Complete the following questions before our next class.
Exercise 6C.3 1be, 2ade, 3bc, 4afg
If you’re interested in looking in more detail at the \(\leq\) relation, here’s a fun puzzle to consider.
The real numbers are totally ordered by the \(\leq\) relation, that is, \[\text{(1) for any } a,b \in \mathbb{R},\text{ we have } a \leq b \text{ or } b \leq a.\] We also have the following properties, which are true for any \(a,b,c \in \mathbb{R}\)
\[\text{(2) if } 0 \leq a \text{ and } b\leq c, \text{ then } ab\leq ac, \text{ and}\]
\[\text{(3) if } b\leq c, \text{ then } b+ a\leq c+a.\]
What happens if we try to extend the \(\leq\) relation to the complex numbers? If we can, we would want to do so without “breaking” properties 1–3. Can we extend our ordering relation to the complex numbers without running into trouble in this way?