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The Chemistry Maths Book
Second Edition
Erich SteinerUniversity of Exeter
without mathematics the sciences cannot be understood, normade
clear, nor taught, nor learned.
(Roger Bacon, 12141292)
1
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3Great Clarendon Street, Oxford OX2 6DP
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E. Steiner 2008
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Oxford University Press (maker)
First published 1996 Second edition published 2008
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Library of Congress Cataloging in Publication Data
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Typeset by Graphicraft Limited, Hong KongPrinted in Great
Britain on acid-free paper by Ashford Colour Press Ltd, Gosport,
Hampshire
ISBN 9780199205356
1 3 5 7 9 10 8 6 4 2
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Preface
This book describes the mathematics required for the full range
of topics that make upa university degree course in chemistry. It
has been designed as a textbook for coursesin mathematics for
chemists.
Structure of the book
The subject is developed in a logical and consistent way with
few assumptions made of prior knowledge of mathematics. The
material is organized in three largelyindependent parts: Chapters 1
to 15 on algebra, the calculus, differential equations,and
expansions in series; Chapters 16 to 19 on vectors, determinants,
and matrices;Chapters 20 and 21 are introductions to the big topics
of numerical analysis andstatistics.
A feature of the book is the extensive use of examples to
illustrate every importantconcept and method in the text. Some of
these examples have also been used todemonstrate applications of
the mathematics in chemistry and several basic conceptsin physics.
The exercises at the end of each chapter are an essential element
of thedevelopment of the subject, and have been designed to give
the student a workingunderstanding of the material in the text. The
text is accompanied by a footnote history of mathematics.
Several topics in chemistry are given extended treatments. These
include the concept of pressurevolume work in thermodynamics in
Chapter 5, periodic systemsin Chapter 8, the differential equations
of chemical kinetics in Chapter 11, and severalapplications of the
Schrdinger equation in Chapters 12 and 14. In addition, the
contents of several chapters are largely dictated by their
applications in the physical sciences: Chapter 9, the mathematics
of thermodynamics; Chapters 10 and 16, thedescription of systems
and processes in three dimensions; Chapter 13 (advanced),some
important differential equations and special functions in
mathematical chemistryand physics; Chapter 15 (advanced),
intermolecular forces, wave analysis, and Fouriertransform
spectroscopy; Chapters 18 and 19, molecular symmetry and
symmetryoperations, molecular orbital theory, molecular dynamics,
and advanced quantummechanics.
Global changes in this edition
1. An overall reorganization has been carried out to link the
text and examples moreclosely to the exercises at the end of each
chapter. The symbol 0 has been placed atappropriate places within
the body of the text as a pointer to the relevant exercises.New
examples and exercises have been inserted to give a more complete
coverageof the development of the mathematics.
2. In addition to the solutions to the numerical exercises given
at the back of thebook, a full set of worked solutions of the
end-of-chapter exercises has been placedon the books companion
website at www.oxfordtextbooks.co.uk/orc/steiner2e
3. The opportunity has been taken to consolidate the several
major and many minorcorrections and improvements that have been
made over the years since publicationof the rst edition in 1996. A
small number of new historical footnotes have been
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vi Preface
added to accompany new material. The material within some
chapters has beenreordered to make the development of the subject
more logical.
Other principal changes
Chapter 1. A new section, Factorization, factors, and
factorials, lls a gap in the coverageof elementary topics. The
rules of precedence for arithmetic operations has beenbrought
forward from chapter 2 and extended with examples and exercises,
providing further revision and practice of the arithmetic that is
so important for the understanding of the material in subsequent
chapters. The biggest change inthe chapter, reected in the change
of title to Numbers, variables, and units, is arewritten and much
enlarged section on units to make it a more authoritative anduseful
account of this important but often neglected topic. It includes
new examplesof the type met in the physical sciences, a brief
subsection on dimensional analysis,and a new example and exercise
on the structure of atomic units.
Chapter 2. Parts of the chapter have been rewritten to
accommodate more discussionof the factorization and manipulation of
algebraic expressions.
Chapter 7. Numerous small changes have been made, including an
introduction tothe multinomial expansion, and revision of the
discussion of the Taylor series.
Chapter 9. Section 9.8 has been rewritten to clarify the
relevance of line integrals tochange of state in
thermodynamics.
Chapter 13. The section on the Frobenius method has been
revised, with new andmore demanding examples and exercises.
Chapter 19. Sections 19.2 and 19.3 on eigenvalues and
eigenvectors have been rewritten, with new examples and exercises,
to improve the ow and clarity of thediscussion.
Acknowledgements
I wish to express my gratitude to colleagues and students at
Exeter University and otherinstitutions for their often helpful
comments on the previous edition of the book, forpointing out
errors and obscurities, and for their suggestions for improvements.
