AN INTRODUCTION TO MATHEMATICS With Applications to Science and Agriculture BY ISATAII LESLIE MILLER Professor of Mathematics, South Dakota State College of Agriculture and Mechanic Arts F. S. CROFTS CO. NEW YORK ----MCMXXX COPYRIGHT, 1930, BY F. S. CROITS Co., INC. MANUFACTURED IN THE UNITED STATES OF AMERICA BY BRAUNWORTH CO., INC., BROOKLYN, NEW YORK PREFACE AFTER some fourteen years of teaching in American colleges and universities the author finds that the average high school graduate has not developed in himself a mathematical type of reasoning. lie therefore hopes that this treatment may in some measure accomplish this purpose. The first few chapters are devoted to a thorough review of high school algebra, for the author is convinced that most college freshmen need considerable drill on the fundamental processes of algebra before attempting a very extensive study of mathematics. In preparing this book the author has kept in mind two types of students first, those who will never take additional work in mathematics, and second, those who will continue the work in science or agriculture for advanced degrees and will doubtless desire to pursue additional courses in mathematics. He has therefore attempted to write a book basic in the fundamental principles of mathematics and at the same time has endeavored to make practical applications to the fields of science and agri culture, wherever possible. He feels that a thorough knowledge of the material covered in this work will enable the second type of student to successfully pursue a course in analytical geometry followed by a course in the calculus. The author gratefully acknowledges his indebtedness to his colleagues, Professor Win. Asker for preparing the chapter on statistics, and Mr. H. B. MacDougal for checking much of the material, to Professor I. W. Smith of the North Dakota Agri cultural College for using the material in mimeographed form and offering many valuable suggestions, to Dean D. A. Roth VI PREFACE rock of Indiana University for reading most of the manuscript and to Professor Wm. Marshall of Purdue University for encouraging him in the work. The author also desires to thank Professor E. S. Crawley of the University of Pennsylvania for his generous permission to use the greater part of his Tables of Logarithms as a portion of this book. I. L. MILLER SOUTH DAKOTA STATE COLLEGE CONTENTS CHAPTER I ALGEBRAIC OPERATIONS ARTICLE PAGE 1. FOUR FUNDAMENTAL OPERATIONS 1 2. ADDITION AND SUBTRACTION 1 3. USE OF PARENTHESES, SIGNS OF AGGREGATION 1 4. MULTIPLICATION 3 5. DIVISION 4 6. DIVISION OF A POLYNOMIAL BY A POLYNOMIAL 4 7. ZERO IN DIVISION 4 CHAPTER II FACTORING 8. IMPORTANT TYPE PRODUCTS i 9. OTHER IMPORTANT PRODUCTS 8 10. HIGHEST COMMON FACTOR 9 11. LOWEST COMMON MULTIPLE 10 CHAPTER III LINEAR EQUATIONS IN ONE UNKNOWN 12. EQUALITIES 12 13. SOLUTION OR ROOT OF AN EQUATION 12 14. EQUIVALENT EQUATIONS 13 15. OPERATIONS ON EQUATIONS 13 16. TYPE FORM OF THE LINEAR EQUATION IN ONE UNKNOWN. . . 13 17. VERIFICATION BY SUBSTITUTION 13 CHAPTER IV FRACTIONS 18. ALGEBRAIC FRACTION 16 19. OPERATIONS 16 vii Vlll CONTENTS ARTICLE PAGE 20. REDUCTION OP A FRACTION TO ITS LOWEST TERMS 17 21. ADDITION AND SUBTRACTION 18 22. MULTIPLICATION AND DIVISION 19 23. COMPLEX FRACTIONS 20 24. FRACTIONAL EQUATIONS 21 CHAPTER V FUNCTIONS 25. CONSTANTS AND VARIABLES 24 26. DEFINITION OF A FUNCTION 24 27. FUNCTIONAL NOTATION 24 28. FUNCTIONAL RELATIONS 25 29. FORMULAS TAKEN FROM GEOMETRY . . . 26 30. GRAPHICAL REPRESENTATION OF FUNCTIONAL RELATIONS. ... 29 31. STATISTICAL DATA 34 CHAPTER VI SYSTEMS OF LINEAR EQUATIONS 32. GRAPHS OP LINEAR EQUATIONS . 39 33. GRAPHICAL SOLUTION 41 34. ALGEBRAIC SOLUTION 43 35. SOLUTION OF THREE LINEAR EQUATIONS IN THREE UNKNOWNS. 44 36. SLOPE OF A STRAIGHT LINE 48 37. DISTANCE BETWEEN Two POINTS 50 38. EQUATION OF A STRAIGHT LINE 50 39. PROBLEM Two POINT FORM OF THE EQUATION OF A LINE . 51 40. PROBLEM SLOPE AND ONE POINT FORM OF THE EQUATION OF A LINE 53 41...