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We state Peano’s axioms for the positive integers and show how all the properties we have stated for the positive integers can be derived from those axioms. We show how to assign a number to any finite set. We discuss various objections to the way we have discussed numbers.
The axiom of induction is central to the study of the positive integers. We obtain equivalent forms of the axiom and use it to obtain definition by induction. We use induction to analyse various algorithms and to study primes and prime factorisation.
We discuss why fractions are important and why the modern notation is better than its various predecessors. We introduce the notion of an equivalence class and use it to define the positive rationals and obtain their properties.
The quaternions of Hamilton show that it possible to generalise the complex numbers, but only at the expense of introducing unexpected behaviour. It is thus reasonable to extend no further than the complex numbers. However the quaternions provided the inspiration for vectors.
We show that the rationals will not support calculus. Using the intermediate value theorem, we discuss the properties required by a number system to allow us to do calculus. We state the fundamental axiom of analysis and obtain various equivalent forms of that axiom.