Iparticularly want to thank Anthony Legon for his valuable comments
on the revisedchapters 1 and 9.
I also wish to thank the reviewers of this book for their
generous response to theproposal of a second edition, and the staff
of Oxford University Press for theirpatience and help.
Above all, I want to thank my wife Mary, without whom nothing
could have beendone.
Erich Steiner, Exeter, June 2007
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Contents
1 Numbers, variables, and units 11.1 Concepts 1
1.2 Real numbers 3
1.3 Factorization, factors, and factorials 7
1.4 Decimal representation of numbers 9
1.5 Variables 13
1.6 The algebra of real numbers 14
1.7 Complex numbers 19
1.8 Units 19
1.9 Exercises 29
2 Algebraic functions 312.1 Concepts 31
2.2 Graphical representation of functions 32
2.3 Factorization and simplication of expressions 34
2.4 Inverse functions 37
2.5 Polynomials 40
2.6 Rational functions 50
2.7 Partial fractions 52
2.8 Solution of simultaneous equations 55
2.9 Exercises 58
3 Transcendental functions 623.1 Concepts 62
3.2 Trigonometric functions 63
3.3 Inverse trigonometric functions 72
3.4 Trigonometric relations 73
3.5 Polar coordinates 77
3.6 The exponential function 80
3.7 The logarithmic function 83
3.8 Values of exponential and logarithmic functions 86
3.9 Hyperbolic functions 87
3.10 Exercises 89
4 Differentiation 934.1 Concepts 93
4.2 The process of differentiation 94
4.3 Continuity 97
4.4 Limits 98
4.5 Differentiation from rst principles 100
4.6 Differentiation by rule 102
4.7 Implicit functions 110
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viii Contents
4.8 Logarithmic differentiation 111
4.9 Successive differentiation 113
4.10 Stationary points 114
4.11 Linear and angular motion 118
4.12 The differential 119
4.13 Exercises 122
5 Integration 1265.1 Concepts 126
5.2 The indenite integral 127
5.3 The denite integral 132
5.4 The integral calculus 142
5.5 Uses of the integral calculus 147
5.6 Static properties of matter 148
5.7 Dynamics 152
5.8 Pressurevolume work 157
5.9 Exercises 160
6 Methods of integration 1636.1 Concepts 163
6.2 The use of trigonometric relations 163
6.3 The method of substitution 165
6.4 Integration by parts 173
6.5 Reduction formulas 176
6.6 Rational integrands. The method of partial fractions 179
6.7 Parametric differentiation of integrals 184
6.8 Exercises 187
7 Sequences and series 1917.1 Concepts 191
7.2 Sequences 191
7.3 Finite series 196
7.4 Innite series 203
7.5 Tests of convergence 204
7.6 MacLaurin and Taylor series 208
7.7 Approximate values and limits 214
7.8 Operations with power series 219
7.9 Exercises 221
8 Complex numbers 2258.1 Concepts 225
8.2 Algebra of complex numbers 226
8.3 Graphical representation 228
8.4 Complex functions 235
8.5 Eulers formula 236
8.6 Periodicity 240
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Contents ix
8.7 Evaluation of integrals 244
8.8 Exercises 245
9 Functions of several variables 2479.1 Concepts 247
9.2 Graphical representation 248
9.3 Partial differentiation 249
9.4 Stationary points 253
9.5 The total differential 258
9.6 Some differential properties 262
9.7 Exact differentials 272
9.8 Line integrals 275
9.9 Multiple integrals 281
9.10 The double integral 283
9.11 Change of variables 285
9.12 Exercises 289
10 Functions in 3 dimensions 29410.1 Concepts 294
10.2 Spherical polar coordinates 294
10.3 Functions of position 296
10.4 Volume integrals 299
10.5 The Laplacian operator 304
10.6 Other coordinate systems 307
10.7 Exercises 312
11 First-order differential equations 31411.1 Concepts 314
11.2 Solution of a differential equation 315
11.3 Separable equations 318
11.4 Separable equations in chemical kinetics 322
11.5 First-order linear equations 328
11.6 An example of linear equations in chemical kinetics 330
11.7 Electric circuits 332
11.8 Exercises 334
12 Second-order differential equations. Constant coefcients
33712.1 Concepts 337
12.2 Homogeneous linear equations 337
12.3 The general solution 340
12.4 Particular solutions 344
12.5 The harmonic oscillator 348
12.6 The particle in a one-dimensional box 352
12.7 The particle in a ring 356
12.8 Inhomogeneous linear equations 359
12.9 Forced oscillations 363
12.10 Exercises 365
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x Contents
13 Second-order differential equations. Some special functions
36813.1 Concepts 368
13.2 The power-series method 369
13.3 The Frobenius method 371
13.4 The Legendre equation 375
13.5 The Hermite equation 381
13.6 The Laguerre equation 384
13.7 Bessel functions 385
