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by I.M. Gelfand and A. Shen

Edition: 93Copyright: 1993

Publisher: Birkhauser Boston, Inc.

Published: 1993

International: No

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The need for improved mathematics education at the high school and college levels has never been more apparent than in the 1990's. As early as the 1960's, I.M. Gelfand and his colleagues in the USSR thought hard about this same question and developed a style for presenting basic mathematics in a clear and simple form that engaged the curiosity and intellectual interest of thousands of high school and college students. These same ideas, this development, are available in the following books to any student who is willing to read, to be stimulated, and to learn.

Algebra is an elementary algebra text from one of the leading mathematicians of the world -- a major contribution to the teaching of the very first high school level course in a centuries old topic -- refreshed by the author's inimitable pedagogical style and deep understanding of mathematics and how it is taught and learned.

**Gelfand, I.M. : Rutgers the State University of New Jersey Central Office **

Shen, A. : Institute of Problems of Information Transportation, Moscow

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 a m by a n , 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 your 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. Polynomials 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 = 0. Square roots

50. Rules for square roots

51. The equation x 2 +px + q =0

52. Vieta's theorem

53. Factoring ax 2 +bx +c

54. A formula for ax 2 +bx +c =0(where a does not equal 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 quadratic 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 several numbers

71. The quadratic mean

72. The harmonic mean

Summary

The need for improved mathematics education at the high school and college levels has never been more apparent than in the 1990's. As early as the 1960's, I.M. Gelfand and his colleagues in the USSR thought hard about this same question and developed a style for presenting basic mathematics in a clear and simple form that engaged the curiosity and intellectual interest of thousands of high school and college students. These same ideas, this development, are available in the following books to any student who is willing to read, to be stimulated, and to learn.

Algebra is an elementary algebra text from one of the leading mathematicians of the world -- a major contribution to the teaching of the very first high school level course in a centuries old topic -- refreshed by the author's inimitable pedagogical style and deep understanding of mathematics and how it is taught and learned.

Author Bio

**Gelfand, I.M. : Rutgers the State University of New Jersey Central Office **

Shen, A. : Institute of Problems of Information Transportation, Moscow

Table of Contents

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 a m by a n , 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 your 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. Polynomials 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 = 0. Square roots

50. Rules for square roots

51. The equation x 2 +px + q =0

52. Vieta's theorem

53. Factoring ax 2 +bx +c

54. A formula for ax 2 +bx +c =0(where a does not equal 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 quadratic 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 several numbers

71. The quadratic mean

72. The harmonic mean

Publisher Info

Publisher: Birkhauser Boston, Inc.

Published: 1993

International: No

Published: 1993

International: No

Algebra is an elementary algebra text from one of the leading mathematicians of the world -- a major contribution to the teaching of the very first high school level course in a centuries old topic -- refreshed by the author's inimitable pedagogical style and deep understanding of mathematics and how it is taught and learned.

**Gelfand, I.M. : Rutgers the State University of New Jersey Central Office **

Shen, A. : Institute of Problems of Information Transportation, Moscow

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 a m by a n , 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 your 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. Polynomials 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 = 0. Square roots

50. Rules for square roots

51. The equation x 2 +px + q =0

52. Vieta's theorem

53. Factoring ax 2 +bx +c

54. A formula for ax 2 +bx +c =0(where a does not equal 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 quadratic 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 several numbers

71. The quadratic mean

72. The harmonic mean