Monday, 11 March 2013

summation - For pmsqrt1pmsqrt2pmsqrt3pmcdotspmsqrt2009, show there is a choice of signs such that it is irrational




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 220091 choices that make the expression irrational. Let r be the rational. Now if you change the sign on 2 you have either r+22 or r22. Each of these will be irrational.



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