13.8 Exercises 389
14 Partial differential equations 39114.1 Concepts 391
14.2 General solutions 392
14.3 Separation of variables 393
14.4 The particle in a rectangular box 395
14.5 The particle in a circular box 398
14.6 The hydrogen atom 401
14.7 The vibrating string 410
14.8 Exercises 413
15 Orthogonal expansions. Fourier analysis 41615.1 Concepts
416
15.2 Orthogonal expansions 416
15.3 Two expansions in Legendre polynomials 421
15.4 Fourier series 425
15.5 The vibrating string 432
15.6 Fourier transforms 433
15.7 Exercises 441
16 Vectors 44416.1 Concepts 444
16.2 Vector algebra 445
16.3 Components of vectors 448
16.4 Scalar differentiation of a vector 453
16.5 The scalar (dot) product 456
16.6 The vector (cross) product 462
16.7 Scalar and vector elds 466
16.8 The gradient of a scalar eld 467
16.9 Divergence and curl of a vector eld 469
16.10 Vector spaces 471
16.11 Exercises 471
17 Determinants 47417.1 Concepts 474
17.2 Determinants of order 3 476
17.3 The general case 481
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Contents xi
17.4 The solution of linear equations 483
17.5 Properties of determinants 488
17.6 Reduction to triangular form 493
17.7 Alternating functions 494
17.8 Exercises 496
18 Matrices and linear transformations 49918.1 Concepts 499
18.2 Some special matrices 502
18.3 Matrix algebra 505
18.4 The inverse matrix 513
18.5 Linear transformations 516
18.6 Orthogonal matrices and orthogonal transformations 521
18.7 Symmetry operations 524
18.8 Exercises 529
19 The matrix eigenvalue problem 53219.1 Concepts 532
19.2 The eigenvalue problem 534
19.3 Properties of the eigenvectors 537
19.4 Matrix diagonalization 543
19.5 Quadratic forms 546
19.6 Complex matrices 551
19.7 Exercises 555
20 Numerical methods 55820.1 Concepts 558
20.2 Errors 558
20.3 Solution of ordinary equations 562
20.4 Interpolation 566
20.5 Numerical integration 573
20.6 Methods in linear algebra 581
20.7 Gauss elimination for the solution of linear equations
581
20.8 GaussJordan elimination for the inverse of a matrix 584
20.9 First-order differential equations 585
20.10 Systems of differential equations 590
20.11 Exercises 592
21 Probability and statistics 59521.1 Concepts 595
21.2 Descriptive statistics 595
21.3 Frequency and probability 601
21.4 Combinations of probabilities 603
21.5 The binomial distribution 604
21.6 Permutations and combinations 607
21.7 Continuous distributions 613
21.8 The Gaussian distribution 615
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xii Contents
21.9 More than one variable 618
21.10 Least squares 619
21.11 Sample statistics 623
21.12 Exercises 624
Appendix. Standard integrals 627
Solutions to exercises 631
Index 653
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1 Numbers, variables, and units
1.1 Concepts
Chemistry, in common with the other physical sciences,
comprises
(i) experiment: the observation of physical phenomena and the
measurement ofphysical quantities, and
(ii) theory: the interpretation of the results of experiment,
the correlation of one setof measurements with other sets of
measurements, the discovery and applicationof rules to rationalize
and interpret these correlations.
Both experiment and theory involve the manipulation of numbers
and of the symbolsthat are used to represent numbers and physical
quantities. Equations containingthese symbols provide relations
amongst physical quantities. Examples of such equations are
1. the equation of state of the ideal gas
pV = nRT (1.1)
2. Braggs Law in the theory of crystal structure
n1= 2d1sin1 (1.2)3. the Arrhenius equation for the temperature
dependence of rate of reaction
(1.3)
4. the Nernst equation for the emf of an electrochemical
cell
(1.4)
When an equation involves physical quantities, the expressions
on the two sides of theequal sign1 must be of the same kind as well
as the same magnitude.
E ERTnF
Q= o ln
k AeE RT
= a /
1 The sign for equality was introduced by Robert Recorde (c.
15101558) in his The whetstone of witte (London,1557); I will sette
as I doe often in woorke use, a paire of paralleles, or Gemowe
(twin) lines of one lengthe, thus:, bicause noe.2. thynges can be
moare equalle.
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2 Chapter 1 Numbers, variables, and units
EXAMPLE 1.1 The equation of state of the ideal gas, (1.1), can
be written as anequation for the volume,
in which the physical quantities on the right of the equal sign
are the pressure p of the gas, the temperature T, the amount of
substance n, and the molar gas constantR = 8.31447 J K1 mol1.
