01 prelims 5. · Types of number 6 The natural numbers – Numerals and place value – Points on a line and order – The integers – Brackets – Addition and subtraction Multiplication

Contents

Preface to the first edition xviiiPreface to the second edition xixPreface to the third edition xxPreface to the fourth edition xxiPreface to the seventh edition xxiiHow to use this book xxiiiUseful background information xxv

Part 1 Foundation topics 1Programme F.1 Arithmetic 3

Learning outcomes 3Quiz F.1 4Types of number 6

The natural numbers – Numerals and place value – Points on a line andorder – The integers – Brackets – Addition and subtractionMultiplication and division – Brackets and precedence rules – Basic lawsof arithmetic – Estimating – Rounding – Review summary – Reviewexercise

Factors and prime numbers 15Factors – Prime numbers – Prime factorization – Fundamental theoremof arithmetic – Highest common factor (HCF) – Lowest commonmultiple (LCM) – Review summary – Review exercise

Fractions, ratios and percentages 18Division of integers – Multiplying fractions – Of – Equivalent fractionsDividing fractions – Adding and subtracting fractions – Fractions on acalculator – Ratios – Percentages – Review summary – Review exercise

Decimal numbers 27Division of integers – Rounding – Significant figures – Decimal placesTrailing zeros – Fractions as decimals – Decimals as fractions – Unendingdecimals – Unending decimals as fractions – Rational, irrational and realnumbers – Review summary – Review exercise

Powers 33Raising a number to a power – The laws of powers – Powers on acalculator – Fractional powers and roots – Surds – Multiplication anddivision by integer powers of 10 – Precedence rules – Standard formWorking in standard form – Using a calculator – Preferred standard form– Checking calculations – Accuracy – Review summary – Review exercise

Number systems 44Denary (or decimal) system – Binary system – Octal system (base 8)Duodecimal system (base 12) – Hexadecimal system (base 16)An alternative method – Review summary – Review exercise

Change of base from denary to a new base 52Binary form – Octal form – Duodecimal form – A denary decimal inoctal form – Use of octals as an intermediate step

Reverse method 56Review summary – Review exercise

v

Copyrighted material - ISBN 9781137031204


Can You? Checklist F.1 58Test exercise F.1 59Further problems F.1 60

Programme F.2 Introduction to algebra 63

Learning outcomes 63Quiz F.2 64Algebraic expressions 65

Symbols other than numerals – Constants – Variables – Rules of algebraRules of precedence – Terms and coefficients – Collecting like termsSimilar terms – Expanding brackets – Nested brackets – Review summaryReview exercise

Powers and logarithms 72Powers – Rules of indices – Logarithms – Rules of logarithms – Base 10and base e – Change of base – Logarithmic equations – Review summaryReview exercise

Algebraic multiplication and division 81Multiplication – Division – Review summary – Review exercise

Algebraic fractions 85Addition and subtraction – Multiplication and division – Reviewsummary – Review exercise

Factorization of algebraic expressions 88Common factors – Common factors by grouping – Useful productsof two simple factors – Quadratic expressions as the product oftwo simple factors – Factorization of a quadratic expression – Test forsimple factors – Review summary – Review exercise


Programme F.3 Expressions and equations 101

Learning outcomes 101Quiz F.3 102Expressions and equations 103

Evaluating expressions – Equations – Evaluating independent variablesTransposition of formulas – The evaluation process – Review summaryReview exercise

Polynomial equations 113Polynomial expressions – Evaluation of polynomials – Evaluation of apolynomial by nesting – Remainder theorem – Factor theoremFactorization of fourth-order polynomials – Review summary – Reviewexercise


Programme F.4 Graphs 127

Learning outcomes 127Quiz F.4 128Graphs of equations 129

Equations – Ordered pairs of numbers – Cartesian axes – Drawinga graph – Review summary – Review exercise

vi Contents



Using a spreadsheet 137Spreadsheets – Rows and columns – Text and number entry – FormulasClearing entries – Construction of a Cartesian graph – Displays – Reviewsummary – Review exercise

Inequalities 147Less than or greater than – Review summary – Review exercise

Absolute values 148Modulus – Graphs – Inequalities – Interaction – Review summaryReview exercise


Programme F.5 Linear equations 161

Learning outcomes 161Quiz F.5 162Linear equations 163

Solution of simple equations – Simultaneous linear equations with twounknowns – Simultaneous equations with three unknownsPre-simplification – Review summary – Review exercise


Programme F.6 Polynomial equations 177

Learning outcomes 177Quiz F.6 178Polynomial equations 179

Quadratic equations – Cubic equations having at least one simple linearfactor – Fourth-order equations having at least two linear factorsReview summary – Review exercise


Programme F.7 Binomials 195

Learning outcomes 195Quiz F.7 196Factorials and combinations 197

Factorials – Combinations – Three properties of combinatorialcoefficients – Review summary – Review exercise

Binomial expansions 205Pascal’s triangle – Binomial expansions – The general term of thebinomial expansion – Review summary – Review exercise

The � (sigma) notation 211General terms – The sum of the first n natural numbers – Rules formanipulating sums – The exponential number e – Review summaryReview exercise


Contents vii



Programme F.8 Partial fractions 223

Learning outcomes 223Quiz F.8 224Partial fractions 225

Review summary – Review exerciseDenominators with repeated and quadratic factors 232

Review summary – Review exerciseCan You? Checklist F.8 239Test exercise F.8 240Further problems F.8 240

Programme F.9 Trigonometry 243

Learning outcomes 243Quiz F.9 244Angles 245

Rotation – Radians – Triangles – Trigonometric ratios – Reciprocal ratiosPythagoras’ theorem – Special triangles – Half equilateral – Reviewsummary – Review exercise

Trigonometric identities 258The fundamental identity – Two more identities – Identities forcompound angles – Trigonometric formulas – Review summaryReview exercise


Programme F.10 Functions 267

Learning outcomes 267Quiz F.10 268Processing numbers 269

Functions are rules but not all rules are functions – Functions and thearithmetic operations – Inverses of functions – Graphs of inversesThe graph of y ¼ x3 – The graph of y ¼ x1=3 – The graphs of y ¼ x3 andy ¼ x1=3 plotted together – Review summary – Review exercise

Composition 280Function of a function – Inverses of compositions – Review summaryReview exercise


Programme F.11 Trigonometric and exponentialfunctions 287

Learning outcomes 287Quiz F.11 288Trigonometric functions 289

Rotation – The tangent – Period – Amplitude – Phase differenceInverse trigonometric functions – Trigonometric equations – Equationsof the form a cos xþ b sin x = c – Review summary – Review exercise

Exponential and logarithmic functions 303Exponential functions – Indicial equations – Review summary – Reviewexercise

viii Contents



Odd and even functions 307Odd and even parts – Odd and even parts of the exponential functionLimits of functions – The rules of limits – Review summary – Reviewexercise


Programme F.12 Differentiation 315

Learning outcomes 315Quiz F.12 316Gradients 317

The gradient of a straight line – The gradient of a curve at a given pointAlgebraic determination of the gradient of a curve – Derivatives ofpowers of x – Differentiation of polynomials – Second and higherderivatives – alternative notation – Review summary – Review exercise

