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The real numbers are the only complete, linearly ordered field. Any other system of numbers we construct will either be incomplete, not linearly ordered, or not a field. Consider these three possibilities briefly.
The most fundamental aspect of any system of numbers is their algebra. Our intention is to consider number systems that can seriously be used in place of the reals. So number systems that are not fields, stray too far, we believe, from what makes the reals the reals. Thus, we avoid systems that are not fields (with one exception).
Linear order is another matter. There are many practical problems to which the geometry of a line does not apply. Therefore, we shall consider number systems that are complete fields, but not linearly ordered. To simplify this situation, we only consider systems with the geometry of n-dimensional space (for some integer n). With this restriction, we give a complete account of all such systems.
Finally, there are several interesting number systems that preserve the algebra and the geometry of the reals, but are incomplete. They appear in Part III.
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