This book takes its readers on an exciting journey through the most important and powerful areas of contemporary numerical mathematics. The route is based on the 10 competition problems of the "SIAM 10 x 10-Digit Challenge," which Nick Trefethen from Oxford used to challenge his colleagues and their doctoral students worldwide in the spring of 2002: each of the highly nontrivial problems, formulated in just a few sentences, requires the computation of the first 10 digits of a real number. For each problem, a variety of solution methods are discussed, touching on nearly every significant technique in modern numerical mathematics. Many practical tips help the reader develop skills in solving numerical problems and learn to make a well-considered choice among competing methods.