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Unformatted text preview: NOTES ON REAL NUMBERS In these notes we will construct the set of real numbers. Why would we want to do this you may ask? Well, mathematicians want mathematics to be based on a solid foundation, such as set theory. Adopting this point of view then requires us to give precise definitions of the different things we use in math; for example, weve seen how to give precise definitions of ordered pair and function in terms of set theory. So somewhere along the way, someone had to give a precise definition of the real numbers, built up from set theory as well. We will give one such construction here. At the end we will have a precise definition of R , but just as we did for ordered pairs and functions, we will then forget about the precise construction and just use real numbers as we are used to them. Still, this is a good thing to go through since it gives some down to earth applications of some of the tools we have been studying, especially equivalence relations, to create some interesting mathematics. As a word of caution, this material is not easy and may seem very confusing at first. Because of this, this material is not something that quizzes nor the rest of the course will focus on. However, this material does give a good glimpse into the kinds of things that modern mathematicians do and the kinds of things that modern mathematical research is concerned with, although this specific material was first developed towards the end of the 19th century and so is not exactly modern. You can definitely expect to see the types of constructions and ideas described here in later upper division courses, and possibly in your future mathematical careers. The only assumptions we will make are that the set of natural numbers N exists and that we know how to add and multiply natural numbers. In fact, that this set and these operations exist follows from commonly accepted axioms of set theory (in particular, the axiom of infinity), but we will just take their existence for granted. It is truly amazing that such a simple set of axioms gives rise to the rich complexity of the real numbers! We will first construct the set of integers. We want to think of ( a,b ) N N as representing the integer a- b . The first problem with this is that there are many ways of representing an integer as a difference a- b ; for example, (2 , 7), (1 , 6), and (95 , 100) all give the same integer, namely- 5. So, to deal with this we will define an equivalence relation on N N in a way so that equivalent pairs correspond to the same integer. Then, even though different pairs give the same integer, there will in fact be a unique equivalence class which gives a certain integer....
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