NotesOnHomework5AndLongAssignment1

# NotesOnHomework5AndLongAssignment1 - Note on Homework 5 and...

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Unformatted text preview: Note on Homework 5 and first long assignment - Math 139 Instructor Mauro Maggioni Office 293 Physics Bldg. Office hours Monday 1:30pm-3:30pm. Web page www.math.duke.edu/˜ mauro/teaching.html This is to clarify the connection between Homework 5, the first long assignment, and why what we constructed in class was indeed the field of real numbers. In class we constructed a space C ( Q ) of equivalence classes of Cauchy sequences of rationals, with respect to the equivalence relation { a n } ˜ { a ′ n } is a n- a ′ n → 0. We can define R to be this set, but in order to be convinced that this matches what “we think R is”, some work is in order. The properties that we think of as characterizing R are that it is a field, it is ordered, it is Archimedean, and it is complete. Moreover, we are used to identify elements in R with decimal representations. At this point C ( Q ) is just a set. In Homework 5 we saw we can define naturally two operations, + and · , that in fact give us a field (...
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NotesOnHomework5AndLongAssignment1 - Note on Homework 5 and...

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