Nick Trefethen FRS, Professor of Numerical Analysis at Oxford University, teaches a course for DPhil (PhD) students across all the science departments at the university. The course is distinctive for its exceptionally strong conceptual basis, focussing on fundamental ideas of numerical algorithms and the world of science and engineering they sustain. Historical and philosophical notes on many subjects make these lectures unusual, offering deeper understanding and new perspectives for anyone involved with numerical analysis or scientific computing.