We suppose that we have one tenth of a mole of gas, n1=10.1 mol,
at temperature T1=1298 K and pressure p1=1105 Pa. Then
The quantities on the right side of the equation have been
expressed in terms of SIunits (see Section 1.8), and the
combination of these units is the SI unit of volume, m3
(see Example 1.17).
Example 1.1 demonstrates a number of concepts:
(i) Function. Given any particular set of values of the pressure
p, temperature T,and amount of substance n, equation (1.1) allows
us to calculate the corresponding volume V. The value of V is
determined by the values of p, T, and n; we say
V is a function of p, T, and n.
This statement is usually expressed in mathematics as
V1=1f ( p, T, n)and means that, for given values of p, T and n,
the value of V is given by the value of afunction f ( p, T, n). In
the present case, the function is f (p, T, n)1=1nRT2p. A
slightlydifferent form, often used in the sciences, is
V1=1V( p, T, n)which means that V is some function of p, T and
n, which may or may not be known.
Algebraic functions are discussed in Chapter 2. Transcendental
functions, includingthe trigonometric, exponential and logarithmic
functions in equations (1.2) to (1.4),are discussed in Chapter
3.
= . 2 478 10 3 3 m
=
. .
0 1 8 31447 298
105
1 1mol J K mol KPa
VnRT
p= =
. . 0 1 8 31447 298
10
1 1
5
mol J K mol K
Pa
V
nRTp
=
-
1.2 Real numbers 3
(ii) Constant and variable. Equation (1.1) contains two types of
quantity:
Constant: a quantity whose value is xed for the present
purposes. The quantity R1=18.31447 J K1 mol1 is a constant physical
quantity.2 A constant number is anyparticular number; for example,
a1=10.1 and 1=13.14159=Variable: a quantity that can have any value
of a given set of allowed values. The quantities p, T, and n are
the variables of the function f( p, T, n)1=1nRT2p.
Two types of variable can be distinguished. An independent
variable is one whosevalue does not depend on the value of any
other variable. When equation (1.1) is written in the form
V1=1nRT2p, it is implied that the independent variables are p,
T,and n. The quantity V is then the dependent variable because its
value depends on thevalues of the independent variables. We could
have chosen the dependent variable to be T and the independent
variables as p, V, and n; that is, T =1pV2nR. In practice, the
choice of independent variables is often one of mathematical
convenience, but itmay also be determined by the conditions of an
experiment; it is sometimes easier to measure pressure p,
temperature T, and amount of substance n, and to calculate Vfrom
them.
Numbers are discussed in Sections 1.2 to 1.4, and variables in
Section 1.5. The algebra of numbers (arithmetic) is discussed in
Section 1.6.
(iii) A physical quantity is always the product of two
quantities, a number and a unit;for example T1=1298.15 K or
R1=18.31447 J K1 mol1. In applications of mathematicsin the
sciences, numbers by themselves have no meaning unless the units of
thephysical quantities are specied. It is important to know what
these units are, but themathematics does not depend on them. Units
are discussed in Section 1.8.
1.2 Real numbers
The concept of number, and of counting, is learnt very early in
life, and nearly everymeasurement in the physical world involves
numbers and counting in some way. Thesimplest numbers are the
natural numbers, the whole numbers or signless integers1,12,13,1=
It is easily veried that the addition or multiplication of two
natural numbers always gives a natural number, whereas subtraction
and division may not.For example 51131=12, but 5116 is not a
natural number. A set of numbers for whichthe operation of
subtraction is always valid is the set of integers, consisting of
all positive and negative whole numbers, and zero:
- 3 2 1 0 +1 +2 +3 - The operations of addition and subtraction
of both positive and negative integers aremade possible by the
rules
m1+1(n)1=1m11nm11(n)1=1m1+1n (1.5)
2 The values of the fundamental physical constants are under
continual review. For the latest recommendedvalues, see the NIST
(National Institute of Standards and Technology) website at
www.physics.nist.gov
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4 Chapter 1 Numbers, variables, and units
so that, for example, the subtraction of a negative number is
equivalent to the additionof the corresponding positive number. The
operation of multiplication is madepossible by the rules
(m)11(n)1=1+(m11n)(m)11(+n)1=1(m11n) (1.6)
Similarly for division. Note that m1=1(1)11m.EXAMPLES 1.2
Addition and multiplication of negative numbers
21+1(3)1=121131=11 211(3)1=121+131=15(2)11(3)1=121131=16
(2)11(3)1=121131=16(6)11(3)1=161131=12 611(3)1=161131=12
0 Exercises 17In equations (1.5) and (1.6) the letters m and n
are symbols used to represent any pair ofintegers; they are integer
variables, whose values belong to the (innite) set of integers.