Standard derivatives and rules 330Limiting value of sin �=� as �! 0 – Standard derivatives – Derivative of aproduct of functions – Derivative of a quotient of functions – Derivativeof a function of a function – Derivative of ax – Review summaryReview exercise

Newton–Raphson iterative method 344Notation – Tabular display of results – Review summary – Reviewexercise


Programme F.13 Integration 353

Learning outcomes 353Quiz F.13 354Integration 355

Constant of integration – Standard integrals – Review summaryReview exercise

Integration of polynomial expressions 358Functions of a linear function of x – Review summary – Review exercise

Integration by partial fractions 363Review summary – Review exercise

Areas under curves 366Review summary – Review exercise

Integration as a summation 370The area between a curve and an intersecting line – Review summaryReview exercise


Part II 383Programme 1 Complex numbers 1 385

Learning outcomes 385Introduction 386

Ideas and symbols

Contents ix



The symbol j 386Quadratic equations

Powers of j 389Positive integer powers – Negative integer powers

Complex numbers 390Addition and subtraction – Multiplication – Division – Equal complexnumbers – Review exercise

Graphical representation of a complex number 399Argand diagram – Graphical addition of complex numbers

Polar form of a complex number 401Exponential form of a complex number 406

Review summaryCan You? Checklist 1 409Test exercise 1 409Further problems 1 410

Programme 2 Complex numbers 2 412

Learning outcomes 412Polar form calculations 413

Review exerciseRoots of a complex number 421Expansions 427

Expansions of sin n� and cos n�, where n is a positive integerExpansions of cosn � and sinn �

Loci problems 430Review summary

Can You? Checklist 2 434Test exercise 2 434Further problems 2 435

Programme 3 Hyperbolic functions 437

Learning outcomes 437Introduction 438Graphs of hyperbolic functions 440

Review exerciseEvaluation of hyperbolic functions 445Inverse hyperbolic functions 446Log form of the inverse hyperbolic functions 448Hyperbolic identities 451Relationship between trigonometric and hyperbolic functions 453Can You? Checklist 3 456Test exercise 3 457Further problems 3 457

Programme 4 Determinants 459

Learning outcomes 459Determinants 460Determinants of the third order 466

Evaluation of a third-order determinantSimultaneous equations in three unknowns 470

Review exerciseConsistency of a set of equations 478Properties of determinants 481

x Contents




Programme 5 Matrices 489

Learning outcomes 489Matrices – definitions 490Matrix notation 491Equal matrices 492Addition and subtraction of matrices 492Multiplication of matrices 493

Scalar multiplication – Multiplication of two matricesTranspose of a matrix 496Special matrices 497Determinant of a square matrix 499

Cofactors – Adjoint of a square matrixInverse of a square matrix 501

Product of a square matrix and its inverseSolution of a set of linear equations 504

Gaussian elimination method for solving a set of linear equationsEigenvalues and eigenvectors 509

Eigenvalues – Eigenvectors – Review summaryCan You? Checklist 5 515Test exercise 5 515Further problems 5 516

Programme 6 Vectors 519

Learning outcomes 519Introduction: scalar and vector quantities 520Vector representation 521

Two equal vectors – Types of vector – Addition of vectors – The sum of anumber of vectors

Components of a given vector 524Components of a vector in terms of unit vectors

Vectors in space 530Direction cosines 532Scalar product of two vectors 533Vector product of two vectors 535Angle between two vectors 537Direction ratios 540


Programme 7 Differentiation 544

Learning outcomes 544Standard derivatives 545Functions of a function 546

Products – QuotientsLogarithmic differentiation 552

Review exerciseImplicit functions 555

Contents xi



Parametric equations 557Can You? Checklist 7 559Test exercise 7 560Further problems 7 560

Programme 8 Differentiation applications 562

Learning outcomes 562Differentiation of inverse trigonometric functions 563

Review exerciseDerivatives of inverse hyperbolic functions 565

Review exerciseMaximum and minimum values 570Points of inflexion 574Can You? Checklist 8 580Test exercise 8 580Further problems 8 580

Programme 9 Tangents, normals and curvature 583

Learning outcomes 583Equation of a straight line 584Tangents and normals to a curve at a given point 587Curvature 592

Centre of curvatureCan You? Checklist 9 601Test exercise 9 602Further problems 9 602

Programme 10 Sequences 605

Learning outcomes 605Functions with integer input 606

Sequences – Graphs of sequences – Arithmetic sequence – Geometricsequence – Harmonic sequence – Recursive prescriptions – Othersequences – Review summary – Review exercise

Difference equations 618Solving difference equations – Second-order homogeneous equationsEqual roots of the characteristic equation – Review summary – Reviewexercise

Limits of sequences 626Infinity – Limits – Infinite limits – Rules of limits – Indeterminate limitsReview summary – Review exercise


Programme 11 Series 1 639

Learning outcomes 639Series 640

Arithmetic series – Arithmetic mean – Geometric series – Geometricmean

Series of powers of the natural numbers 645Sum of natural numbers – Sum of squares – Sum of cubes

Infinite series 648

xii Contents



Limiting values 650Convergent and divergent series – Tests for convergence – Absoluteconvergence – Review summary


Programme 12 Series 2 663

Learning outcomes 663Power series 664

Introduction – Maclaurin’s series – Standard series – The binomial seriesApproximate values 674

Taylor’s seriesLimiting values – indeterminate forms 677

L’Hopital’s rule for finding limiting valuesCan You? Checklist 12 684Test exercise 12 685Further problems 12 685

Programme 13 Curves and curve fitting 688

Learning outcomes 688Introduction 689Standard curves 689

Straight line – Second-degree curves –Third-degree curves – CircleEllipse – Hyperbola – Logarithmic curves – Exponential curvesHyperbolic curves – Trigonometrical curves

Asymptotes 697Determination of an asymptote – Asymptotes parallel to the x- andy- axes

Systematic curve sketching, given the equation of the curve 702Symmetry – Intersection with the axes – Change of origin – AsymptotesLarge and small values of x and y – Stationary points – Limitations

Curve fitting 707Straight line law – Graphs of the form y ¼ axn, where a and n areconstants – Graphs of the form y ¼ aenx

Method of least squares 712Fitting a straight line graph

Correlation 718Correlation – Measures of correlation – The Pearson product-momentcorrelation coefficient – Spearman’s rank correlation coefficientReview summary


Programme 14 Partial differentiation 1 731

Learning outcomes 731Partial differentiation 732

Review exerciseSmall increments 744Can You? Checklist 14 749Test exercise 14 749Further problems 14 750

Contents xiii



xiv Contents

Programme 15 Partial differentiation 2 752

Learning outcomes 752Partial differentiation 753Rate-of-change problems 755Change of variables 763Can You? Checklist 15 765Test exercise 15 765Further problems 15 766

Programme 16 Integration 1 768

Learning outcomes 768Introduction 769

Standard integralsFunctions of a linear function of x 772Integrals of the forms

Ðf 0ðxÞ=f ðxÞdx and

Ðf ðxÞf 0ðxÞdx 774

Integration of products – integration by parts 778Integration by partial fractions 782Integration of trigonometric functions 787Can You? Checklist 16 791Test exercise 16 792Further problems 16 792

Programme 17 Integration 2 794

Learning outcomes 794Can You? Checklist 17 819Test exercise 17 819Further problems 17 820

