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Can you get a rational nonzero number adding or subtracting 2 irrational numbers? If so, do you have examples?
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 $\displaystyle 2+\sqrt{2}, 3+\sqrt{2}$).
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 $\displaystyle A=\alpha.\alpha_0\alpha_1\alpha_2... , B=\beta.\beta_0\beta_1\beta_2... $, then for the sum A+B we have the following possibilities:
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a) A+B is non-terminating and unperiodical.
Then, it is irrational.
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b) A+B is terminating.
Then, there exists a minimum index k, such that $\displaystyle \alpha_n =-\beta_n, \ \forall \ n\geq k.$ This means that
$\displaystyle A+B=(\alpha+\beta).(\alpha_0+\beta_0)(\alpha_1+ \beta_1)... =
\gamma.\gamma_0\gamma_1\gamma_2...\gamma_{k-1}=C,$
this last number $\displaystyle C$ being rational (as terminating). Then, $\displaystyle A=-B+C$, and so the two irrationals were related at first hand!
(Example: $\displaystyle A=1+\sqrt{2}=2.141421..., B=1-\sqrt{2}=-0.141421...$. Then $\displaystyle A+B=2.141421... -0.141421... =2$.)
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c) A+B is periodical.
Then, there exists a minimum index k and a maximum m, such that the decimal expansion
$\displaystyle A+B= \gamma.\gamma_0\gamma_1\gamma_2...=\gamma.\gamma_0 \gamma_1...\gamma_{k-1}\gamma_k...\gamma_m\gamma_k...\gamma_m...$.
This means that for all naturals t, $\displaystyle \gamma_k=\gamma_{tm-k}$ or $\displaystyle \alpha_k+\beta_k=\alpha_{tm-k}+\beta_{tm-k},$ and so
$\displaystyle \alpha_k -\alpha_{tm-k}=-(\beta_k -\beta_{tm-k})$, for all t. This (again) sais that A and B are forehand related, as promised.
(Example: Consider the numbers $\displaystyle A=0.1212212221... $ and $\displaystyle B=0.1010010001... $They are unperiodical by construction, and so irrational. If $\displaystyle A=0.\alpha_0\alpha_1...$ and $\displaystyle B=0.\beta_0\beta_1...$, then $\displaystyle \alpha_0=\beta_0=\alpha_2=\beta_2=\alpha_5=\beta_5 ...=1,$ that is they are quite related. And, as if by magic, $\displaystyle A+B= 0.222...$, which is periodical, and so very rational.)
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Sorry for the long answer, hope it was worth it.