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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 ).
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:
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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 This means that
this last number being rational (as terminating). Then, , and so the two irrationals were related at first hand!
(Example: . Then .)
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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
.
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.)
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Sorry for the long answer, hope it was worth it.