Programme 18 Reduction formulas 821

Learning outcomes 821Can You? Checklist 18 831Test exercise 18 831Further problems 18 832

Programme 19 Integration applications 1 834

Learning outcomes 834Basic applications 835

Areas under curves – Definite integralsParametric equations 843Mean values 844Root mean square (rms) value 846



Learning outcomes 851Introduction 852Volume of a solid of revolution 852Centroid of a plane figure 856



Contents xv

Centre of gravity of a solid of revolution 859Length of a curve 860

Parametric equationsSurface of revolution 864

Parametric equationsRules of Pappus 867



Learning outcomes 873Moments of inertia 874

Radius of gyration – Parallel axes theorem – Perpendicular axes theorem (for thinplates) – Useful standard results

Second moments of area 888Composite figures

Centre of pressure 892Pressure at a point P, depth z below the surface – Total thrust on a vertical plateimmersed in a liquid – Depth of the centre of pressure – Review summary


Programme 22 Approximate integration 902

Learning outcomes 902Introduction 903Approximate integration 904

Series – Simpson’s ruleProof of Simpson’s rule 916Can You? Checklist 22 918Test exercise 22 918Further problems 22 919

Programme 23 Polar coordinate systems 921

Learning outcomes 921Introduction to polar coordinates 922Polar curves 924Standard polar curves 926Applications 929


Programme 24 Multiple integrals 943

Learning outcomes 943Summation in two directions 944Double integrals 947Triple integrals 949Applications 951

Review exercise



xvi Contents

Alternative notation 956Determination of areas by multiple integrals 960Determination of volumes by multiple integrals 962Can You? Checklist 24 965Test exercise 24 965Further problems 24 966

Programme 25 First-order differential equations 968

Learning outcomes 968Introduction 969Formation of differential equations 970Solution of differential equations 972

By direct integration – By separating the variables – Review exerciseSolution of differential equations 979

Homogeneous equations – by substituting y ¼ vx – Review exerciseSolution of differential equations 985

Linear equations – use of integrating factor – Review exerciseBernoulli’s equation 993


Programme 26 Second-order differential equations 1004

Learning outcomes 1004Homogeneous equations 1005

Review exerciseInhomogeneous equations 1013

Review summaryParticular solution 1020


Programme 27 Introduction to Laplace transforms 1027

Learning outcomes 1027The Laplace transform 1028

The inverse Laplace transform, Table of Laplace transformsReview summary – Review exercise

The Laplace transform 1032Laplace transform of a derivative – Two properties of Laplace transformsTable of Laplace transformsReview summary – Review exercise

The Laplace transform 1036Generating new transforms – Laplace transforms of higher derivativesTable of Laplace transforms – Linear, constant coefficient,inhomogeneous differential equations – Review summary – Reviewexercise




Contents xvii

Programme 28 Data handling and statistics 1045

Learning outcomes 1045Introduction 1046Arrangement of data 1046

Tally diagram – Grouped data – Grouping with continuous dataRelative frequency – Rounding off data – Class boundaries

Histograms 1053Frequency histogram – Relative frequency histogram

Measures of central tendency 1055Mean – Coding for calculating the mean – Decoding – Coding with agrouped frequency distribution – Mode – Mode with grouped dataMedian – Median with grouped data

Measures of dispersion 1063Mean deviation – Range – Standard deviation – Alternative formula forthe standard deviation

Distribution curves 1066Frequency polygons – Frequency curves – Normal distribution curve

Standardized normal curve 1069Review summary


Programme 29 1076

Learning outcomes 1076Probability 1077

Random experiments – Events – Sequences of random experimentsCombining events

Events and probabilities 1079Probability – Assigning probabilities – Review summary

Probabilities of combined events 1082Or – Non-mutually exclusive events – And – Dependent eventsIndependent events – Probability trees – Review summary

Conditional probability 1087Probability distributions 1090

Random variables – Expectation – Variance and standard deviationBernoulli trials – Binomial probability distribution – Expectation andstandard deviation – The Poisson probability distribution – Binomialand Poisson compared

Continuous probability distributions 1106Normal distribution curve

Standard normal curve 1106Review summary


Answers 1117Index 1145



Programme F.1 Frames 1 to 154

Arithmetic

Learning outcomes

When you have completed this Programme you will be able to:

Carry out the basic rules of arithmetic with integers

Check the result of a calculation making use of rounding

Write a whole as a product of prime numbers

Find the highest common factor and lowest common multiple of two wholenumbers

Manipulate fractions, ratios and percentages

Manipulate decimal numbers

Manipulate powers

Use standard or preferred standard form and complete a calculation to the requiredlevel of accuracy

Understand the construction of various number systems and convert from onenumber system to another.

If you already feel confident about these why not try the quiz over the page?You can check your answers at the end of the book.

3



Quiz F.1Frames

1 Place the appropriate symbol < or > between each of the following pairs ofnumbers:(a) �3 �2 (b) 8 �13 (c) �25 0 1 to 4

2 Find the value of each of the following:(a) 13þ 9� 3� 2� 5 (b) ð13þ 9Þ � ð3� 2Þ � 5 5 to 12

3 Round each number to the nearest 10, 100 and 1000:(a) 1354 (b) 2501 (c) �2452 (d) �23 625 13 to 15

4 Write each of the following as a product of prime factors:(a) 170 (b) 455 (c) 9075 (d) 1140 19 to 22

5 Find the HCF and the LCM of each pair of numbers:(a) 84, 88 (b) 105, 66 23 to 24

6 Reduce each of the following fractions to their lowest terms:

(a)1218

(b)14421

(c) �4914

(d)644

28 to 36

7 Evaluate the following:

(a)37� 2

3(b)

1130� 5

6(c)

37þ 4

13(d)

516� 4

337 to 46

8 Write the following proportions as ratios:

(a)12

of A,15

of B and3

10of C

(b)14

of P,13

of Q ,15

of R and the remainder S 47 to 48

9 Complete the following:

(a)45¼ % (b) 48% of 50¼

(c)9

14¼ % (d) 15% of 25¼ 49 to 52

10 Round each of the following decimal numbers, first to 3 significant figuresand then to 2 decimal places:(a) 21.355 (b) 0.02456(c) 0.3105 (d) 5134.555 56 to 65

11 Convert each of the following to decimal form to 3 decimal places:

(a)4

15(b) � 7

13(c)

95

(d) �2813

66 to 67

12 Convert each of the following to fractional form in lowest terms:(a) 0:8 (b) 2:8 (c) 3: _3 _2 (d) �5:5 68 to 73

13 Write each of the following in abbreviated form:(a) 1.010101 . . . (b) 9.2456456456 . . . 70 to 71

14 Write each of the following as a number raised to a power:

(a) 36 � 33 (b) 43 � 25 (c) 92� �3 (d) 70

� ��878 to 89

Questions marked with this icon can be foundat www.palgrave.com/stroud. There you can gothrough the question step by step and followonline hints, with full working provided.

