arrow_back The real problem we're grappling with here is that we don't have a clear idea in our minds of how to answer the question:

What is a real number?

That's exactly what we're going to be concentrating on through Chapter 1. It isn't going to be real easy, either, so you should hold onto the feelings of frustration you had while you were trying to prove Archimedes's theorem. The pay-off of the work we're going to do in Chapter 1 is that we'll be able to get a completely convincing argument to prove that theorem, and we can go through it step by step and see exactly which of our axioms is being used for each step.

Models for the Real Numbers

If you came up with any argument to prove the theorem, you needed along the way to come up with a model for the real numbers. Chances are it was either

• The Real number line, or else
• Real numbers as infinite decimals

Both of those are useful ways of thinking of the real numbers, and we use both of them when they are appropriate to help us understand the behavior of the real numbers.