NUMERICAL MATHEMATICS EXERCISES IN COMPUTING WITH A DESK CALCULATOR INTRODUCTORY MONOGRAPHS IN MATHEMATICS General Editor A. J. MOAKEs, M.A. N umerical Mathematics A. J. Moakes Exercises in computing with a desk calculating machine Mathematics for Circuits W. Chellingsworth The Co re of Mathematics A. J. Moakes An introduction to 'modern' mathematics A Boolean Algebra: Abstract and Concrete A. P. Bowran Other volumes in preparation Palgrave Macmillan ISBN 978-1-349-81612-5 ISBN 978-1-349-81610-1 (eBook) DOI 10.1007/978-1-349-81610-1 Copyright © A. J. Moakes 1963 Reprint of the original edition 1963 First Edition I963 Second Edition I965 MACMILLAN AND COMPANY LIMITED St Martin's Street London WCz also Bombay Calcuua Madras Melbourne THE MACMILLAN COMPANY OF CANADA L1MITED 70 Bond Street Toronto z ST MARTIN'S PRESS INC I75 Fifth Avenue New York IO NY PREFACE TO SECOND EDITION I HAVE taken the opportunity to correct so me errors and obscurities. I wish to thank correspondents and others who have made suggestions. Remaining imperfections are of course my own. 80me new material has been added, notably flow diagrams and fuHer references. A. J. MOAKES 8t. Paul's 8chool 1965 PREFACE TO FIRST EDITION THE view is gaining ground in the educational world that to free the learner (after a certain point) from laborious calculation is to help hirn to grasp and develop the underlying ideas. These exercises are therefore devised to use the machine so as to stimulate mathematical thought. This aim has been pursued from the beginning but only becomes fuHy apparent after the fuH range of machine operations has been mastered. This coHection of exercises, along with the necessary minimum of explanation, was devised for the use of sixth-formers. It should be he1pful to those teachers who want their pupils to be introduced to machines but who are not clear how to set about it. In the first five chapters the technique of handling a machine is developed stage by stage through progressive examples. For each stage it is necessary first to learn certain manual operations, preferably by seeing them demonstrated and immediately imitating them. For a course in being, one pupil can demonstrate to another - to their mutual advantage. A teacher wishing to initiate such a course can acquire the skills with aid of (a) a full demonstration by the maker's representa tive, foHowed by (b) thorough practice, with the maker's handbook for reference. The author wishes to express gratitude, for their encouragement and he1p, to Mr. M. 8hoenberg (educational adviser of Olympia Business Machines, who make the Brunsviga), to Mr. F. W. Russell (education manager, Monroe Calculating Machines), to Dr. J. Crank (Head of Mathematics Department, Brunel College), to Mr. M. N. Horsman of v VI PREFACE TO FIRST EDITION Brunel College, as weIl as to Mr. D. Grisewood of Macrnillan & Co. Ltd. and to Mr. M. Bridger of Leicester College of Technology. Acknowledgement must also be made to the author's pupils at St. Paul's School, who have greeted machines with enthusiasm, treated them with respect, and made shrewd comments on the exercises. Standing orders for pupils doing the course are reproduced as an appendix, together with some hints which may prove useful to teachers who wish to use a machine occasionally for junior class-teaching. A. J. M. CONTENTS PAGE Introduction IX Chapter 1. Addition, subtraction. Difference tables. 2. Multiplication. Accumulation of products, includ- ing negative terms. Summation of a frequency table. 4 3. Conversion of units. Running totals. Compound interest. Errors. 7 A note on Flow Diagrams. 11 4. Sequences. Evaluation of polynomial function. Use of differences for evaluation of terms of certain sequences (or their sum). Use of differences for interpolating values. 13 5. Division (a) by tear-down method (b) by build-up method. Applications, including evaluation of some power-series. 17 A note on Checks. 22 6. Iterative method for square-root. Newton's method for a root of an equation: methods of finding the required first approximation. 23 7. Statistical and allied calculations: total number, mean, standard deviation. 30 8. Solution of simultaneous linear equations: pivotal method, iterative method, relaxation method. 40 9. Numerical integration by Simpson's rule: estimate of error. 50 10. Numerical differentiation. Tabulation of certain functions defined by differential equations. 54 Appendix I. Programming and the Computer. 58 11. Instructors' notes. 61 111. Machines. 62 Solution Notes 63 Bibliography 66 Index 67 m Setting register Forward [ direction Setting I 0 0 0 5 2.3 7 \11 Levers / f ,J / :' .. oftum Cycle reversal ,r " ! " f , : : " ; r ~ ; f ! T ! fI :I :I :• :I iI :, Cycle clearance t : : : ; ! : Crank in ./000002110000010461 position of rest \ /' Cycle register Accumulator / Shift ~ Carriage (in extreme Ieft position) Basic fonn of desk machine (lever-setting type) INTRODUCTION The Desk Calculating Machine MACHINES operated entirely by hand invariably incorporate the following registers, where the numbers which are involved in the cal culation are exhibited: (1) The input or setting register. A number is set on the machine by the pressing of keys or the setting of levers; and the setting register indicates this number, as a check that (a) its digits have been set correctly and (b) it is correctly located. (2) The cycle register or multiplier register. The processing of the input number is done entirely by turns of the operating handle, e.g. a single forward turn of this handle causes the input number N to be added in to the accumulator once (while still remaining set for further use). This turn is recorded as a unit figure in the cyde register. Seven such turns would show the number 7 and give 7N in the accumulator: this fact explains the name 'multiplier register' often employed. (3) The accumulator, already mentioned, can also be called the pro duct register as the above example explains; but the former term explains more dearly its usual function, viz.: to accumulate (so long as no action is taken to dear it) all numbers which, singly or as multiples, are brought into it by the action of the operating handle. The Operation of the Machine This is described in general terms which apply to any machine except where special reference is made to a particular model. The basic function of the operating handle has been described; but its operation would be very limited without the shift lever, a device which by lateral movement of the accumulator causes the numbers to be fed into it at any position. E.g. with the input one place to the left (relatively) the operation described in (2) would feed in not 7N but 70N. ix