Am 04.03.2012 14:09, schrieb prateek papriwal:
also the addition of two positive irrational number is also irrational .
A trivial counterexample:
2 +/- sqrt(2) are positive and irrational, yet their sum is 4, which is
rational.
There are less trivial cases.
Such as the sum of 1/(sqrt2-1) and 2-sqrt(2), which is 3.
(Taken from Wikipedia and trivially modified, but unvalidated.)
In more generality, I'm a bit concerned that we're investing a lot of
effort into building a rationality test that works only for a small
class of numbers. It would probably be better to make this extensible,
so that people can add more algorithms as we pick up techniques.
Background: Testing for rationality in general is an undecidable
problem. It is proven to be impossible to have an algorithm that will
work for arbitrary formulae. There are two possible failure modes:
- The algorithm is correct but may run into an endless loop.
- The algorithm is incorrect.
- The algorithm returns "rational", "irrational", or "don't know".
The third behaviour is not ideal, but I doubt the other two are acceptable.
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