4



Frames

15 Find the value of each of the following to 3 dp:

(a) 1513 (b)

ffiffiffi55p

(c) ð�27Þ13 (d) ð�9Þ

12 90 to 94

16 Write each of the following as a single decimal number:(a) 3:2044� 103 (b) 16:1105� 10�2 95 to 98

17 Write each of the following in standard form:(a) 134.65 (b) 0.002401 99 to 101

18 Write each of the following in preferred standard form:(a) 16:1105� 10�2 (b) 9.3304 102 to 104

19 In each of the following the numbers have been obtained bymeasurement. Evaluate each calculation to the appropriate level ofaccuracy:(a) 11:4� 0:0013� 5:44� 8:810

(b)1:01� 0:00335

9:12� 6:342105 to 108

20 Express the following numbers in denary form:(a) 1011:012 (b) 456:7218

(c) 123:�2912 (d) CA1:B2216 112 to 126

21 Convert 15:60510 to the equivalent octal, binary, duodecimal andhexadecimal forms. 127 to 149

5



Types of number

1 The natural numbers

The first numbers we ever meet are the whole numbers. These, together with zero, arecalled the natural numbers, and are written down using numerals.

Numerals and place value

The natural numbers are written using the ten numerals 0, 1, . . ., 9 where the positionof a numeral dictates the value that it represents. For example:

246 stands for 2 hundreds and 4 tens and 6 units. That is 200þ 40þ 6

Here the numerals 2, 4 and 6 are called the hundreds, tens and unit coefficientsrespectively. This is the place value principle.

Points on a line and order

The natural numbers can be represented by equally spaced points on a straight linewhere the first natural number is zero 0.

0 1 2 3 4 5 6 87

The natural numbers are ordered – they progress from small to large. As we movealong the line from left to right the numbers increase as indicated by the arrow at theend of the line. On the line, numbers to the left of a given number are less than (<)the given number and numbers to the right are greater than (>) the given number. Forexample, 8 > 5 because 8 is represented by a point on the line to the right of 5.Similarly, 3 < 6 because 3 is to the left of 6.

Now move on to the next frame

2 The integers

If the straight line displaying the natural numbers is extended to the left we can plotequally spaced points to the left of zero.

–5 –4 3210–1–2–3

These points represent negative numbers which are written as the natural numberpreceded by a minus sign, for example �4. These positive and negative wholenumbers and zero are collectively called the integers. The notion of order still applies.For example, �5 < 3 and �2 > �4 because the point on the line representing �5 is tothe left of the point representing 3. Similarly, �2 is to the right of �4.

The numbers �10, 4, 0, �13 are of a type called . . . . . . . . . . . .

You can check your answer in the next frame

6 Programme F.1



3Integers

They are integers. The natural numbers are all positive. Now try this:

Place the appropriate symbol < or > between each of the following pairs of numbers:

(a) �3 �6(b) �2 �4(c) �7 12

Complete these and check your results in the next frame

4(a) � 3 > �6

(b) 2 > �4

(c) � 7 < 12

The reasons being:

(a) �3 > �6 because �3 is represented on the line to the right of � 6

(b) 2 > �4 because 2 is represented on the line to the right of � 4

(c) �7 < 12 because �7 is represented on the line to the left of 12

Now move on to the next frame

5Brackets

Brackets should be used around negative numbers to separate the minus signattached to the number from the arithmetic operation symbol. For example, 5��3should be written 5� ð�3Þ and 7��2 should be written 7� ð�2Þ. Never write twoarithmetic operation symbols together without using brackets.

Addition and subtraction

Adding two numbers gives their sum and subtracting two numbers gives theirdifference. For example, 6þ 2 ¼ 8. Adding moves to the right of the first number andsubtracting moves to the left of the first number, so that 6� 2 ¼ 4 and 4� 6 ¼ �2:

4 – 66 – 2

6 + 2

–2 0 2 4 6 8 10

Adding a negative number is the same as subtracting its positive counterpart. Forexample 7þ ð�2Þ ¼ 7� 2. The result is 5. Subtracting a negative number is the sameas adding its positive counterpart. For example 7� ð�2Þ ¼ 7þ 2 ¼ 9.

So what is the value of:

(a) 8þ ð�3Þ(b) 9� ð�6Þ(c) ð�4Þ þ ð�8Þ(d) ð�14Þ � ð�7Þ?

When you have finished these check your results with the next frame

Arithmetic 7



6 (a) 5

(b) 15

(c) � 12

(d) � 7

Move now to Frame 7

7 Multiplication and division

Multiplying two numbers gives their product and dividing two numbers gives theirquotient. Multiplying and dividing two positive or two negative numbers gives apositive number. For example:

12� 2 ¼ 24 and ð�12Þ � ð�2Þ ¼ 24

12� 2 ¼ 6 and ð�12Þ � ð�2Þ ¼ 6

Multiplying or dividing a positive number by a negative number gives a negativenumber. For example:

12� ð�2Þ ¼ �24, ð�12Þ � 2 ¼ �6 and 8� ð�4Þ ¼ �2

So what is the value of:

(a) ð�5Þ � 3

(b) 12� ð�6Þ(c) ð�2Þ � ð�8Þ(d) ð�14Þ � ð�7Þ?

When you have finished these check your results with the next frame

8 (a) � 15

(b) � 2

(c) 16

(d) 2

Move on to Frame 9

9 Brackets and precedence rules

Brackets and the precedence rules are used to remove ambiguity in a calculation. Forexample, 14� 3� 4 could be either:

11� 4 ¼ 44 or 14� 12 ¼ 2

depending on which operation is performed first.

To remove the ambiguity we rely on the precedence rules:

In any calculation involving all four arithmetic operations we proceed as follows:

(a) Working from the left evaluate divisions and multiplications as they areencountered;

this leaves a calculation involving just addition and subtraction.

(b) Working from the left evaluate additions and subtractions as they areencountered.

8 Programme F.1



For example, to evaluate:

4þ 5� 6� 2� 12� 4� 2� 1

a first sweep from left to right produces:

4þ 30� 2� 3� 2� 1

a second sweep from left to right produces:

4þ 15� 6� 1

and a final sweep produces:

19� 7 ¼ 12

If the calculation contains brackets then these are evaluated first, so that:

ð4þ 5� 6Þ � 2� 12� 4� 2� 1 ¼ 34� 2� 6� 1

¼ 17� 7

¼ 10

This means that:

14� 3� 4 ¼ 14� 12

¼ 2

because, reading from the left we multiply before we subtract. Brackets must be usedto produce the alternative result:

ð14� 3Þ � 4 ¼ 11� 4

¼ 44

because the precedence rules state that brackets are evaluated first.

So that

34þ 10� ð2� 3Þ � 5 ¼ . . . . . . . . . . . .

Result in the next frame

10�16

Because

34þ 10� ð2� 3Þ � 5 ¼ 34þ 10� ð�1Þ � 5 we evaluate the bracket first¼ 34þ ð�10Þ � 5 by dividing¼ 34þ ð�50Þ by multiplying¼ 34� 50 finally we subtract¼ �16

Notice that when brackets are used we can omit the multiplication signs and replacethe division sign by a line, so that:

5� ð6� 4Þ becomes 5ð6� 4Þ

and

ð25� 10Þ � 5 becomes ð25� 10Þ=5 or25� 10

5

Arithmetic 9



When evaluating expressions containing nested brackets the innermost brackets areevaluated first. For example:

3ð4� 2½5� 1�Þ ¼ 3ð4� 2� 4Þ evaluating the innermost bracket ½. . .� first

¼ 3ð4� 8Þ

¼ 3ð�4Þ

¼ �12 multiplication completes the calculation

so that

5� f8þ 7½4� 1� � 9=3g ¼ . . . . . . . . . . . .

