# Intermediate Analysis: Order Properties

During this morning’s class I surprised the students by letting them know that today was my turn to answer homework questions at the board. As I have mentioned in previous posts, the students are being randomly selected to present some of the homework problems at the board. For many of them this is a considerably stressful event and with today’s assignment being particularly challenging, I noticed that many were very relieved when I offered to answer some of the questions at the board. Plus, it saves me needing to grade those particular problems.

For the lecture portion of today’s class, we went over the order properties of the Real Numbers. I recall the first time I learned this material and was actually quite perplexed by the definition of the set of positive numbers. Of course, I had an intuitive concept of what I meant when I said a number was positive but that was dependent on an ordering that was "inherited" from concept of the number line. Instead of starting with a number line or a given ordering, we simply define a set $P$ that has the following properties:

1. $\forall \ a, b \in P, \ \ a+b \in P$
2. $\forall \ a, b \in P, \ \ a\cdot b \in P$
3. $\forall \ a \in \mathbb{R},$ exactly one of the following hold:
(i) $a \in P$
(ii) $-a \in P$
(iii) $a=0$

This set, which we define to be the positive real numbers, provides a complete ordering on the set of reals. One of the students asked a question that I remember asking myself. Why can’t the set $P$ be any other set, say, the negatives. The answer is that (ii) is violated when we consider the negative reals.

Next time, we begin considering the topology of the reals, looking at $\varepsilon$-neighborhoods, open and closed sets, cluster points, etc.