Division of one integer by another does not always give an
integer; for example 61131=12, but 6114 is not an integer. A set of
numbers for which the operation of division is always valid is the
set of rational numbers, consisting of all the numbersm2n1=1m11n
where m and n are integers (m2n, read as m over n, is the more
commonly used notation for m divided by n). The denition excludes
the case n1=10because division by zero is not dened (see Section
1.6), but integers are includedbecause an integer m can be written
as m21. The rules for the combination of rationalnumbers (and of
fractions in general) are
(1.7)
(1.8)
(1.9)
where, for example, mq means m11q.
EXAMPLES 1.3 Addition of fractions
(1) Add and .14
12
mn
pq
mn
qp
mqnp
= =
mn
pq
mpnq
=
mn
pq
mq npnq
+ =+
-
1.2 Real numbers 5
The number one half is equal to two quarters and can be added to
one quarter togive three quarters:
The value of a fraction like 122 is unchanged if the numerator
and the denominatorare both multiplied by the same number:
and the general method of adding fractions is (a) nd a common
denominator forthe fractions to be added, (b) express all the
fractions in terms of this commondenominator, (c) add.
(2) Add and .
A common denominator is 31151=115. Then
(3) Add and .
A common denominator is 41161=124, but the lowest (smallest)
common denominator is 12:
0 Exercises 813EXAMPLE 1.4 Multiplication of fractions
This can be interpreted as taking two thirds of 425 (or four
fths of 223).0 Exercises 1417
23
45
2 43 5
815
=
=
14
56
312
1012
1312
+ = + =
56
14
23
45
2 53 5
3 43 5
1015
1215
10 1215
2215
+ =
+
= + =
+=
45
23
12
1 22 2
24
=
=
12
14
24
14
34
+ = + =
-
6 Chapter 1 Numbers, variables, and units
EXAMPLE 1.5 Division of fractions
The number 10212 can be simplied by dividing top and bottom by
the commonfactor 2: 102121=1526 (see Section 1.3).0 Exercises
1821
Every rational number is the solution of a linear equation
mx1=1n (1.10)where m and n are integers; for example, 3x1=12 has
solution x1=1223. Not all numbersare rational however. One solution
of the quadratic equation
x21=12is , the positive square root of 2 (the other solution is
), and this numbercannot be written as a rational number m2n; it is
called an irrational number. Other irrational numbers are obtained
as solutions of the more general quadraticequation
ax21+1bx1+1c1=10where a, b, and c are arbitrary integers, and of
higher-order algebraic equations; forexample, a solution of the
cubic equation
x31=12is , the cube root of 2. Irrational numbers like and are
called surds.
The rational and irrational numbers obtained as solutions of
algebraic equationsof type
a01+1a1x1+1a2x21+1a3x31+1-1+1anxn1=10 (1.11)where a0, a1, =, an
are integers, are called algebraic numbers; these numbers canbe
expressed exactly in terms of a nite number of rational numbers and
surds. There exist also other numbers that are not algebraic; they
are not obtained as solutions of any nite algebraic equation. These
numbers are irrational numbers called transcendental numbers; they
transcend the power of algebraic methods
232x = 23
2x = 2
23
45
23
54
1012
= =
-
1.3 Factorization, factors, and factorials 7
(Euler).3 The best known and most important of these are the
Euler number e and theArchimedean number .4 These are discussed in
Section 1.4.
The rational and irrational numbers form the continuum of
numbers; togetherthey are called the real numbers.
1.3 Factorization, factors, and factorials
Factorization is the decomposition of a number (or other
quantity) into a product ofother numbers (quantities), or factors;
for example
301=12113115shows the decomposition of the natural number 30
into a product of prime numbers;that is, natural numbers that
cannot be factorized further (the number 1 is not countedas a prime
number). The fundamental theorem of arithmetic is that every
naturalnumber can be factorized as a product of prime numbers in
only one way.5
EXAMPLES 1.6 Prime number factorization
(1) 41=121121=122(2) 121=121121131=122113(3)
3151=131131151171=132115117(4)
51201=121121121121121121121121121121151=1210115
0 Exercises 2225
Factorization and cancellation of common factors can be used for
the simplicationof fractions. For example, in
1242
6 26 7
27
=
=
3 Leonhard Euler (17071783). Born in Switzerland, he worked most
of his life in St Petersburg and in Berlin. One of the worlds most
prolic mathematicians, he wrote voluminous papers and huge
textbooks. He contributed to nearly all branches of mathematics and
its application to physical problems, including the calculus,
differential equations, innite series, complex functions,
mechanics, and hydrodynamics, and his nameis associated with many
theorems and formulas. One of his important, if unspectacular,
contributions was to mathematical notation. He introduced the
symbol e, gave the trigonometric functions their modern
denition,and by his use of the symbols sin, cos, i, and made them
universally accepted.
4 The symbol was rst used by William Jones (16751749) in a
textbook on mathematics, Synopsis palmariorum matheseos (A new
introduction to the mathematics) in 1706. Eulers adoption of the
symbol ensuredits acceptance.
