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# Class Contents

###
What is Analysis and Why Study It?

Analysis is one of the large divisions of modern mathematics. (Algebra,
applied mathematics, topology, and discrete mathematics are some others.)### Mathematical Rigor and Axioms

### Group Exercise: Archimedes's Property

- What is Analysis and why study it?
- Mathematical rigor and axioms
- Group exercise: a theorem of Archimedes
- Class discussion: How we think of the real numbers

Analysis is the study of *limits*. Anything in mathematics which
has limits in it uses ideas of analysis. Some of the examples which
will be important in
this course are sequences, infinite series, derivatives of functions,
and integrals.

As you know from calculus, limits are the basis of understanding integration and differentiation, and, as you also know from calculus, these things are the basis of everything in the world you could ever need to know.

Probably, when you took calculus, you spent a lot of time learning
techniques of how to integrate and differentiate, and actually learning
about limits was dealt with in a fairly short time.
`(see note)`
But limits are a very
rich, and very subtle concept.
So, if you like, you can think of a theme in this course as "lift up the hood of calculus
and take a look at what's inside". But you'll also find a lot of ideas
that are completely new, and that are tied in with thinking about what we
mean by a limit.

The idea of taking limits is very old, and just as old are some of the problems which limits can raise. The ancient Greeks calculated the area of a circle using a limiting argument. But they also knew of Zeno's Paradox , which uses a suble misunderstanding of limits to "prove" that all motion is impossible. The modern study of analysis grew out of a dissatisfaction with the early intellectual basis of calculus.

When calculus was first discovered, it enabled a very thorough
understanding of many different physical phenomena, from the motion of
the planets to a bouncing ball. What was unsatisfactory wasn't the
*results* it produced, but the mathematical arguments that calculus
was based on. These appealed to ideas of
*infinitesimals*
, which
mathematicians found impossible to tie down to a clear meaning.

Because the idea of limits was notorious for leading to paradoxes (like
Zeno's Paradox
), or else to two different answers, with
equally plausible justifications
, analysis was developed to put calculus and limits on a
firm footing. The heritage of analysis, for this reason, is *rigor*.
The way you can be confident that you are on the right path, is that
every little step can be checked carefully, and every assumption can be
traced back to a few simple rules (axioms).
You will probably find that this course puts more emphasis on rigor than
any previous math courses which you've had.

**Axioms** are a set of rules for a mathematical theory. Everything
you do in that branch of mathematics has to be able to be broken down
into steps which are all applications of the axioms.

The benefits of this are that you can analyze an argument in a proof carefully, and be completely confident that it's right, by checking that every step conforms to the axioms.

Thus, the purpose of having axioms in mathematics is to enable you to be confident that your proofs are correct. The next topic highlights some of the difficulties you can get into with even quite simple facts when you don't have clear basic principles (which is what axioms are) to appeal to.

As an illustration of the need for firm principles to base our work with real numbers on, consider the following theorem (which we will prove later on in Chapter 1).

Theorem. (The Archimedean Property of the Reals)Letxbe a real number. Then there always exists a whole numbernwhich is bigger thanx

`In groups of two or three, talk about the theorem, and how you would
prove it. Take turns to explain why it's true, while the others try to
spot what hidden assumptions you're making.`

The *truth* of the theorem isn't in doubt for a minute! The point
is that when you start to think about it, you begin to see how difficult
it is to prove it to your complete satisfaction.

Question: What is it that makes it so difficult to tie down this (let's face it, obvious) fact?

The next couple of sections will deal with building up the basic rules for the real numbers, but we'll come back to the Archimedean Property, and prove it in Proposition 2.3.