Section 1.1 Lesson One
Get a pad of paper and write down the answer to this question: What is . . . No, really. We’re serious. Get a writing pad. We’ll wait.
Got it? Good. Now write down your answer to this question: What is a number? Don’t think about it. Don’t analyze it. Don’t consider it. Just write down the best answer you can without thinking. You are the only person who ever needs to see what you’ve written.
Done? Good.
Now consider this: All of the objects listed below are “numbers” in a sense we will not make explicit here. How many of them does your definition include?
- \(\displaystyle 1\)
- \(\displaystyle -1\)
- \(\displaystyle 0\)
- \(\displaystyle 3/5\)
- \(\displaystyle \sqrt{2}\)
- \(\displaystyle \sqrt{-1}\)
- \(\displaystyle i^i\)
- \(\displaystyle e^{5i}\)
- \(4+3i-2j+6k\) (this is called a quaternion)
- \(\dx{x}\) (this is the differential you learned all about in calculus)
- \(\begin{pmatrix} 1\amp 2\\-2\amp 1 \end{pmatrix}\) (yes, matrices can be considered numbers).
Surely you included \(1\text{.}\) Almost surely you included \(3/5\text{.}\) But what about \(0?\) \(-1?\) Does your definition include \(\sqrt{2}?\) Do you consider \(\dx{x}\) a number? Leibniz did. Any of the others? (And, yes, they really are all “numbers.”)
The lesson in this little demonstration is this: You don’t really have a clear notion of what we mean when we use the word “number.” And this is fine. Not knowing is acceptable.
A principal goal of this course of study is to rectify this, at least a little bit. When the course is over you may or may not be able to give a better definition of the word “number” but you will have a deeper understanding of the real numbers at least. That is enough for now.
