irrational + irrational = rational

tonebone10

Can someone prove that there exists x and y which are elements of the reals such that x and y are irrational but x+y is rational? Certainly, there are an infinite number of examples (pi/4 + -pi/4 for example) to show this, but how would you prove the general case?

Jameson

[tex]\frac{\pi}{4} + \frac{3\pi}{4} = \pi[/tex]

This is not a rational number. I don't believe you can add two irrational numbers and get a rational result. The only case I could think of would be something like [tex]\sqrt{5} - \sqrt{5} = 0[/tex].

I know you can multiply two irrational numbers to get a rational one, but like I said, I don't think addition can do this.


Jameson

tongos

ofcourse you can,
how about
(In6)/(In3) and (In1.5)/(In3) they add to give 2.

Jameson

Hmmm... good example tongos. Can you give an example though that doesn't have any division to express the irrational number?

tongos

how about 2 sqrts that add to equal one.

sqrt(x)+sqrt(y)=1
y=1-sqrtx+x if x is rational, and not a perfect square then y cant be rational.
sqrty is irrational

Icebreaker

Jameson,

What about [tex](\pi -1)+1[/tex]?

Although I don't know if [tex](\pi -1)[/tex] can be considered as an irrational number, or a "number".

Jameson

Pi is irrational.

Tongos - I don't understand what you mean... please explain and give an example.

Icebreaker

I wrote it wrong. I meant [tex](1-\pi)+\pi[/tex]

Jameson

Yeah, I agree. If you look at my first post I gave an example like that. You're basically just cancelling out the irrational number... can you show me one where you add them and they don't result in zero?

Icebreaker

[tex](1-\pi)+\pi=1[/tex]

The question is whether [tex](1-\pi)[/tex] can be considered as an irrational number.

Jameson

That is an irrational number and I agree it is a case that proves the inital topic. But what you're doing is saying [tex] a + C -a = C[/tex]

This is where "a" is the irrational number and C is any rational constant. Can you show me an example where you don't simply subtract the irrational number to get a rational one?

tongos

if there is no pattern in the numbers after the decimal point in x,
then when you add it to y to obtain a rational number, y would also have to be irrational, or have a no pattern in its numbers.

Jameson

Very true.

tongos

jameson, my example about the sqrts shows it

Jameson

Yeah, I get it now. Thank you for explaining.

Icebreaker

What about this for a general case?

Assume that the difference between a rational number and an irrational number is an irrational number. The sum of any such two irrational numbers would be a rational number.

All we have to do now is prove the assumption, which I think is easier.

edit: I just saw tongos' proposition, which I think is better than mine.

mathwonk

this was a homerwork problem freshman year in about my first homework. My idea was to take any decimal with non repeating entries, hence irrational, and then add to it thed ecimal whose entries were 9-the entry in the firstd ecimal. that gives .9999999 .... =1 a rational number. finally i realixzed that this mjust emant thnat for any rational number r and irrational number, we have r-x and x both irrational, hence (r-x) + x = r is rational.