Arithmetic and Algebra: Numbers and the beginnings of Algebra is written for students of mathematics in classes 7 to 10. It can be used by middle school and high school mathematics teachers who wish to take their students to a deeper level of the subject; it can also be studied by those who have a general interest in the subject. With the help of revealing examples and exercises, the book aims to help students journey into a world of pattern, power and beauty—a journey which can enrich their life greatly.
This is the first book in the MPOWER series of books on arithmetic, algebra and geometry and is included in the Ramanujan Mathematical Series: Little Mathematical Treasures.
Shailesh Shirali is the Head of the Community Mathematics Centre housed in Rishi Valley School (Andhra Pradesh). He has been deeply involved in the field of mathematics education for three decades, and has been closely associated with the Math Olympiad movement in India. The
1. The World of Numbers 1.1 The unending supply of numbers 1.2 Roman numerals 1.3 Hindu–Arabic numerals 1.4 Positive and negative 1.5 Addition and subtraction 1.6 Multiplication and division 1.7 Number patterns 1.8 Palindromic entertainment 1.9 Miscellaneous problems I 1.10 Miscellaneous problems II 1.11 Miscellaneous problems III
2. Beginnings of Algebra 2.1 Framing sentences 2.2 Expressing patterns 2.3 Laws of algebra 2.4 Using algebra to explain patterns 2.5 Using algebra to discover properties 2.6 Factorisation 2.7 Equations 2.8 Closing quotes
3. Families of Integers 3.1 Odd and even 3.2 Multiples of 3 3.3 Multiples of 5 3.4 Prime and composite 3.5 GCD and LCM 3.6 Euclid’s algorithm 3.7 Perfect numbers 3.8 Squares and cubes 3.9 Triangular numbers 3.10 Powers of 2 3.11 Miscellaneous problems
4. Fractions, Decimals and Percentages 4.1 Fractions as numbers 4.2 Equivalent fractions 4.3 Arithmetic with fractions 4.4 Decimal fractions 4.5 Percentages 4.6 Fractions on the number line 4.7 Rational numbers 4.8 Miscellaneous exercises
5. Fractions: Special Topics 5.1 Recurring decimals 5.2 Powers of integers---on a clothes line! 5.3 Cyclic numbers 5.4 Decimal expansion of 1/p 5.5 Irrational numbers 5.6 Unit fractions 5.7 Mischievous fractions 5.8 Farey sequences 5.9 Miscellaneous exercises
6. Ratio and Proportion 6.1 Ratios for unit conversion 6.2 Rate 6.3 Dimensionless ratios 6.4 Proportion 6.5 Results concerning proportion 6.6 Continued proportion 6.7 Historical note 6.8 Approximations to pi 6.9 The tangram puzzle
7. Indices and Exponents 7.1 Introduction 7.2 The laws of indices 7.3 Big numbers 7.4 The four twos 7.5 Visualising big numbers 7.6 Negative exponents 7.7 Fractional exponents 7.8 Growth patterns 7.9 Exponential notation in science 7.10 Three diversions
8. Polynomials 8.1 Notion of a polynomial 8.2 Degree of the zero polynomial 8.3 Alternate definition 8.4 Multiplication 8.5 Division 8.6 GCD 8.7 Roots of a polynomial 8.8 The remainder theorem 8.9 Symmetric polynomials
9. Expansions in Algebra 9.1 Homogeneous expressions 9.2 Expansions using shortcuts 9.3 Binomial expansions I 9.4 Using the Pascal triangle 9.5 Binomial expansions II 9.6 Squares of multinomials 9.7 Sum and product formulas 9.8 Difference of two squares 9.9 Visual representations 9.10 Miscellaneous expansions 9.11 A claim of Ramanujan’s
10. Factorisation of Polynomials 10.1 Difference of two squares 10.2 Application to integer factorisation 10.3 Factorisation by breaking up an expression 10.4 Factorisation of quadratic trinomials 10.5 A more complex example 10.6 Using the factor theorem 10.7 A square investigation
11. Linear Equations 11.1 Introduction to equations 11.2 Genesis of equations 11.3 Solution sets 11.4 Solving an equation 11.5 Linear equations in two unknowns 11.6 Application I: Clock problems 11.7 Application II: Pythagorean triples 11.8 Application III: Squares in AP 11.9 Application IV: Unit fractions
12. Problem Corner 12.1 Appetisers 12.2 For the junior mathlete
13. Investigations 13.1 Mischievous cancellations 13.2 Mischievous square roots 13.3 Prime numbers 13.4 Digital sums 13.5 Digital reversals 13.6 Squares and 2006 . . . 13.7 Squares and cubes 13.8 Still more squares 13.9 And yet more squares . . . 13.10 Accommodating numbers
Answers to Exercises Index
Shirali’s book is a superb introduction to some substantial material for students of mathematics in classes 7, 8 and 9 in Indian schools. If this book had come twenty years ago, there would have been an entire generation of school students who would have ‘journeyed into a world of pattern power and beauty’,to quote the author. This book is far from being routine and needs dedication both from the teacher and the pupil. It is time that such a book is available at a reasonable price to our students.
Alladi Sitaram