Why do we need the real numbers? How should we construct them?
These questions arose in the nineteenth century, along with the
ideas and techniques needed to address them. Nowadays it is
commonplace for apprentice mathematicians to hear 'we shall assume
the standard properties of the real numbers' as part of their
training. But exactly what are those properties? And why can we
assume them? This book is clearly and entertainingly written for
those students, with historical asides and exercises to foster
understanding. Starting with the natural (counting) numbers and
then looking at the rational numbers (fractions) and negative
numbers, the author builds to a careful construction of the real
numbers followed by the complex numbers, leaving the reader fully
equipped with all the number systems required by modern
mathematical analysis. Additional chapters on polynomials and
quarternions provide further context for any reader wanting to
delve deeper.
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