X Prefaces to the German Editions If we apply this formula to our example, we obtain the two solu- tions xi = 3 + 2\/2 and x2 = 3 - 2\/2. If you are interested in a numerical solution, you can pull out your handy pocket calculator (or perhaps you know how to com- pute square roots by hand) and obtain the decimal representations x\ — 5.828427... and x2 = 0.171572 — You could also use your cal- culator to verify that these values are in fact solutions to the original equation. A skeptic who wished to verify that the solutions derived from the formula are the exact solutions would have to substitute the expressions containing the square roots into the equation and demonstrate that the quadratic polynomial x2 — 6x + 1 = 0 actually vanishes—that is, assumes the value zero—at the values x = x\ and x = x2- The Solution of Equations of Higher Degree. It has long been known how to solve cubic equations such as x3 - 3x2 - 3x - 1 = 0 by means of a formula similar to the quadratic formula. Indeed, such formulas were first published in 1545 by Cardano (1501-1576) in his book Ars Magna. However, they are quite complicated, and have little use for numerical calculation. In an age of practically unlimited computing power, we can do without such explicit formulas in practical applications, since it suffices completely to determine the solutions by means of numeric algorithms. Indeed, for every such equation in a single variable there exist approximation methods that iteratively, that is, step by step, compute the desired solution more and more precisely. Such a procedure is run until the solution has reached an accuracy suitable for the given application. However such iterative numeric procedures are unsuitable when not only the numerical value of a solution is sought, such as x\ — 3.847322... in the previous example, but the "exact" value x1 = 1 + ^ 2 + ^ 4 . It is not only that such an algebraic representation possesses a certain aesthetic quality, but in addition, a numeric solution is insufficient if

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2006 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.