Introduction

For many years mathematicians used the "Real number line" with an imperfect understanding of how the system worked.
The Greeks (and earlier civilisations) used numbers to parametrise the straight line (this is called "measurement") but ran into problems because of their lack of a rigorous basis for the real numbers. The followers of Pythagoras (569BC to 475BC) ran into difficulies with what we now call irrationals (as we will see later) and Zeno of Elea (490BC to 425BC) published a collection of forty "paradoxes" including "The Arrow" and "Achilles" which highlighted the problems with this way of viewing the reals.

Following the invention of the calculus by Newton, Leibniz and others in the 17th Century and its application in many areas of mathematics and science it became necessary to put the system on to a "safer foundation". This was done by 19th century mathematicians who formulated rigorously the ideas of convergence and continuity as well as tightening up concepts like "function" which were previously only vaguely defined.

This course aims to show much of how they did it.

(See Some definitions of the concept of a function.)