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18.014ESG Problem Set 2 Pramod N. Achar

Summary: 18.014­ESG Problem Set 2
Pramod N. Achar
Fall 1999
1. Exercise 2 in Section I 3.5 of Apostol. (You may use Axioms 1­9 and
Theorems I.1­I.25.)
2. Let C denote the field of complex numbers. (Of course, we haven't defined
C rigorously in class; this problem requires you to use what you know
about complex numbers in "real life.") Show that there is no subset
C with respect to which the order axioms are satisfied. (Hint:
Suppose there is such a C+
, and consider i =

-1. If i C+
, where must
-i = i · i · i lie? What if i / C+
? Derive a contradiction to one of the
order axioms.)
3. (Optional) We have assumed in this class that there exists a certain set


Source: Achar, Pramod - Department of Mathematics, Louisiana State University


Collections: Mathematics