Real and complex numbers
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 verybasic 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 being finite 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.
You maynd the foundational questions raised above interesting or you may grow impatient with all the introspection as we seem to be making trouble for ourselves when previously all was smooth sailing. There is a serious point, however. Mathematicians appreciate that, whenever new mathematical objects are introduced, it important to construct them from known mathematical objects, the way, for instance, fractions can be thought of as pairs of ordinary integers. In this way, we may carefully build up the rules that govern the new extended system and know where we stand. If we neglect foundations completely, it will come back to haunt us later. For example, the rapid development of calculus, which was born out of the study of motion, led to spectacular results, such as prediction of the movement of the planets. However, manipulation of innite things as if they werenite sometimes provided amazing insights and at other times patent nonsense. By putting your mathematical systems on arm foundation, we can learn how to tell the difference. In practice, mathematicians often indulge in ‘formal’manipulations in order to see ifsome sparkling new theorem is in the offing. Ifthe outcome is worthy of attention, the result can be proved rigorously by going back to basics and by invoking results that have been properly established earlier.