New numbers between real numbers
Source: Iranian National Olympiad (3rd Round) 2007
September 10, 2007
algebrapolynomialquadraticscomplex numbersalgebra proposed
Problem Statement
Scientist have succeeded to find new numbers between real numbers with strong microscopes. Now real numbers are extended in a new larger system we have an order on it (which if induces normal order on ), and also 4 operations addition, multiplication,... and these operation have all properties the same as .
http://i14.tinypic.com/4tk6mnr.png
a) Prove that in this larger system there is a number which is smaller than each positive integer and is larger than zero.
b) Prove that none of these numbers are root of a polynomial in .