5 A version of the fundamental theorem of arithmetic is given by
Propositions 31 and 32 in Book VII of EuclidsStoichia (Elements).
Euclid was one of the rst teachers at the Museum and Library of
Alexandria founded byPtolemy I in about 300 BC after he had gained
control of Egypt when Alexanders empire broke up in 323 BC.
-
8 Chapter 1 Numbers, variables, and units
cancellation of the common factor 6 is equivalent to dividing
both numerator anddenominator by 6, and such an operation does not
change the value of the fraction.
EXAMPLES 1.7 Simplication of fractions
(1)
(2)
(3)
0 Exercises 2629
In general, the purpose of factorization is to express a
quantity in terms of simplerquantities (see Section 2.3 for
factorization of algebraic expressions).
Factorials
The factorial of n is the number whose factors are the rst n
natural numbers:
n!1=1111211311-11n(1.12)
=1n11(n111)11(n112)11-112111(read as n factorial). Consecutive
factorials are related by the recurrence relation
(n1+11)!1=1(n1+11)11n!for example, 3!1=131121111=16 and
4!1=141131121111=14113!1=124. In addition, the factorial of zero is
dened as 0!1=11.
EXAMPLES 1.8 Factorials
(1) 1!1=11110!1=111111=11(2)
5!1=15114!1=15114113!1=15114113112!1=151141131121=1120(3)
(4)
0 Exercises 3036
73 4
7 6 5 43 2 4
7 5 35!
! !!
!=
= =
53
5 4 33
5 4 20!!
!!
=
= =
1051470
3 5 7
2 3 5 7
12 7
1142
=
=
=
1525
3 5
5
352
=
=
424
2
2 3
2 22 2 2 3
16
2
3=
=
=
-
1.4 Decimal representation of numbers 9
1.4 Decimal representation of numbers
These are the nine gures of the Indians
9 8 7 6 5 4 3 2 1
With these nine gures, and with this sign 0 which in Arabic
iscalled zephirum, any number can be written, as will below
bedemonstrated.
(Fibonacci)6
In the decimal system of numbers, the ten digit symbols 0 to 9
(Hindu-Arabic numerals)7
are used for zero and the rst nine positive integers; the tenth
positive integer is denotedby 10. A larger integer, such as three
hundred and seventy-two is expressed in the form
3001+1701+121=13111021+1711101+12and is denoted by the symbol
372, in which the value of each digit is dependent on itsposition
in the symbol for the number. The decimal system has base 10, and
is theonly system in common use.
Although rational numbers can always be expressed exactly as
ratios of integers,this is not so for irrational numbers. For
computational purposes, a number that is not an integer is
conveniently expressed as a decimal fraction;8 for
example,5241=11.25. The general form of the decimal fraction
(integral part).(fractional part)
consists of an integer to the left of the decimal point, the
integral part of the number,and one or more digits to the right of
the decimal point, the decimal or fractional partof the number. The
value of each digit is determined by its position; for example
=12111021+13111011+14111001+15111011+16111021+1711103234 567 200
30 4
510
6100
71000
. = + + + + +
6 Leonardo of Pisa, also called Fibonacci (c. 1170after 1240).
The outstanding mathematician of the LatinMiddle Ages. In his
travels in Egypt, Syria, Greece, and Sicily, Fibonacci studied
Greek and Arabic (Muslim) mathematical writings, and became
familiar with the Arabic positional number system developed by the
Hindumathematicians of the Indus valley of NW India. Fibonaccis rst
book, the Liber abaci, or Book of the Abacus,(1202, revised 1228)
circulated widely in manuscript, but was published only in 1857 in
Scritti di Leonardo Pisano.The rst chapter opens with the quotation
given above in the text.
7 One of the principal sources by which the Hindu-Arabic decimal
position system was introduced into (Latin)Europe was Al-Khwarizmis
Arithmetic. Muhammad ibn Musa Al-Khwarizmi (Mohammed the son of
Mosesfrom Khorezm, modern Khiva in Uzbekistan) was active in the
time of the Baghdad Caliph Al-Mamun (813833),and was probably a
member of his House of Wisdom (Academy) at a time when Baghdad was
the largest city inthe world. Al-Khwarizmis Algebra was widely used
in Arabic and in Latin translation as a source on linear
andquadratic equations. The word algorithm is derived from his
name, and the word algebra comes from the title,Liber algebrae et
almucabala, of Robert of Chesters Latin translation (c. 1140) of
his work on equations.
