Teach Yourself: Algebra: A Complete Introduction

Editor/Author Neill, Hugh
Publication Year: 2013
Publisher: Hodder & Stoughton

Price: Core Collection Only
ISBN: 978-1-44-419106-6
Category: Mathematics & Statistics - Mathematics
Image Count: 76
Book Status: Available
Table of Contents

Algebra--A Complete Introduction provides everything you need to broaden your skills and gain confidence. Assuming only a basic level of arithmetic, this carefully graded and progressive book guides you through the basic principles of the subject with the help of exercises and fully worked examples.

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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 The graph 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 numbers17.10 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,
  • 20.12 Inverse variation:
  • 20.13 Graph of
  • 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 Arithmetical and geometrical 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