The book covers all the key areas of algebra including elementary operations, linear equations, formulae, simultaneous equations, quadratic equations, logarithms, variation, laws and sequences.

Table of Contents

• Introduction
• 1 The meaning of algebra
• 1.1 An illustration from numbers
• 1.2 Substitution
• 1.3 Examples of generalizing patterns
• 1.4 Letters represent numbers, not quantities
• 1.5 Examples of algebraic forms
• 2 Elementary operations in algebra
• 2.1 Use of symbols
• 2.2 Symbols of operation
• 2.3 Algebraic expression – terms
• 2.4 Brackets
• 2.5 Coefficient
• 2.6 Addition and subtraction of like terms
• 2.7 Worked examples
• 2.8 The order of addition
• 2.9 Evaluation by substitution
• 2.10 Multiplication
• 2.11 Powers of numbers
• 2.12 Multiplication of powers of a number
• 2.13 Power of a product
• 2.14 Division of powers
• 2.15 Easy fractions
• 2.16 Addition and subtraction
• 2.17 Multiplication and division
• 3 Brackets and operations with them
• 3.1 Removal of brackets
• 3.2 Addition and subtraction of expressions within brackets
• 3.3 Worked examples
• 3.4 Systems of brackets
• 3.5 Worked examples
• 4 Positive and negative numbers
• 4.1 The scale of a thermometer
• 4.2 Motion in opposite directions
• 4.3 Positive and negative numbers
• 4.4 Negative numbers
• 4.5 Graphical representation of the number line
• 4.6 Addition of positive and negative numbers
• 4.7 Subtraction
• 4.8 Graphical illustrations
• 4.9 Multiplication
• 4.10 Division
• 4.11 Summary of rules of signs for multiplication and division
• 4.12 Powers, squares and square roots
• 5 Equations and expressions
• 5.1 Understanding expressions
• 5.2 Using function machines
• 5.3 Function notation
• 5.4 Inverse functions
• 5.5 An introduction to solving equations
• 6 Linear equations
• 6.1 Meaning of an equation
• 6.2 Solving an equation
• 6.3 Worked examples
• 6.4 Problems leading to simple equations
• 7 Formulae
• 7.1 Practical importance of formulae
• 7.2 Treatment of formulae
• 7.3 Worked examples
• 7.4 Transformation of formulae
• 7.5 Worked examples
• 7.6 Literal equations
• 7.7 Worked examples
• 8 Simultaneous equations
• 8.1 Simple equations with two unknown quantities
• 8.2 Solution of simultaneous equations
• 8.3 Worked examples
• 8.4 Problems leading to simultaneous equations
• 8.5 Worked examples
• 9 Linear inequalities
• 9.1 The idea of an inequality
• 9.2 Representing inequalities
• 9.3 Solving inequalities
• 9.4 A trap for the unwary
• 9.5 Simultaneous inequalities
• 10 Straight-line graphs; coordinates
• 10.1 The straight-line graph
• 10.2 The law represented by a straight-line graph
• 10.3 Graph of an equation of the first degree
• 10.4 Worked examples
• 10.5 Position in a plane; coordinates
• 10.6 A straight line as a locus
• 10.7 Equation of any straight line passing through the origin
• 10.8 Graphs of straight lines not passing through the origin
• 10.9 Graphical solution of simultaneous equations
• 11 Using inequalities to define regions
• 11.1 Defining regions
• 11.2 Regions above and below straight lines
• 11.3 Greatest or least values in a region
• 11.4 Linear programming
• 12 Multiplying algebraical expressions
• 12.1 Multiplying expressions when one factor consists of one term
• 12.2 Product of expressions with two terms
• 12.3 When the coefficients of the first terms are not unity
• 12.4 Multiplication of an expression with three terms
• 12.5 Square of an expression with two terms
• 12.6 Square of an expression with three terms
• 12.7 Cube of an expression with two terms
• 12.8 Product of sum and difference
• 13 Factors
• 13.1 The process of finding factors
• 13.2 Factors consisting of one term only
• 13.3 Worked examples
• 13.4 Factors with two terms
• 13.5 Worked examples
• 13.6 The form x2 + ax + b
• 13.7 Worked examples
• 13.8 The form ax2 + bx + c
• 13.9 Expressions which are squares
• 13.10 Difference of two squares
• 13.11 Worked examples
• 13.12 Evaluation of formulae
• 13.13 Sum and difference of two cubes
• 13.14 Worked examples
• 14 Fractions
• 14.1 Algebraic fractions
• 14.2 Laws of fractions
• 14.3 Reduction of fractions
• 14.4 Multiplication and division