Work this out, the result is in the following frame

11 �21

Because

5� f8þ 7½4� 1� � 9=3g ¼ 5� f8þ 7� 3� 9� 3g¼ 5� f8þ 21� 3g¼ 5� f29� 3g¼ 5� 26

¼ �21

Now move to Frame 12

12 Basic laws of arithmetic

All the work that you have done so far has been done under the assumption that youknow the rules that govern the use of arithmetic operations as, indeed, you no doubtdo. However, there is a difference between knowing the rules innately and beingconsciously aware of them, so here they are. The four basic arithmetic operations are:

addition and subtraction

multiplication and division

where each pair may be regarded as consisting of ‘opposites’ – in each pair oneoperation is the reverse operation of the other.

1 Commutativity

Two integers can be added or multiplied in either order without affecting the result.For example:

5þ 8 ¼ 8þ 5 ¼ 13 and 5� 8 ¼ 8� 5 ¼ 40

We say that addition and multiplication are commutative operations

The order in which two integers are subtracted or divided does affect the result. Forexample:

4� 2 6¼ 2� 4 because 4� 2 ¼ 2 and 2� 4 ¼ �2

Notice that 6¼ means is not equal to. Also

4� 2 6¼ 2� 4

We say that subtraction and division are not commutative operations

multiplication before subtraction inside theð. . .Þ bracketsubtraction completes the evaluation of theð. . .Þ bracket

10 Programme F.1



2 Associativity

The way in which three or more integers are associated under addition ormultiplication does not affect the result. For example:

3þ ð4þ 5Þ ¼ ð3þ 4Þ þ 5 ¼ 3þ 4þ 5 ¼ 12

and

3� ð4� 5Þ ¼ ð3� 4Þ � 5 ¼ 3� 4� 5 ¼ 60

We say that addition and multiplication are associative operations

The way in which three or more integers are associated under subtraction or divisiondoes affect the result. For example:

3� ð4� 5Þ 6¼ ð3� 4Þ � 5 because

3� ð4� 5Þ ¼ 3� ð�1Þ ¼ 3þ 1 ¼ 4 and ð3� 4Þ � 5 ¼ ð�1Þ � 5 ¼ �6

Also

24� ð4� 2Þ 6¼ ð24� 4Þ � 2 because

24� ð4� 2Þ ¼ 24� 2 ¼ 12 and ð24� 4Þ � 2 ¼ 6� 2 ¼ 3

We say that subtraction and division are not associative operations

3 Distributivity

Consider the equations:

3� ð4þ 5Þ ¼ 3� 9 ¼ 27

and

ð3� 4Þ þ ð3� 5Þ ¼ 12þ 15 ¼ 27

From this we can deduce that:

3� ð4þ 5Þ ¼ ð3� 4Þ þ ð3� 5Þ

We say that multiplication distributes itself over addition from the left. Multiplication isalso distributive over addition from the right. For example:

ð3þ 4Þ � 5 ¼ ð3� 5Þ þ ð4� 5Þ ¼ 35

The same can be said of multiplication and subtraction: multiplication is distributiveover subtraction from both the left and the right. For example:

3� ð4� 5Þ ¼ ð3� 4Þ � ð3� 5Þ ¼ �3 and ð3� 4Þ � 5 ¼ ð3� 5Þ � ð4� 5Þ ¼ �5

Division is distributed over addition and subtraction from the right but not from theleft. For example:

ð60þ 15Þ � 5 ¼ ð60� 5Þ þ ð15� 5Þ because

ð60þ 15Þ � 5 ¼ 75� 5 ¼ 15 and ð60� 5Þ þ ð15� 5Þ ¼ 12þ 3 ¼ 15

However, 60� ð15þ 5Þ 6¼ ð60� 15Þ þ ð60� 5Þ because

60� ð15þ 5Þ ¼ 60� 20 ¼ 3 and ð60� 15Þ þ ð60� 5Þ ¼ 4þ 12 ¼ 16

Also:

ð20� 10Þ � 5 ¼ ð20� 5Þ � ð10� 5Þ because

ð20� 10Þ � 5 ¼ 10� 5 ¼ 2 and ð20� 5Þ � ð10� 5Þ ¼ 4� 2 ¼ 2

but 20� ð10� 5Þ 6¼ ð20� 10Þ � ð20� 5Þ because

20� ð10� 5Þ ¼ 20� 5 ¼ 4 and ð20� 10Þ � ð20� 5Þ ¼ 2� 4 ¼ �2

Arithmetic 11



13 Estimating

Arithmetic calculations are easily performed using a calculator. However, by pressinga wrong key, wrong answers can just as easily be produced. Every calculation madeusing a calculator should at least be checked for the reasonableness of the final resultand this can be done by estimating the result using rounding. For example, using acalculator the sum 39þ 53 is incorrectly found to be 62 if 39þ 23 is entered bymistake. If, now, 39 is rounded up to 40, and 53 is rounded down to 50 thereasonableness of the calculator result can be simply checked by adding 40 to 50 togive 90. This indicates that the answer 62 is wrong and that the calculation should bedone again. The correct answer 92 is then seen to be close to the approximationof 90.

Rounding

An integer can be rounded to the nearest 10 as follows:

If the number is less than halfway to the next multiple of 10 then the number isrounded down to the previous multiple of 10. For example, 53 is rounded downto 50.

If the number is more than halfway to the next multiple of 10 then the number isrounded up to the next multiple of 10. For example, 39 is rounded up to 40.

If the number is exactly halfway to the next multiple of 10 then the number isrounded up. For example, 35 is rounded up to 40.

This principle also applies when rounding to the nearest 100, 1000, 10 000 or more.For example, 349 rounds up to 350 to the nearest 10 but rounds down to 300 to thenearest 100, and 2501 rounds up to 3000 to the nearest 1000.

Try rounding each of the following to the nearest 10, 100 and 1000 respectively:

(a) 1846(b) �638(c) 445

Finish all three and check your results with the next frame

14 (a) 1850, 1800, 2000

(b) � 640, � 600, 1000

(c) 450, 400, 0

Because

(a) 1846 is nearer to 1850 than to 1840, nearer to 1800 than to 1900 and nearerto 2000 than to 1000.

(b) �638 is nearer to �640 than to �630, nearer to �600 than to �700 and nearerto �1000 than to 0. The negative sign does not introduce any complications.

(c) 445 rounds to 450 because it is halfway to the next multiple of 10, 445 is nearerto 400 than to 500 and nearer to 0 than 1000.