8 The use of decimal fractions was introduced into European
mathematics by the Flemish mathematician andengineer Simon Stevin
(15481620) in his De Thiende (The art of tenths) in 1585. Although
decimal fractions wereused by the Chinese several centuries
earlier, and the Persian astronomer Al-Kashi used decimal and
sexagesimalfractions in his Key to Arithmetic early in the fteenth
century, the common use of decimal fractions in Europeanmathematics
can be traced directly to Stevin, especially after John Napier
modied the notation into the presentone with the decimal point (or
decimal comma as is used in much of continental Europe). It greatly
simplied theoperations of multiplication and division.
-
10 Chapter 1 Numbers, variables, and units
where 10n1=11210n and 1001=11 (see Section 1.6).0 Exercises
3742
A number with a nite number of digits after (to the right of )
the decimal point can always be written in the rational form m2n;
for example 1.2341=1123421000. Theconverse is not always true
however. The number 123 cannot be expressed exactly asa nite
decimal fraction:
the dots indicating that the fraction is to be extended
indenitely. If quoted to fourdecimal places, the number has lower
and upper bounds 0.3333 and 0.3334:
where the symbol < means is less than; other symbols of the
same kind are for isless than or equal to, > for is greater
than, and for is greater than or equal to.Further examples of
nonterminating decimal fractions are
In both cases a nite sequence of digits after the decimal point
repeats itselfindenitely, either immediately after the decimal
point, as the sequence 142857 in127, or after a nite number of
leading digits, as 3 in 1212. This is a characteristicproperty of
rational numbers.
EXAMPLE 1.9 Express 1213 as a decimal fraction. By long
division,
The rational number 12131=10.076923 076923= is therefore a
nonterminating decimalfraction with repeating sequence 076923 after
the decimal point.
0 Exercises 4346
0 0769230713 1 00
919078120117
30264039100
. ...
.)
17
0 142857 1428571
120 083333 333333= . , = .
0 3333
13
0 3334. < < .
13
0 333= .
-
1.4 Decimal representation of numbers 11
An irrational number cannot be represented exactly in terms of a
nite number ofdigits, and the digits after the decimal point do not
show a repeating sequence. Thenumber has approximate value to 16
signicant gures,
=11.414213 562373 095=and can, in principle, be computed to any
desired accuracy by a numerical methodsuch as the NewtonRaphson
method discussed in Chapter 20.9
The Archimedean number
The number is dened as the ratio of the circumference of a
circle to its diameter. It is a transcendental number,10 and has
been computed to many signicant gures; it was quoted to 127 decimal
places by Euler in 1748. Its value to 16 signicant gures is
1=13.14159 26535 897931=The value of has been of practical
importance for thousands of years. For example,an Egyptian
manuscript dated about 1650 BC (the Rhind papyrus in the
BritishMuseum) contains a prescription for the calculation of the
volume of a cylindricalgranary from which the approximate value
256281113.160 can be deduced. Amethod for generating accurate
approximations was rst used by Archimedes11 whodetermined the
bounds
and the upper bound has an error of only 2 parts in a
thousand.
22371
227
<
-
12 Chapter 1 Numbers, variables, and units
The Euler number e
The number e is dened by the innite series (see Chapter 7)
=12.71828 18284 59045=The value of e can be computed from the
series to any desired accuracy. The numberwas shown to be a
transcendental number by Hermite in 1873.12
EXAMPLE 1.10 Show that the sum of the rst 10 terms of the series
gives anapproximate value of e that is correct to at least 6
signicant gures.
111+111+10.51+10.1666671+10.0416671+10.0083331+10.0013891+10.000198+10.0000251+10.0000031+10.0000003
12.71828The value is correct to the 6 gures quoted because every
additional term in the seriesis at least ten times smaller than the
preceding one.
Signicant gures and rounding
In practice, arithmetic involving only integers gives exact
answers (unless the numbers are too large to be written). More
generally, a number in the decimal system is approximated either
with some given number of decimal places or with agiven number of
signicant gures, and the result of an arithmetic operation is
alsoapproximate. In the xed-point representation, all numbers are
given with a xednumber of decimal places; for example,
3.142, 62.358, 0.013, 1.000
have 3 decimal places. In the oating-point representation, used
more widely in thesciences, the numbers are given with a xed number
of signicant gures, with zeroson the left of a number not counted.
For example,
32101=10.321011104, 003.2101=10.321011101,
0.0032101=10.321011102all have 4 signicant gures.
e = + + + + + + + + +1 1
12
16
124
1120
1720
15040
140320
13628880
13628800
+ +
e = +
!+
!+
!+
!+1
11
12
13
14
12 Charles Hermite (18221901). French mathematician, professor
at the Sorbonne, is known for his work inalgebra and number theory.
His work on the algebra of complex numbers (Hermitian forms) became
importantin the formulation of quantum theory. The Hermite
differential equation and the Hermite polynomials areimportant in
the solution of the Schrdinger equation for the harmonic
oscillator.
