:confused: :confused: Help :confused: :confused:

Can you get a rational nonzero number adding or subtracting 2 irrational numbers? If so, do you have examples?

Printable View

- Jul 27th 2005, 08:27 AMnotamathnerdMore Irrational Number ??>
:confused: :confused: Help :confused: :confused:

Can you get a rational nonzero number adding or subtracting 2 irrational numbers? If so, do you have examples? - Jul 27th 2005, 10:17 AMRebesques
Well, if you substract the irrationals , , you 'll get (the very rational) 1 :p

- Jul 27th 2005, 10:24 AMnotamathnerd
you didn't use 2 irrational numbers, 3 is rational

- Jul 27th 2005, 10:34 AMRebesques
Do not judge too fast: :p

Cheap!!?? :cool: - Jul 27th 2005, 10:50 AMnotamathnerd
but you need to add the irrationals together

- Jul 27th 2005, 11:00 AMRebesquese?
Just did :)

Sorry, but I see kinda stress here...

is very irrational, and so is the other fellow. Tell me what you do not understand. - Jul 27th 2005, 12:57 PMnotamathnerd
in the number model 2 + sqrt 2, only sqrt 2 is irrational. 2 as an addend is rational.

- Jul 27th 2005, 01:06 PMRebesques
...So? The number is irrational, still. What troubles you? :)

- Jul 27th 2005, 01:13 PMMathGuru
the point is you are simply saying that

I don't think that is the point of the question. - Jul 27th 2005, 02:43 PMnotamathnerd
the point of the question is to use two addends, both irrational, and get a sum that is rational.

can it be done? - Jul 27th 2005, 03:32 PMMathGurue^pi*i
I don't know if you can do

irrational + irrational = rational

but you can do

(irrational)^(irrational*i)=rational

for example

- Jul 27th 2005, 11:55 PMRebesquese?
I think I get the idea. The lad asks whether two -irrelevant- irrationals, can add up to a rational. The

**general**answer is**no**. Only with trickery can this happen (like the example with ).

To see this, we remember that a number is rational, if and only if it has a terminating or a periodical decimal expansion. Therefore, if we have two irrationals , then for the sum A+B we have the following possibilities:

------------------------------------------------------------------

**a)**A+B is non-terminating and unperiodical.

Then, it is irrational.

--------------------------------------------------------------------

**b)**A+B is terminating.

Then, there exists a minimum index k, such that This means that

this last number being rational (as terminating). Then, , and so**the two irrationals were related at first hand**!

(**Example**: . Then .)

---------------------------------------------------------------------

**c)**A+B is periodical.

Then, there exists a minimum index k and a maximum m, such that the decimal expansion

.

This means that for all naturals t, or and so

, for all t. This (again) sais that**A and B are forehand related**, as promised.

(**Example**: Consider the numbers and They are unperiodical by construction, and so irrational. If and , then that is they are quite related. And, as if by magic, , which is periodical, and so very rational.)

------------------------------------------------------------------------

Sorry for the long answer, hope it was worth it. :eek: - Jul 31st 2005, 12:22 AMRebesquese?
Nobody agrees/disagrees? :) :(

- Jul 31st 2005, 11:55 AMMathGuru
I like the unperiodical by construction trick. But it is still an artificial construction.

So I agree, and I appreciate the trick, and you have answered the question well.