How about estimating each of the following using rounding to the nearest 10:

(a) 18� 21� 19� 11(b) 99� 101� 49� 8

Check your results in Frame 15

12 Programme F.1



Index

A priori 1080Absolute values 148Accuracy 42Acute angle 245Addition 7

Algebraic fractions 85Complex numbers 391Fractions 22Graphical 400Matrices 492Powers 33Vectors 522

Adjoint matrix 500Algebra

Brackets 69Constants 66Factorization 68, 88Fractions 85Introduction 63Symbols 65Variables 66

Algebraic equations 65, 101Algebraic expressions

Factorization 88Algebraic fractions 85Algebraic symbols 65Amplitude 294And 1084Angle between two vectors 537Angles 245

Rotation 245Applications of multiple integrals 951Applications of polar curves 929

Area of surface of revolution 937Length of arc 934Volume of rotation 932

Approximate integration 902By series 904By Simpson’s rule 908

Area between a curve and a line 374Areas under curves 366, 835Argand diagram 399Arithmetic 3

Addition 7Brackets 7, 8Calculations 13

Decimal numbers 27Division 8Estimating 12Factors 15Fractions 18Fundamental theorem 16Highest common factor (HCF) 16Laws 10Lowest common multiple (LCM) 16Multiplication 8Negative numbers 6Number systems 44Percentages 18Precedence rules 8, 13Prime factorization 15Prime numbers 15Quotient 8Rational number 18Ratios 18Rounding 12Subtraction 7Types of number 6

Arithmetic operations 272Arithmetic sequence 608

Common difference 608Arrangement of data 1046Arrangements 200Assigning equations 105Assigning probabilities 1080Associativity 11, 67Asymptote 134, 697, 703

Determination 698Parallel to axes 699

Augmented matrix 507Auxiliary equation 1006

Complex roots 1009Real and different roots 1007Real and equal roots 1008

Bernoulli trials 1094Bernoulli’s equation 993Binary system 45Binomial expansions 205, 207

General term 209Binomial expectation 1099Binomial probability distribution 1095, 1102

1145



Binomial standard deviation 1099Binomials 195

Arrangements 200Binomial expansions 205, 207, 209Combinations 200Combinatorial coefficients 203Factorials 197Pascal’s triangle 205Permutations 200

Brackets 7, 8Expanding 69Nested 69

Calculations 13Accuracy 42Checking 42Decimal places 27Significant figures 27Standard form 39

Calculator 23Powers 36Standard form 40

Cancelling 20Cartesian axes 130Cartesian graph 139Catenary 438Cause-and-effect 718Centre of curvature 594, 598Centre of gravity of a solid of revolution 859Centre of pressure 892

Depth of centre of pressure 895Pressure at a point 892Total thrust 892

Centroid of a plane figure 856Certainty 1079Chain rule 338Change of base of logarithm 77Change of number base

Denary decimal to duodecimal 54Denary decimal to octal 53Denary to binary 52Denary to duodecimal 52Octals as an intermediate step 55Reverse method 56

Change of variables 763Characteristic determinant 510Characteristic equation 510, 620

Equal roots 623Characteristic values 509Characteristic vectors 509

Checking calculations 42Circle 692Class boundaries 1051Class interval 1051Co-domain 271Coding for a grouped distribution 1058Coding for calculating the mean 1057Coefficients 6, 68, 72Cofactor 500Collecting like terms 68Combinations 200Combinatorial coefficients 203

Properties 203Combining events 1079Common denominator 22Common difference 608Common factors 68

Algebra 88Common ratio 609Commutativity 10, 67Complementary function 1013Completing the square 181Complex conjugate 393Complex numbers 385, 412

Addition and subtraction 391Argand diagram 399Complex conjugate 393Complex plane 399De Moivre’s theorem 422Division 394Equal 396Exponential form 406Graphical representation 399Imaginary part 390Loci problems 430Multiplication 392Polar form 401, 413Principal root 425Real part 390Roots 421Trigonometric expansions 427

Complex plane 399Components of a vector 524Composition 280Conditional equations 104Conditional probability 1087Consistency of a set of equations 478Constant of integration 355Constants 66Continuous variable 1046

1146 Index



Index 1147

Convergence 628Correlation 718

Measures 719Pearson product-moment coefficient 719Spearman’s rank coefficient 723

Cosecant 251Cosine 249Cosine function 290Cotangent 251Cube root 36Cubic equations 184Curvature 592

Centre of curvature 594. 598Parametric equations 596Radius of curvature 594

Curve fitting 707Straight-line law 707

Curve sketching 702Asymptotes 703Change of origin 703Intersection with axes 702Large and small values of the variables 704Limitations 705Stationary points 704Symmetry 702

Curves 688Asymptotes 697Circle 692Ellipse 693Exponential curves 695Hyperbola 693Hyperbolic curves 696Logarithmic curves 695Second-degree 690Standard curves 689Third-degree 691Trigonometrical curves 696

Data handling 1045Arrangement of data 1046Class boundaries 1051Class interval 1051Frequency distribution 1047Grouped data 1048Grouping with continuous data 1049Histograms 1053Relative frequency 1050Rounding off data 1050Tally diagram 1047

De Moivre’s theorem 422

Trigonometric expansions 427Decimal numbers 27

As fractions 29Standard form 38Trailing zeros 29Unending 30

Decimal places (dp) 27, 28Decimal point 27Decoding 1058Defining equations 104Definite integrals 838Definition of a Laplace transform 1028Degrees 245Denary system 44Denominator 18

Common 22Dependent events 1084Dependent variables 104Depth of centre of pressure 895Derivative

Implicit functions 759Partial 732

Derivative of a function of a function 337Derivative of a product 333Derivative of a quotient 334Derivative of powers of x 323Derived curve 572Determinants 459

Properties 481Second-order 461Simultaneous equations in three

unknowns 470Square matrix 499Third-order 466

Diagonal matrix 497Difference equations 618

Characteristic equation 620Homogeneous equation 619Second-order, homogeneous 621Solving 619

Difference of two numbers 7Differential equations

First-order 968Formation 970Solution 972

Differentials 322Differentiation 315, 544

Function of a function 546Implicit functions 555Inverse hyperbolic functions 565



Differentiation (cont.)Inverse trigonometric functions 563Logarithmic differentiation 552Maximum and minimum values 570Partial differentiation 731Point of inflexion 571Products 549Quotients 550Standard derivatives 545Stationary points 571Parametric equations 557

Differentiation of polynomials 326Direction cosines 532Direction ratios 540Discontinuity 135Discrete variable 1046Distribution curves 1066

Frequency curves 1067Frequency polygons 1066Normal distribution curve 1067

Distributivity 11, 67Divergence 629Division 8

Algebraic expressions 82Algebraic fractions 86Complex numbers 394Fractions 21By integer powers of 10 37Subtraction of powers 34

Division of integers 27Domain 271Double integrals 947Double suffix notation 491Drawing a graph 131Duodecimal system 46

e 217Eigenvalues 509Eigenvectors 509, 511Ellipse 693Equal complex numbers 396Equal matrices 492Equal roots of characteristic equation 623Equal vectors 521Equation of a straight line 584Equations 104

Assigning 105Conditional 104Consistency of a set 478Cubic equations 184

Defining 104Formula 105Fourth-order polynomials 187Graphs of 129Homogeneous equation 619Identity 104Linear equations 163Logarithmic 78Newton–Raphson method 344Polynomial equations 113, 177Simultaneous equations 165Solution of simple equations 163

Equivalent fractions 20Estimating 12Evaluating

Expressions 103Hyperbolic functions 445Independent variables 106Third-order determinant 467

Evaluation by nesting 114Evaluation of areas 960Evaluation of polynomials 114Evaluation of volumes 962Evaluation process 111Even functions 307Events 1077Events and probabilities 1079

A priori 1080Assigning probabilities 1080Certainty 1079Impossibility 1079Probability 1079Probability distribution 1080Statistical regularity 1080

