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V.14. The Fundamental Theorem of Arithmetic 699 enough to stop the path of P (r e i ) going around zero four times. Next, let us consider what happens when r is very small. Then P (r e i ) is very close to 9, whatever the value of , since (r e i ) 4 , (r e i ) 2 , and (r e i ) are all small. But this means that the path traced out by P (r e i ) does not go around zero at all. For any r we can ask how many times the path traced out by P (r e i ) goes around zero. What we have just established is that for very large r the answer is four and for very small r it is zero. It follows that at some intermediate r the answer changes. But if you gradu- ally shrink r , the path traced out by P (r e i ) varies in a continuous way, so the only way this change can come about is if for some r the path crosses 0. This gives us the root we are looking for, since the path consists of points of the form P (r e i ) and one of these points is 0. of evidence that the complex number system is, in fact, natural, and natural in a profound way. It states that, within the complex number system, every poly- nomial has a root. In other words, once we introduce the number i, then not only can we solve the equation x 2 + 1 = 0, we can solve all polynomial equations (even if the coefficients are themselves complex). Thus, when one defines the complex numbers, one gets much more out of them than one puts in. It is this that makes them seem not an artificial construction but a wonderful discovery. For many polynomials it is not hard to see that they have roots. For example, if P (x) = x d - u for some pos- itive integer d and some complex number u, then a root of P will be a dth root of u. One can write u in the form r e i , and then r 1/d e i/d will be such a root. This means that any polynomial that can be solved by a