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I am currently doing a project on irrational and transcendental numbers and part of this project requires me to look at sums and products of irrational numbers.

I am aware that the sum of 2 irrational numbers can be rational or irrational but was wondering if anyone knew of a definite way to look at the numbers and say if their sum/product will be rational/irrational. Is there some sort of theorem than can be applied or is the only way of knowing just working it out?

Thanks in advance for any help.

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"requires me to look at sums and products of irrational numbers." so you don't ask for transcendental?

The sum or product of two irrational algebraic numbers is not necessarily irrational. Counterexamples:

$$ (2 +\sqrt 2)+(2 -\sqrt 2)= 4\\ (2 +\sqrt 2)(2 -\sqrt 2)= 2 $$

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  • $\begingroup$ If you have any input on the sum/product of two transcendental numbers then that will be very much appreciated too !! $\endgroup$
    – user610274
    Commented Jan 30, 2019 at 15:55
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    $\begingroup$ For the sum/product of two transcendental numbers there are no known rules. See Joses answer. $\endgroup$
    – Andreas
    Commented Jan 30, 2019 at 16:28
  • $\begingroup$ I thought that would be the case, any idea on how to show that though? $\endgroup$
    – user610274
    Commented Jan 30, 2019 at 16:29
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    $\begingroup$ Well, there are examples with transcendental numbers in both ways, e.g. for multiplication of two transcendental numbers: $\pi \cdot \frac{2}{\pi} = 2$ and $\pi \cdot \pi = \pi^2$ where the first result is non-transcendental and the second one is transcendental. So there are no clear rules. $\endgroup$
    – Andreas
    Commented Jan 30, 2019 at 16:41
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No, there is not. If there was, we would know whether $e+\pi$ is rational or not. But, in fact, that's an open problem.

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  • $\begingroup$ Oh yeah, that seems so obvious now. Thank you! $\endgroup$
    – user610274
    Commented Jan 30, 2019 at 15:43

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