Excel 137Expanding brackets 69Expectation 1092Explicit function 555Exponent 38Exponential curves 695Exponential form of a complex number 406Exponential functions 287, 303Exponential number e 217Expressions

Algebraic 65Evaluating 103Polynomial 113

Factor theorem 116Factorials 197

1148 Index



Index 1149

Factorization 68Algebraic expressions 88Fourth-order polynomials 119Perfect square 94Quadratic expressions 91, 93Test for simple factors 94

Factors 15Fibonacci sequence 615Fitting a straight-line graph 712Formula 105Fourth-order polynomials 187Fractional powers 36Fractions 18

Addition 22Algebraic 85Calculator 23Cancelling 20As decimals 29Division 21Equivalent 20Lowest terms 20Reciprocal 21Subtraction 22

Free vector 522Frequency curves 1067Frequency distribution 1047Frequency polygons 1066Full angle 245Function of a function 280, 546Functions 267

Amplitude 294Arithmetic operations 272Co-domain 271Composition 280Cosine function 290Domain 271Even functions 307Exponential 287, 303Function of a function 280Functions as rules 270Graphs of inverse functions 274Indicial equations 304Inverse functions 272Inverse of a composition 282Inverse trigonometric functions 296Limits of functions 309Logarithmic functions 303Odd and even parts 307Odd functions 307Period 292

Phase difference 295Processing numbers 269Range 271Sine function 290Tangent function 291Trigonometric 287

Functions as rules 270Functions of a linear function of x 772

Integration 360Functions with integer input 606Fundamental theorem of arithmetic 16Fundamental trigonometric identity 258

Gaussian elimination 507Generating new Laplace transforms 1036Geometric sequence 609

Common ratio 609Gradient determined algebraically 322Gradient of a curve 320Gradient of a straight line 317Gradient of a tangent 320Gradients 317Gradients of perpendicular lines 585Graphical addition of complex numbers 400Graphs 127

Absolute values 149Drawing 131Hyperbolic functions 440Inverse functions 274Sequences 606

Grouped data 1048Grouping with continuous data 1049

Half equilateral triangle 255Harmonic sequence 610Hexadecimal system 47Higher derivatives 327Highest common factor (HCF) 16Histograms 1053

Frequency histogram 1054Relative frequency histogram 1055

Homogeneous equation 619, 979Second-order 621

Hyperbola 693Hyperbolic cosine 438Hyperbolic curves 696Hyperbolic functions 437

Cosine 438Evaluation 445Graphs 440



Hyperbolic functions (cont.)Identities 451Inverse functions 446Sine 438Tangent 439And trigonometric functions 453

Hyperbolic identities 451Hyperbolic sine 438Hyperbolic tangent 439Hypotenuse 249

Identities 104Identities for compound angles 260Imaginary number 386Imaginary part of a complex number 390Implicit functions 555Impossibility 1079Independent events 1085Independent variables 104Indeterminate limits 633Index 33Indicial equations 304Inequalities 147Infinite limits 629Infinity 627Input – process – output 111Integers 6Integrating factor 985Integration 353, 768, 794

Area between a curve and a line 374Areas under curves 366, 835Centre of gravity of a solid of

revolution 859Centroid of a plane figure 856Change of variable 776Definite integrals 838Difference of squares in the

denominator 795Functions of a linear function of x 360, 772Length of a curve 860Mean value 845Numerator derivative of denominator 774Parametric equations 843, 862, 865Partial fractions 363, 782By parts 778Root mean square (rms) value 846Rules of Pappus 867Square roots in the denominator 804Square roots in the numerator 810Standard integrals 769

As a sum 370, 847Sum of squares in the denominator 802Surface of revolution 864Trigonometric functions 787Volume of revolution 852

Integration as a sum 370, 847Integration by partial fractions 782Integration by parts 778Integration of polynomials 358Integration by series 904Integration of trigonometric functions 787Inverse functions 272Inverse hyperbolic functions 446, 565

Log form 448Inverse of a composition 282Inverse of a square matrix 501Inverse transform 1029Inverse trigonometric functions 296, 563Irrational numbers 31Isosceles triangle 253

j 386Powers of 389

Laplace transform of a derivative 1032Laplace transforms 1027

Definition 1028Generating new transforms 1036Inverse transform 1029Properties 1032Table of transforms 1030, 1034, 1039Transform of a derivative 1032Transforms of higher derivatives 1037

Laplace transforms of higherderivatives 1037

Latent roots 509Laws of arithmetic 10Laws of powers 33Length of a curve 860Like terms 68Limits 310Limits of functions 309Limits of sequences 626, 628

Indeterminate 633Infinity 627

Linear equations 161Pre-simplification 169Solution 504

Linear, constant coefficient, inhomogeneousdifferential equations 1039

1150 Index



Loci problems 430Log form of inverse hyperbolic

functions 448Logarithmic curves 695Logarithmic differentiation 552Logarithmic equations 78Logarithmic functions 303Logarithms 72

Bases 10 and e 76Change of base 77Natural 76

Lowest common multiple (LCM) 16Lowest terms 20

Mantissa 38Matrix 489

Addition and subtraction 492Augmented 507Cofactor 500Diagonal 497Double suffix notation 491Eigenvalues 509Eigenvectors 509, 511Equal 492Gaussian elimination 507Inverse 501Multiplication 493Notation 491Null 498Order 490Scalar multiplication 493Single element 491Singular 499Skew-symmetric 497Square 497Symmetric 497Transpose 496Unit 497

Maximum values 570Mean 1055Mean deviation 1063Mean values 844Measures of central tendency 1055

Coding for a grouped distribution 1058Coding for calculating the mean 1057Decoding 1058Mean 1055Median 1061; with grouped data 1061Mode 1059; with grouped data 1059

Measures of correlation 719

Measures of dispersion 1063Mean deviation 1063Range 1063Standard deviation 1063, 1064Variance 1063

Median 1061With grouped data 1061

Method of least squares 712Using a spreadsheet 714

Minimum values 570Mode 1059

With grouped data 1059Modulus 148Moments of inertia 874

Circular disc 883Parallel axes theorem 881Perpendicular axes theorem 884Radius of gyration 877Rectangular plate 878Rod 877Standard results 885

Multiple integrals 943Applications 951Areas 960Double integrals 947Triple integrals 949Volumes 962

Multiplication 8Addition of powers 33Algebraic expressions 81Algebraic fractions 86Complex numbers 392Fractions 19By integer powers of 10 37Matrices 493Powers 35Scalar 493

Mutually exclusive 1077

Natural logarithms 76Natural numbers 6Negative numbers 6Negative powers 35Nested brackets 69Nesting 114Newton–Raphson iteration 344Non-mutually exclusive events 1083Normal distribution curve 1067, 1106Normal to a curve 587

Parametric equations 590

Index 1151



1152 Index

Null matrix 498Number systems 44

Binary 45Change of base 52Denary 44Duodecimal 46Hexadecimal 47Octal 45

Numerals 6Numerator 18

Obtuse angle 245Octal system 45Odd and even parts 307Odd functions 307Of 19Or 1082Order 6Ordered pairs of numbers 130Outcome tree 1078

Parabola 690Parallel axes theorem 881Parameter 557Parametric equations 557, 843, 862, 865Partial derivative 732

