Friday, 16 December 2016

elementary number theory - Prove that sqrt2+sqrt3 is irrational




I have proved in earlier exercises of this book that 2 and 3 are irrational. Then, the sum of two irrational numbers is an irrational number. Thus, 2+3 is irrational. My first question is, is this reasoning correct?



Secondly, the book wants me to use the fact that if n is an integer that is not a perfect square, then n is irrational. This means that 6 is irrational. How are we to use this fact? Can we reason as follows:




6 is irrational



23 is irrational.



23 is irrational



2 or 3 or both are irrational.



2+3 is irrational.




Is this way of reasoning correct?


Answer



If 2+3 is rational, then so is (2+3)2=5+26. But this is absurd since 6 is irrational.


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