What is Mathematical Analysis?

Frontmatter

1. Numbers, Lines and Holes

Abstract

In school we learn about numbers, connections between numbers, and how to manipulate them to solve problems — that’s arithmetic and algebra. At the same time we learn about shapes and space, shapes being made from straight and curved lines — that’s geometry. Then at some future stage (earlier rather than later according to current trends) these two major threads are woven together with the realisation that equations can describe lines — and we have coordinate geometry. How’s that for a whirlwind summary?

John Baylis

2. Curves — Continuous, Discontinuous and Unimaginable

Abstract

In Chapter I we mentioned the usefulness of representing a relationship between two variables by its graph. It is worth spelling out exactly what this representation entails in a specific example, so let us return to y = x². We assume for the moment that x is allowed to be any real number, then y has to be the square of x.

John Baylis

3. Adding up Forever — Paradoxes at Infinity

Abstract

In Chapter 1 we mentioned sequences of rational numbers, in particular null sequences, during our discussion of the completeness property of the real number system. The main idea of this section is to say something about sequences of real numbers. We begin with a variety of examples:

(a)

1, 2, 4, 8, 16, 32,...

 

(b)

3/4, 4/5, 5/6, 6/7,...

 

(c)

0, 1, −1, 0, 1, −1, 0, 1, −1,...

 

(d)

1, −2, 3, −4, 5, −6,...

 

(e)

1, 1, 2, 3, 5, 8, 13, 21,...

 

(f)

9, −3, 1, −1/3, 1/9, −1/27, 1/81,...

 

(g)

1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13,...

 

(h)

√2 − √1, √3 − √2, √4 − √3, √5 − √4,...

 

(i)

(1 + 1/1)¹, (1 + 1/2)², (1 + 1/3)³,...

 

(j)

1.0001, (1.0001)²/2, (1.0001)³/3,...

 

John Baylis

4. Smooth or Spiky? — Differentiation

Abstract

In this chapter we assume that you have met the idea of differentiation, some of its techniques, and some of its uses like finding maximum values of functions. Our aim is to put examples of all three of these items on a firm logical footing, building on the ideas of Chapter 2. Like Chapter 2, it will yield some suitably wild examples which can be tamed by analytic techniques but which would be very difficult to cope with by purely descriptive informal methods.

John Baylis

5. Putting it all Together — Integration

Abstract

You may have met integration in two different guises — as something to do with differentiation in reverse and as something to do with areas under curves. The purposes of this chapter are:

(i)

to explain the connection between these two ideas;

 

(ii)

to show that area is a rather more problematic idea than an integral;

 

(iii)

to make a precise definition of the Riemann integral and explore some of its consequences.

 

John Baylis

6. A Brief Look at Further Developments of Analysis, and Suggestions for Further Reading

Abstract

Now that this book has given you a taste of analysis, and hopefully a taste for analysis, a natural question is ‘where next?’

John Baylis

Backmatter