PRACTICAL MATHEMATICS PRACTICAL MATHEMATICS INCLUDING - TRIGONOMETRY AND AN INTRODUCTION TO THE CALCULUS A Class-Rook for Higher Elementary Secondary and Technical Schools BY A. H. BELL, B. Sc. Hailing Scholar of Oivens Col cge, Manchester Director and Secretary for Higher Education, Sheer ness BLACKIE AND SON LIMITED 50 OLD BAILEY LONDON GLASGOW AND BOMBAY 1919 PREFACE So many text-books on Algebra have appeared during the last few years that the publication of yet another requires some justification. In the present volume, the processes of Algebra are established in a practical way, and their application to Science and Mensura tion is made a prominent feature Although some boys find a certain satisfaction in juggling with meaningless symbols, the application of Algebra to practical problems rouses their interest and fills them with enthusiasm. The author finds encouragement in a recent address to the Mathematical Association by Professor Whitehead, F. R. S., which contained the following Algebra is the intellectual instrument which has been created for rendering clear the quantitative aspects of the world. Graphs are an important feature of this book, and they arc not abandoned as soon as their algebraic significance becomes evident, but are employed to illustrate-and establish many important relations. The treatment is in accord with the recent suggestions of the Board of Education. Ratio and Proportion being of such importance in the teaching of Science, arc given an early place in the scheme. Trigonometry is introduced. Experience shows that when students recognise that trigonometrical terms are merely algebraic symbols, the terrors of the subject disappear. Other special characteristics are I. Academic treatment has been avoided as far as possible. II. The requirements of pupils who will engage in technical work on leaving school, but who will not have the opportunity to attend a University, have been borne in mind. III. Artificial problems have here given place to more useful applications. The calculus is included for its usefulness, and also with a view 6 PREFACE to showing the pupil that a great realm lies beyond elementary Algebra. Chapter XXV is added for those who wish to examine more closely the Binomial and Exponential Theorems, and the deter mination of logarithms. Some may prefer to take this chapter before Chapter XXII. The exercises are of reasonable length, and in many cases are chosen with the idea of preparing the way to future work. The student should therefore work as many as possible. No examination tests are included. A wise teacher prepares or selects these himself. While the majority of the exercises are original, my acknowledgments are due to the Delegates of the Oxford Local Examinations, who have kindly allowed me to include several questions set at various examinations. I have received valuable help from my staff, among whom should be mentioned by name, Mr. R. Hawksworth, B. Sc., Mr. S. Chatwin, B. Sc., and Mr. C. P. Le Huray, B. A., who have taken great interest in the work. A. H. B. August 1916. CONTENTS CHAPTER PAGE I. POSITIVE AND NEGATIVE NUMBERS, ZERO, SIGNS 9 II. SCTM AND DIFFERENCE 14 III. MULTIPLICATION AND DIVISION OF POSITIVE AND NEGATIVE NUMBKIIS 20 IV. SYMBOLS, COEFFICIENTS, COMMON PROCESSES WITH SYMBOLS 25 V. COMMON PROCESSES continitr. d, BRACKETS - - - - 47 VI. SIMPLE EQUATIONS 55 VTT. MULTIPLICATION AND DIVISION OF EXPRESSIONS, SQUARE AND SQUARE ROOT 61 VIII. RATIO AND PROPORTION ....... 71 IX. SPECIAL RATIOS, TRIGONOMETRY 85 REVISION EXERCISE I 92 X. GRAPHS 93 XI. SIMULTANEOUS SIMPLE EQUATIONS, LITERAL EQUATIONS, PROBLEMS 113 XII. EACTOHS, HIGHEST COMMON FACTOR, LOWEST COMMON MULTIPLE, FRACTIONS 124 XIII. SURDS - 137 XIV. LOGARITHMS, THE SLIDE RULE 144 XV. GRAPHS OF QUADRATIC EXPRESSIONS - - - . - - 159 XVI. QUADRATIC EQUATIONS 169 XVII. THE PROPERTIES OF QUADRATIC EXPRESSIONS AND EQUA TIONS, SIMULTANEOUS EQUATIONS, PROBLEMS - - 178 REVISION EXERCISE II 190 XVIIT...