A Variable in Algebra

• 1. Preface
  • 1.1. Why I Wrote This Book?
  • 1.2. How to Read This Book?
  • 1.3. Who Should Read This Book?
  • 1.4. Prerequisite
  • 1.5. Goals for Readers
  • 1.6. Acknowledgments
• 2. Number System
  • 2.1. Definitions and Notations
  • 2.2. Binary Number System
  • 2.3. A Generic Positional Number System
• 3. Ratio
  • 3.1. Comparing Two Ratios
  • 3.2. Ratio of Ratios
  • 3.3. Surds and Ratios
  • 3.4. Compounding Ratios
  • 3.5. Duplicating Ratios
  • 3.6. Antecedent vs Consequent
  • 3.7. Common Properties of Ratios
  • 3.8. Problems
• 4. Solutions for Ratios
• 5. Proportions
  • 5.1. Continued Proportion
  • 5.2. Problems
• 6. Solutions for Proportions
• 7. Variations
  • 7.1. Problems
• 8. Solutions for Variations Problems
• 9. Logarithm
  • 9.1. Important Results
  • 9.2. Important Formulae
  • 9.3. Bases of Logarthims
  • 9.4. Characteristics and Mantissa
  • 9.5. Inequality of Logarithms
  • 9.6. Expansion of Logarithm and its Graph
• 10. Logarithm Problems Part 1
• 11. Logarithm Solutions Part 1
• 12. Logarithm Problems Part 2
• 13. Logarithm Solutions Part 2
• 14. Logarithm Problems Part 3
• 15. Logarithm Solutions Part 3
• 16. Complex Numbers
  • 16.1. Imaginary Numbers
  • 16.2. Problems
  • 16.3. Definitions Related to Complex Numbers
  • 16.4. Simple Operations
  • 16.5. Conjugate of a Complex Number
  • 16.6. Modulus of a Complex Number
  • 16.7. Geometrical Representation
  • 16.8. Vector Representation
  • 16.9. Algebraic Operation’s Representation
  • 16.10. More Roots
  • 16.11. De Moivre’s Therorem
  • 16.12. Some Important Geometrical Results and Equations
• 17. Complex Numbers Problems Part 1
• 18. Complex Numbers Solutions Part 1
• 19. Complex Numbers Problems Part 2
• 20. Complex Numbers Solutions Part 2
• 21. Complex Numbers Problems Part 3
• 22. Complex Numbers Solutions Part 3
• 23. Complex Numbers Problems Part 4
• 24. Complex Numbers Solutions Part 4
• 25. Complex Numbers Problems Part 5
• 26. Complex Numbers Solutions Part 5
• 27. Complex Numbers Problems Part 6
• 28. Complex Numbers Solutions Part 6
• 29. Arithmetic Progressions
  • 29.1. nth Term of Arithmetic Progression
  • 29.2. Sum of An Arithmetic Progression
  • 29.3. Arithmetic Mean
  • 29.4. Deducing Number of Terms
  • 29.5. Some Observations on an A. P.
  • 29.6. Sum of Squares and Cubes and More
• 30. Arithmetic Progression Problems Part 1
• 31. Arithmetic Progression Solutions Part 1
• 32. Arithmetic Progression Problems Part 2
• 33. Arithmetic Progression Solutions Part 2
• 34. Geometric Progressions
  • 34.1. Properties of a G. P.
  • 34.2. Sum of the First n Terms of a G. P.
  • 34.3. Sum of Infinite Terms of a G. P., when \(|r| < 1\)
  • 34.4. Recurring Decimals
  • 34.5. Geometric Mean
  • 34.6. Notes
  • 34.7. Arithmetico Geometric Series
  • 34.8. Sum of \(n\) terms of an Arithmetico-Geometric Series
• 35. Geometric Progressions Problems Part 1
• 36. Geometric Progressions Solutions Part 1
• 37. Geometric Progressions Problems Part 2
• 38. Geometric Progressions Solutions Part 2
• 39. Summation of Series Problems
• 40. Summation of Series Solutions
• 41. Harmonic Progression
  • 41.1. Problems
• 42. Harmonic Progression Solutions
• 43. AM, GM, HM Relation and Problems
  • 43.1. Arithmetic Means(A. M.)
  • 43.2. Geometric Means(G. M.)
  • 43.3. Harmonic Means(H. M.)
  • 43.4. Problems
• 44. AM, GM, HM Solutions
• 45. Quadratic Equations
  • 45.1. Rational expression or Rational function
  • 45.2. Roots of a quadratic equation
  • 45.3. Sum and product of the roots
  • 45.4. Discriminant of the quadratic equation
  • 45.5. Nature of roots
  • 45.6. Symmetric functions of roots
  • 45.7. Representing the equation in terms of roots
  • 45.8. Condition for common roots
  • 45.9. Sign of quadratic expression \(ax^2 + bx + c\)
  • 45.10. Maximum and minimum values of \(ax^2 + bx + c\)
  • 45.11. To find the condition that the general quadratic equation \(ax^2 + 2hxy + by^2 + 2gx + 2fy + c\) in \(x\) and \(y\) may be resolved into two linear rational factors.
• 46. Quadratic Equations Problems Part 1
• 47. Quadratic Equations Solutions Part 1
• 48. Quadratic Equations Problems Part 2
