The classical theory of arithmetic arose in the works of Goedel, Turing, Church, and Kleene, and was an exceptionally successful framework for theoretical computer science. However, the thesis of this book is that it provides an unsuitable basis for modern scientific calculus where most of the algorithms are real digital algorithms.

The aim of this book is to develop a formal account theory that integrates the main themes of classical theory and which applies more directly to problems in mathematics, numerical analysis and scientific computing. Along the way, the authors consider fundamental problems such as: * Is the appointment of Mandelbrot alarming? * For simple quadratic maps, is Julia set as a stop group? * What is the real complexity of the Newton method? * Is there an algorithm to determine the backpack problem in the plolnomial number of steps? * Is Hilbert Nullstellensatz insurmountable? * Is the problem of locating the real zero-class four intractable polynomial? Is linear programming detectable? The book is divided into three parts: the first part provides an extensive introduction and then confirms the basic NP-complementeness theories of Cook-Karp and extends them to more general number fields as real and complex numbers.

The later parts of the book develop a formal mathematical theory that integrates the main themes of classical theory and which apply more directly to problems in mathematics, numerical analysis, and scientific computing.