Realandcomplexnumbers
The construction of the complex numbers is much simpler and goes much more smoothly than the construction of the real numbers. Thefirst stage in producing the reals is development of the rationals, at which point we have to explain what is meant by a fraction. A fraction, such as 23 is just a pair of integers, which we represent in this familiar but peculiar manner. The idea of fractional parts is not difficult to understand, although the corresponding arithmetic takes real effort to master. Along the way your teachers explain in passing that such fractions as 23,46,69etc. are‘equal’–they are not the same number pairs but they do represent equal slices of pie. This is not hard to accept but it does draw our attention to the fact that a rational number is in reality an infinite set of equivalent fractions, each represented by a pair of integers. This sounds intimidating and we might prefer not to think too much about this, for the prospect of manipulating infinite collections of pairs of integers might leave us feeling uneasy. There is one saving grace in that any fraction has a unique reduced representation where the numerator and denominator are coprime, which can be got by cancelling any common factors in the fraction with which you originally began. Nonetheless, once you are familiar with the properties of fractions and the rules for using them, nothing should go wrong even though closer examination reveals that, as you do your sums, you are implicitly manipulating infinite collections of integer pairs.
It is tempting to cut through all this fretting about particular equations and simply declare that we already know what the real numbers are–they are the collection of all possible decimal expansions, both positive and negative. These are very familiar, in practice we know how to use them, and so we feel on safe ground.At least until we ask some very basic questions. The main feature of numbers is that you can add, subtract, multiply, and divide. But,for example, how are you supposed to multiply two infinite non-recurring decimals? We depend on decimals beingfinite in length so that you‘start from the right-hand end’, but there is no such thing with an infinite decimal expansion. It can be done, but it is complicated both in theory and in practice. A number system where you struggle to explain how to add and multiply does not seem satisfactory.