Algebra · A Complete Introduction by Abbott, P. -- Read -- Imperial Library of Trantor

Index

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

Appendix Answers Copyright

← Prev
Back
Next →

← Prev
Back
Next →