• 14.5 Addition and subtraction
• 14.6 Simple equations involving algebraical fractions
• 15 Graphs of quadratic functions
• 15.1 Constants and variables
• 15.2 Dependent and independent variables
• 15.3 Functions
• 15.4 Graph of a function
• 15.5 Graph of a function of second degree
• 15.6 Some properties of the graph of y = x2
• 15.7 The graph of y = −x2
• 15.8 The graphs of y = ax2
• 15.9 The graphs of y = x2 ± a, where a is any number
• 15.10 Graph of y = (x − 1)2
• 15.11 Graph of y = (x − 1)2 − 4
• 15.12 Graph of y = x2 − 2x − 3
• 15.13 Solution of the equation x2 − 2x − 3 = 0 from the graph
• 15.14 Graph of y = 2x2 − 3x − 5
• 15.15 Graph of y = 12 − x − x2
• 15.16 Using graphics calculators
• 15.17 Using graphs to solve quadratic inequalities
• 15.18 Using quadratic inequalities to describe regions
• 16 Quadratic equations
• 16.1 Algebraical solution
• 16.2 The method of solution of any quadratic
• 16.3 Solution of 2x2 + 5x − 3 = 0
• 16.4 Worked examples
• 16.5 Solution of quadratic equations by factorization
• 16.6 Worked examples
• 16.7 General formula for the solution of a quadratic equation
• 16.8 Solution of the quadratic equation ax2 + bx + c = 0
• 16.9 Worked examples
• 16.10 Problems leading to quadratics
• 16.11 Simultaneous equations of the second degree
• 16.12 When one of the equations is of the first degree
• 16.13 Solving quadratic inequalities
• 17 Indices
• 17.1 The meaning of an index
• 17.2 Laws of indices
• 17.3 Extension of the meaning of an index
• 17.4 Graph of 2x
• 17.5 Algebraical consideration of the extension of the meaning of indices
• 17.6 Fractional indices
• 17.7 To find a meaning for a0
• 17.8 Negative indices
• 17.9 Standard forms of numbers
• 17.10 Operations with standard forms
• 18 Logarithms
• 18.1 A system of indices
• 18.2 A system of logarithms
• 18.3 Rules for the use of logarithms
• 18.4 Change of base of a system of logarithms
• 19 Ratio and proportion
• 19.1 Meaning of a ratio
• 19.2 Ratio of two quantities
• 19.3 Proportion
• 19.4 Theorems on ratio and proportion
• 19.5 An illustration from geometry
• 19.6 Constant ratios
• 19.7 Examples of equal ratios
• 20 Variation
• 20.1 Direct variation
• 20.2 Examples of direct variation
• 20.3 The constant of variation
• 20.4 Graphical representation
• 20.5 To find the law connecting two variables
• 20.6 Worked example
• 20.7 y partly constant and partly varying as x
• 20.8 Worked example
• 20.9 y varies as the square of x – that is, y ∝ x2
• 20.10 y varies as the cube of x – that is, y ∝ x3
• 20.11 y varies as or , that is, y ∝
• 20.12 Inverse variation: y ∝
• 20.13 Graph of y =
• 20.14 Other forms of inverse variation
• 20.15 Worked examples
• 20.16 Functions of more than one variable
• 20.17 Joint variation
• 20.18 Worked examples
• 21 The determination of laws
• 21.1 Laws which are not linear
• 21.2 y = axn + b. Plotting against a power of a number
• 21.3 Worked example
• 21.4 y = axn. Use of logarithms
• 21.5 Worked example
• 22 Rational and irrational numbers and surds
• 22.1 Rational and irrational numbers
• 22.2 Irrational numbers and the number line
• 22.3 Geometrical representation of surds
• 22.4 Operations with surds
• 23 Arithmetic and geometric sequences
• 23.1 Meaning of a sequence
• 23.2 The formation of a sequence
• 23.3 Arithmetic sequences, or arithmetic progressions
• 23.4 Any term in an arithmetic sequence
• 23.5 The sum of any number of terms of an arithmetic sequence
• 23.6 Arithmetic mean
• 23.7 Worked examples
• 23.8 Harmonic sequences or harmonic progressions
• 23.9 Geometric sequences or geometric progressions
• 23.10 Connection between a geometric sequence and an arithmetic sequence
• 23.11 General term of a geometric sequence
• 23.12 Geometric mean
• 23.13 The sum of n terms of a geometric sequence
• 23.14 Worked examples
• 23.15 Increasing geometric sequences
• 23.16 Decreasing geometric sequences
• 23.17 Recurring decimals
• 23.18 A geometrical illustration
• 23.19 The sum to infinity
• 23.20 Worked examples
• 23.21 Simple and compound interest
• 23.22 Accumulated value of periodical payments
• 23.23 Annuities
• Appendix
• Answers