Hugh Neill P Abbott Algebra A Complete Introduction Teach Yourself 2013

Cover
Title
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 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 x or x1/2, that is, y = x
	20.12 Inverse variation: y = 1/x
	20.13 Graph of y = k/x
	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
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