1 Introduction.- 2 Exchange of terms in addition.- 3 Exchange of terms in multiplication.- 4 Addition in the decimal number system.- 5 The multiplication table and the multiplication algorithm.- 6 The division algorithm.- 7 The binary system.- 8 The commutative law.- 9 The associative law.- 10 The use of parentheses.- 11 The distributive law.- 12 Letters in algebra.- 13 The addition of negative numbers.- 14 The multiplication of negative numbers.- 15 Dealing with fractions.- 16 Powers.- 17 Big numbers around us.- 18 Negative powers.- 19 Small numbers around us.- 20 How to multiply am by an, or why our definition is convenient.- 21 The rule of multiplication for powers.- 22 Formula for short multiplication: The square of a sum.- 23 How to explain the square of the sum formula to our younger brother or sister.- 24 The difference of squares.- 25 The cube of the sum formula.- 26 The formula for (a + b)4.- 27 Formulas for (a + b)5, (a + b)6,... and Pascal’s triangle.- 28 Polynomials.- 29 A digression: When are polynomials equal?.- 30 How many monomials do we get?.- 31 Coefficients and values.- 32 Factoring.- 33 Rational expressions.- 34 Converting a rational expression into the quotient of two polynomials.- 35 Polynomial and rational fractions in one variable.- 36 Division of polynomials in one variable; the remainder.- 37 The remainder when dividing by x - a.- 38 Values of polynomials, and interpolation.- 39 Arithmetic progressions.- 40 The sum of an arithmetic progression.- 41 Geometric progressions.- 42 The sum of a geometric progression.- 43 Different problems about progressions.- 44 The well-tempered clavier.- 45 The sum of an infinite geometric progression.- 46 Equations.- 47 A short glossary.- 48 Quadratic equations.- 49 The case p =. Square roots.- 50 Rules forsquare roots.- 51 The equation x2 + px + q =.- 52 Vieta’s theorem.- 53 Factoring ax2 + bx + c.- 54 A formula for ax2 + bx + c = (where a ? 0).- 55 One more formula concerning quadratic equations.- 56 A quadratic equation becomes linear.- 57 The graph of the quadratic polynomial.- 58 Quadratic inequalities.- 59 Maximum and minimum values of a qua ratic polynomial.- 60 Biquadratic equations.- 61 Symmetric equations.- 62 How to confuse students on an exam.- 63 Roots.- 64 Non-integer powers.- 65 Proving inequalities.- 66 Arithmetic and geometric means.- 67 The geometric mean does not exceed the arithmetic mean.- 68 Problems about maximum and minimum.- 69 Geometric illustrations.- 70 The arithmetic and geometric means of everal numbers.- 71 The quadratic mean.- 72 The harmonic mean.
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"The idea behind teaching is to expect students to learn why things are true, rather than have them memorize ways of solving a few problems, as most of our books have done. [This] same philosophy lies behind the current text by Gelfand and Shen. There are specific 'practical' problems but there is much more development of the ideas ! [The authors] have shown how to write a serious yet lively look at algebra." --The American Mathematics Monthly "Were 'Algebra' to be used solely for supplementary reading, it could be wholeheartedly recommended to any high school student of any teacher ! In fact, given the long tradition of mistreating algebra as a disjointed collection of techniques in the schools, there should be some urgency in making this book compulsory reading for anyone interested in learning mathematics." --The Mathematical Intelligencer
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