Considering
±√1±√2±√3±⋯±√2009
where you can replace each ± with + or −. Prove that there is at least one choice of signs such that the number is irrational.
How can I prove that?
Answer
Assume there is a choice of signs that makes the expression rational. If not, there are 22009≥1 choices that make the expression irrational. Let r be the rational. Now if you change the sign on √2 you have either r+2√2 or r−2√2. Each of these will be irrational.
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