponding alternating group.)

(ii) Rotation × reflection = reflection. (Take $\displaystyle G$ to be a dihedral group, $\displaystyle H$ to be the subgroup of rotations.)