chapter  20
10 Pages

Two Dimensional Numbers

WithMalcolm E Lines

Let us pause once again to rethink exactly what we mean by number. We recognized at the outset of this book that number is a rather abstract idea, but one which is useful in answering questions like ‘how many?’, ‘how long?’, or ‘how far?’ Through the pages we have increased the degree of sophistication of the concept of number until now, with integers, fractions, and irrationals, we can express the position of any point on a ‘number line’ at least in principle. Unfortunately this ‘linear’ system of numbers is still not satisfactory. It is not complete. By using it we can always add, subtract, multiply and divide, but we cannot always form powers or roots. This means that there are still arithmetic questions we can ask which do not have answers in terms of the number system we have so far developed. Look back at Table 12 in the last chapter, for example, and you will see that there are no geometric numbers (or powers) which correspond to negative numbers. We might therefore ask ‘what is the geometric number (or logarithm) of —1?’ At an even more elementary level we might also ask ‘what number, when multiplied by itself, makes —2?’ In other words if we search for quantities like ln(— 1) and ( − 2 ) or, equivalently, for solutions of equations like e x = − 1 ,