Tuesday, April 3, 2018

algebra precalculus - Sum of irrational numbers



Well, in this question it is said that 1003+2+10032, and the owner asks for "alternative proofs" which do not use rational root theorem. I wrote an answer, but I just proved 1003+2Q and 10032Q, not the sum of them. I got (fairly) downvoted, because I didn't notice that the sum of two irrational can be either rational or irrational, and I deleted my (incorrect) answer. So, I want help in proving things like 5+7Q, and (1+π)πQ, if there is any "trick" or rule to these cases of summing two (or more) known irrational numbers (without rational root theorem).




Thanks.


Answer



To prove that 5+7 is irrational:



5+7=ab



a2b2=12+35



a212b2b2=35




35=(a212b2)2(b2)2



35|a212b2



352|(a212b2)2



352|(b2)2



Both the numerator and denominator are multiples of an even power of 2. Contradiction.




The method can be extended to many other sums of nth roots.


No comments:

Post a Comment

analysis - Injection, making bijection

I have injection f:AB and I want to get bijection. Can I just resting codomain to f(A)? I know that every function i...