Second derivative 738Partial differentiation 731, 753

Change of variables 763Rate-of-change problems 755Small increments 744

Partial fractions 223Denominators with quadratic

factors 232Denominators with repeated factors 234Integration 363

Particular integral 1013Particular solution 1020Pascal’s triangle 205Pearson product-moment correlation

coefficient 719Percentages 18, 24Perfect square 94Period 292Permutations 200Perpendicular axes theorem 884Phase difference 295Place value 6Points of inflexion 571, 574

Points on a line 6Poisson mean 1101Poisson probability distribution 1101, 1102Poisson standard deviation 1101Poisson variance 1101Polar coordinate systems 921

Applications 929Polar curves 924Standard polar curves 926

Polar curves 924Polar form calculations 413Polar form of a complex number 401Polynomial equations 113, 177

Quadratic equations 179Polynomials 113

Differentiation 326Factor theorem 116Integration 358Remainder theorem 115

Position vector 522Power unity 33Power zero 34Powers 33, 72

On a calculator 36Fractional 36Of j 389Negative 35Roots 36Surds 37

Precedence rules 8, 13, 38, 68Preferred standard form 40Prescription 606Pressure at a point in a fluid 892Prime factorization 15Prime numbers 15Principal root of a complex number 425Probabilities of combined events 1082

And 1084Dependent events 1084Independent events 1085Non-mutually exclusive events 1083Or 1082Probability trees 1085

Probability 1076, 1079Combining events 1079Events 1077Mutually exclusive 1077Outcome tree 1078Random experiments 1077Sequences of random experiments 1078



Probability distributions 1080, 1090Bernoulli trials 1094Binomial expectation 1099Binomial probability distribution 1095, 1102Binomial standard deviation 1099Expectation 1092Normal distribution curve 1106Poisson mean 1101Poisson probability distribution 1101, 1102Poisson standard deviation 1101Poisson variance 1101Random variables 1091Standard deviation 1093Standard normal curve 1106Variance 1093

Probability trees 1085Processing numbers 269Product of a square matrix and its inverse 504Product of simple factors 89Properties of determinants 481Properties of Laplace transforms 1032Pythagoras’ theorem 252

Quadratic equations 179Solution 386Solution by completing the square 181Solution by factors 179Solution by formula 183

Quadratic expressions 91Factorization 91

Quotient 8

Radians 245, 246Radius of curvature 594Radius of gyration 877Random experiments 1077Random variables 1091Range 271, 1063Rate-of-change problems 755Rational numbers 18, 31Ratios 18, 24Real numbers 31, 386Real part of a complex number 390Reciprocal 21Reciprocal trigonometric ratios 251Recurrence relation 618Recursive prescriptions 611Recursive process 611Reduction formulas 821

Wallis’s formula 830Relative frequency 1050

Remainder theorem 115Right angle 245Right-hand rule 530Root mean square (rms) value 846Roots 36Roots of a complex number 421Rotation 289Rounding 12, 27Rounding off data 1050Rules for manipulating sums 215Rules of algebra 67Rules of derivatives 330Rules of indices 72Rules of limits 310, 629Rules of logarithms 75Rules of Pappus 867

Scalar multiplication 493Scalar product of vectors 533Scalar quantity 520Secant 251Second-degree curves 690Second derivatives 327Second moment of area 886

Composite figures 892Rectangle 889Triangle 891

Second-order differential equations 1004Auxiliary equation 1006Complementary function 1013Homogeneous equations 1005Inhomogeneous equations 1013Particular integral 1013Particular solution 1020

Second partial derivative 738Separating variables 973Sequences 605

Arithmetic sequence 608Convergence 628Divergence 629Fibonacci 615Geometric sequence 609Graphs 606Harmonic sequence 610Indeterminate limits 633Infinite limits 629Limits 626Prescription 606Recursive prescriptions 611Rules of limits 629

Index 1153



Sequences of random experiments 1078Sigma notation 211Significant figures (sig fig) 27, 28Similar terms 68Similar triangles 247Simpson’s rule 908

Proof 916Simultaneous equations 460

Three unknowns 470Simultaneous linear equations 165, 167

Solution by equating coefficients 166Solution by substitution 165

Sine 249Sine function 290Single element matrix 491Singular matrix 499Skew-symmetric matrix 497Small increments 744Solution of a set of linear equations 504

Gaussian elimination 507Solution of first-order differential

equations 972Bernoulli’s equation 993Direct integration 973Homogeneous equations 979Integrating factor 985Separating the variables 973

Spearman’s rank correlationcoefficient 723

Special matrices 497Special triangles 253Spreadsheet 137, 714

Absolute values 149Active cell 138Cartesian graph 139Clearing entries 139Columns 138Cursor 138Displays 141Formulas 139Inequalities 147, 150Interaction 154Mouse pointer 138Number entry 138Rows 138Text entry 138

Square matrix 497Adjoint 500Determinant 499Inverse 501

Square root 36Negative number 387

Standard curves 689Standard derivatives 330, 545Standard deviation 1063, 1064, 1093Standard form 38

Preferred standard form 40Standard integrals 355, 769Standard normal curve 1069, 1106

Table of values 1108Standard polar curves 926Stationary points 571, 704Statistical regularity 1080Statistics 1045

Distribution curves 1066Measures of central tendency 1055Measures of dispersion 1063Standardized normal curve 1069

Straight angle 245Straight line 689Straight-line law 707Subtraction 7

Algebraic fractions 85Complex numbers 391Fractions 22Matrices 492Powers 34

Sum of first n natural numbers 215Sum of two numbers 7Sums 215Surds 37Surface of revolution 864Symmetric matrix 497Symmetry 702System 111

Table of Laplace transforms 1039Tally diagram 1047Tangent 249Tangent to a curve 587

Parametric equations 590Tangent function 291Tangents, normals and curvature 583Terms 68Test for simple factors 94Third-degree curves 691Total thrust 892Trailing zeros 29Transpose of a matrix 496Transposing variables 106, 107

1154 Index



Transposition of formulas 107Triangles 247

Half equilateral 255Isosceles 253Special triangles 253

Trigonometric equations 297Trigonometric formulas 262Trigonometric functions 287

And hyperbolic functions 453Rotation 289

Trigonometric identities 258Trigonometric ratios 249

Double angles 262Products of ratios 262Sums and differences of angles 262Sums and differences of ratios 262

Trigonometrical curves 696Trigonometry 243

Identities for compound angles 260Triple integrals 949Types of number 6

Complex 385Decimal 27Fractions 18Imaginary 386Integers 6Irrational numbers 31Natural numbers 6Negative numbers 6Rational numbers 18, 31

Real number 31Whole numbers 6

Unending decimals 30Unit matrix 497Unit vectors 528

Variables 66Variance 1063, 1093Vector product of vectors 535Vector quantity 520Vector representation 521Vectors 519

Addition 522Angle between 537Components 524Direction cosines 532Direction ratios 540Equal 521Representation 521Scalar product 533In space 530Types 522In terms of unit vectors 528Vector product 535

Volumes of revolution 852

Wallis’s formula 830Whole numbers 6

Index 1155





Documents

01 prelims 5. · Types of number 6 The natural numbers – Numerals and place value – Points on a line and order – The integers – Brackets – Addition and subtraction Multiplication