• 49. Quadratic Equations Solutions Part 2
• 50. Quadratic Equations Problems Part 3
• 51. Quadratic Equations Solutions Part 3
• 52. Quadratic Equations Problems Part 4
• 53. Quadratic Equations Solutions Part 4
• 54. Quadratic Equations Problems Part 5
• 55. Quadratic Equations Solutions Part 5
• 56. Quadratic Equations Problems Part 6
• 57. Quadratic Equations Solutions Part 6
• 58. Quadratic Equations Problems Part 7
• 59. Quadratic Equations Solutions Part 7
• 60. Permutations and Combinations
  • 60.1. Fundamental Principle of Counting
  • 60.2. Addition Rule
  • 60.3. Factorial of \(n\)
  • 60.4. Notations
  • 60.5. Establishing the Permutation Formula
  • 60.6. \(^nP_n = n!\) and \(0!\)
  • 60.7. Meaning of \(\frac{1}{(-k)!},\) where \(k\) is a positive integer
  • 60.8. Permutation of Similar Objects
  • 60.9. Repititive Permutation
  • 60.10. Prove that \(^nP_r = r^{n - 1}P_{r - 1} + ^{n - 1}P_r\)
  • 60.11. Circular Permutation
  • 60.12. Clockwise and Anticlockwise Arrangements
  • 60.13. Find the number of combinations of \(n\) different things taken \(r\) at a time(\(r\le n\))
  • 60.14. Properties of \(^nC_r\)
  • 60.15. Restricted Combinations
  • 60.16. Selection of things
  • 60.17. Distribution of things among sets(groups) and persons
• 61. Permutations and Combinations Problems Part 1
• 62. Permutations and Combinations Solutions Part 1
• 63. Permutations and Combinations Problems Part 2
• 64. Permutations and Combinations Solutions 2
• 65. Permutations and Combinations Problems Part 3
• 66. Permutations and Combinations Solutions Part 3
• 67. Permutations and Combinations Problems Part 4
• 68. Permutations and Combinations Solutions Part 4
• 69. Permutations and Combinations Problems Part 5
• 70. .Permutations and Combinations Solutions Part 5
• 71. Permutations and Combinations Problems Part 6
• 72. Permutations and Combinations Solutions Part 6
• 73. Permutations and Combinations Problems Part 7
• 74. Permutations and Combinations Solutions Part 7
• 75. Determinants
  • 75.1. Minors
  • 75.2. Cofactors
  • 75.3. Theorems on Determinants
  • 75.4. System of Linear Equations
  • 75.5. Cramer’s Rule
  • 75.6. Use of Determinants in Coordinate Geometry
  • 75.7. Product of Two Determinants
  • 75.8. An Important Result
  • 75.9. Differential Coefficient of Determinant
• 76. Determinants Problems Part 1
• 77. Determinants Solutions Part 1
• 78. Determinants Problems Part 2
• 79. Determinants Solutions Part 2
• 80. Determinants Problems Part 3
• 81. Determinants Solutions Part 3
• 82. Determinants Problems Part 4
• 83. Determinants Solutions Part 4
• 84. Matrices
  • 84.1. Introduction
  • 84.2. Definition
  • 84.3. Classification of Matrices
  • 84.4. Algebra of Matrices
  • 84.5. Properties of adjoint and inverse matrices
  • 84.6. Solution of Simultaneous Linear Equations
  • 84.7. Elementary Operations/Transformations of a Matrix
  • 84.8. Echelon Form of a Matrix
  • 84.9. Rank of a Matrix
  • 84.10. Application of Matrices to Geometry or Computer Graphics
• 85. Matrices Problems Part 1
• 86. Matrices Solutions Part 1
• 87. Matrices Problems Part 2
• 88. Mathematical Induction
  • 88.1. Principle of Finite Mathematical Induction
  • 88.2. Extended Form of Mathematical Induction
• 89. Mathematical Induction Problems Part 1
• 90. Mathematical Induction Solutions Part 1
• 91. Binomials, Multinomials and Expansions
  • 91.1. Binomial Theorem
  • 91.2. Special Forms of Binomial Expansion
  • 91.3. General Term of a Binomial Expansion
  • 91.4. Middle Term of a Binomial Expansion
  • 91.5. Largest Coefficient in a Binomial Expansion
  • 91.6. Coefficients Equidistant from Start and End
  • 91.7. Properties of Binomial Coefficients
  • 91.8. Multinomial Theorem
  • 91.9. Some Results on Multinomial Expansions
  • 91.10. Binomial Theorem for any Index
  • 91.11. General Term in Binomial Theorem of Any Index
  • 91.12. General Term of \((1 - x)^{-n}\)
  • 91.13. Exponential and Logarithmic Series
• 92. Binomials, Multinomials and Expansions Problems
• 93. Binomials, Multinomials and Expansions Solutions
• 94. Miscellaneous Problems Part 1

Indices and tables

• Index