Professor Hua Loo-Keng is the first person to have undertaken the task of popularizing mathematical methods in China. As early as 1958, he proposed that the application of operations research methods be initiated in industrial production. With his students, Yu Ming-I, Wan Zhe Xian and Wang Yuan, Professor Hua visited various transportation departments to promote mathematical methods for dealing with transportation problems, and a mass campaign was organized by them and other mathematicians to advance and apply linear programming methods to industrial production in Beijing and in Shandong province. However, due to the fact that these methods have limited applications and their computation is rather complex, their popularization and utilization in China have so far been restricted to a small number of sectors such as the above mentioned transportation departments. In 1958 Hua Loo--Keng proposed the use of Input-Output methods in the formulation of national economic plans. Apart from publicizing this method, he carried out in-depth research on the subject. He also gave lectures on related non-negative matrix theory, pointing out the economic significance of various theoretical results.

to Number Theory Translated from the Chinese by Peter Shiu With 14 Figures Springer-Verlag Berlin Heidelberg New York 1982 HuaLooKeng Institute of Mathematics Academia Sinica Beijing The People's Republic of China PeterShlu Department of Mathematics University of Technology Loughborough Leicestershire LE 11 3 TU United Kingdom ISBN -13 : 978-3-642-68132-5 e-ISBN -13 : 978-3-642-68130-1 DOl: 10.1007/978-3-642-68130-1 Library of Congress Cataloging in Publication Data. Hua, Loo-Keng, 1910 -. Introduc- tion to number theory. Translation of: Shu lun tao yin. Bibliography: p. Includes index. 1. Numbers, Theory of. I. Title. QA241.H7513.5 12'.7.82-645. ISBN-13:978-3-642-68132-5 (U.S.). AACR2 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, reuse of illustra- tions, broadcasting, reproductiOli by photocopying machine or similar means, and storage in data banks. Under 54 of the German Copyright Law where copies are made for other than private use a fee is payable to "VerwertungsgeselIschaft Wort", Munich. (c) Springer-Verlag Berlin Heidelberg 1982 Softcover reprint of the hardcover 1st edition 1982 Typesetting: Buchdruckerei Dipl.-Ing. Schwarz' Erben KG, Zwettl. 214113140-5432 I 0 Preface to the English Edition The reasons for writing this book have already been given in the preface to the original edition and it suffices to append a few more points.

It is with great pleasure that I am writing the preface for my little book, "Starting with the Unit Circle," in the office of Springer Verlag in Heidel berg. This is symbolic of the fact that I have once again joined in the main stream of scientific exchange between East and West. Since the establishment of the People's Republic of China, I have written "An Introduction to Number Theory" for the young people studying Number Theory: for the young people studying algebra, Prof. Wan Zhe-xian (Wan Che-hsien) and I have written "Classical Groups"; for those studying the theory of functions of several complex variables, I have written "Har monic Analysis of Functions of Several Complex Variables in the Classical Domains," * and for university students I have written "Introduction to Higher Mathematics." The present volume had been written for those who were beginning to engage in research at the Chinese University of Science and Technology and at the Guangdong Zhongshan University. Its purpose is none other than to make the students see the crucial ideas in their simplest manifestations, so that when they go on to the more complex parts of modem mathematics, they will not be without guidance. For example, in the first chapter when I point out that the Poisson kernel is just the Jacobian of some transformation, I am merely revealing the source of one of the main tools in my work on harmonic analysis in the classical domains."

Owing to the developments and applications of computer science, ma thematicians began to take a serious interest in the applications of number theory to numerical analysis about twenty years ago. The progress achieved has been both important practically as well as satisfactory from the theoretical view point. It'or example, from the seventeenth century till now, a great deal of effort was made in developing methods for approximating single integrals and there were only a few works on multiple quadrature until the 1950's. But in the past twenty years, a number of new methods have been devised of which the number theoretic method is an effective one. The number theoretic method may be described as follows. We use num ber theory to construct a sequence of uniformly distributed sets in the s dimensional unit cube G , where s ~ 2. Then we use the sequence to s reduce a difficult analytic problem to an arithmetic problem which may be calculated by computer. For example, we may use the arithmetic mean of the values of integrand in a given uniformly distributed set of G to ap s proximate the definite integral over G such that the principal order of the s error term is shown to be of the best possible kind, if the integrand satis fies certain conditions.