On Thursday, we began the chapter on Real Numbers. We are taking the axiomatic approach to defining the set of real numbers. I demonstrated the difficulty of defining Real Numbers in the techniques that have been typically used in their earlier classes, such as by decimal expansion or as points on the real number line. One of the things we discussed was the ambiguity of decimal expansions, for example
I also alluded to the method of actually constructing Real Numbers through the method of Dedekind cuts but I have chosen to follow the text by describing the properties that uniquely define the set.
We began with with the field axioms and proved all many of the common results that follow from them, such as the fact that the product of the additive identity with any other real number must be the additive identity. This is the reason that the additive identity cannot have a multiplicative inverse. It was a fairly difficult concept to grasp for the students that we are not necessarily talking about “numbers” at this point but simply a set of objects that obey a certain set of properties. When I write 1 or 0 on the board, it is very helpful to keep in mind that these aren’t just the typical numbers that we’ve always thought them to be, but instead they are symbols used to represent the multiplicative identity and the additive identity, respectively. They are simply the objects that when you multiply or add to another element of the set of real numbers, you get what you started with. I find it very exciting to be developing their ability to think abstractly. I really do love this course, even if a few of them seem to get glossy-eyed at times.
Nevertheless, I am a little saddened by the fact that this is the last undergrad math course for many of them. I know at least a few should end up at graduate school but some may not have any other math courses beyond this point. Plus, most of them have heard all my jokes and stories so they can quote the punchline before I finish setting them up. I sometimes feel like a stand-up comedian with an audience that’s heard all my material. I guess I really need to expand my repertoire.
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