An informative and useful account of complex numbers that includes historical anecdotes, ideas for further research, outlines of theory and a detailed analysis of the ever-elusory Riemann hypothesis. Stephen Roy assumes no detailed mathematical knowledge on the part of the reader and provides a fascinating description of the use of this fundamental idea within the two subject areas of lattice simulation and number theory.Complex Numbers offers a fresh and critical approach to research-based implementation of the mathematical concept of imaginary numbers. Detailed coverage includes: - Riemann's zeta function: an investigation of the non-trivial roots by Euler-Maclaurin summation. - Basic theory: logarithms, indices, arithmetic and integration procedures are described. - Lattice simulation: the role of complex numbers in Paul Ewald's important work of the I 920s is analysed. - Mangoldt's study of the xi function: close attention is given to the derivation of N(T) formulae by contour integration. - Analytical calculations: used extensively to illustrate important theoretical aspects. - Glossary: over 80 terms included in the text are defined. - Offers a fresh and critical approach to the research-based implication of complex numbers - Includes historical anecdotes, ideas for further research, outlines of theory and a detailed analysis of the Riemann hypothesis - Bridges any gaps that might exist between the two worlds of lattice sums and number theory