-
1.5 Variables 13
A number whose exact (decimal) representation involves more than
a given numberof digits is reduced most simply by truncation; that
is, by removing or replacing by zeros all superuous digits on the
right. For example, to 4 decimal places or 5signicant gures,
3.14159 is truncated to 3.1415. Truncation is not
recommendedbecause it can lead to serious computational errors. A
more sensible (accurate)approximation of to ve gures is 3.1416,
obtained by rounding up. The mostwidely accepted rules for rounding
are:
(i) If the rst digit dropped is greater than or equal to 5, the
preceding digit isincreased by 1; the number is rounded up.
(ii) If the rst digit dropped is less than 5, the preceding
digit is left unchanged; thenumber is rounded down. For example,
for 4, 3, 2, and 1 decimal places,
7.36284 is 7.3628, 7.363, 7.36, 7.4
Errors arising from truncation and rounding are discussed in
Section 20.2.
0 Exercises 47491.5 Variables
In the foregoing sections, symbols (letters) have been used to
represent arbitrarynumbers. A quantity that can take as its value
any value chosen from a given setof values is called a variable. If
{x1, x2, x3, =, xn} is a set of objects, not necessarily numbers,
then a variable x can be dened in terms of this set such that x can
have asits value any member of the set; the set forms the domain of
the variable. In (real)number theory, the objects of the set are
real numbers, and a real variable can have asits domain either the
whole continuum of real numbers or a subset thereof. If thedomain
of the variable x is an interval a to b,
a11x11bthen x is a continuous variable in the interval, and can
have any value in thecontinuous range of values a to b (including a
and b). If the domain consists of a discrete set of values, for
example the n numbers x1,1x2,1x3,1=,1xn, then x is called a
discrete variable. If the domain consists of integers, x is an
integer variable. If the set consists of only one value then the
variable is called a constant variable, or simply a constant.
In the physical sciences, variables are used to represent both
numbers and physicalquantities. In the ideal-gas example discussed
in Section 1.1, the physical quantitiesp, V, n, and T are
continuous variables whose numerical values can in principle be
anypositive real numbers. Discrete variables are normally involved
whenever objects are counted as opposed to measured. Typically, an
integer variable is used for thecounting and the counted objects
form a sample of some discrete set. In some caseshowever a physical
quantity can have values, some of which belong to a discrete setand
others to a continuous set. This is the case for the energy levels
and the observedspectral frequencies of an atom or molecule.
-
14 Chapter 1 Numbers, variables, and units
EXAMPLE 1.11 The spectrum of the hydrogen atom
The energy levels of the hydrogen atom are of two types:
(i) Discrete (quantized) energy levels with (negative) energies
given by the formula(in atomic units, see Section 1.8)
n1=11, 2, 3, =The corresponding states of the atom are the bound
states, in which the motion ofthe electron is conned to the
vicinity of the nucleus. Transitions between the energylevels give
rise to discrete lines in the spectrum of the atom.
(ii) Continuous energy levels, with all positive energies,
E1>10. The correspondingstates of the atom are those of a free
(unbound) electron moving in the presence of theelectrostatic eld
of the nuclear charge. Transitions between these energy levels
andthose of the bound states give rise to continuous ranges of
spectral frequencies.
1.6 The algebra of real numbers
The importance of the concept of variable is that variables can
be used to make statements about the properties of whole sets of
numbers (or other objects), and itallows the formulation of a set
of rules for the manipulation of numbers. The set ofrules is called
the algebra.
Let a, b, and c be variables whose values can be any real
numbers. The basic rules forthe combination of real numbers, the
algebra of real numbers or the arithmetic, are
1. a1+1b1=1b1+1a (commutative law of addition)2. ab1=1ba
(commutative law of multiplication)3. a1+1(b1+1c)1=1(a1+1b)1+1c
(associative law of addition)4. a(bc)1=1(ab)c (associative law of
multiplication)5. a(b1+1c)1=1ab1+1ac (distributive law)
The operations of addition and multiplication and their
inverses, subtraction anddivision, are called arithmetic
operations. The symbols +, , and (or 2) are calledarithmetic
operators. The result of adding two numbers, a1+1b, is called the
sum of aand b; the result of multiplying two numbers,
ab1=1a11b1=1a11b, is called the productof a and b.13
En
n = ,1
2 2
13 In 1698 Leibniz wrote in a letter to Johann Bernoulli: I do
not like as a symbol for multiplication, as it easily confounded
with x = often I simply relate two quantities by an interposed dot.
It is becoming accepted practiceto place the dot in the high
position to denote multiplication (2115 = 2 5) and in the low
position, on the line,for the decimal point (2.5 = 522). An
alternative convention, still widely used, is to place the dot on
the line formultiplication (2.5 = 2 5) and high for the decimal
point (2115 = 522).
-
1.6 The algebra of real numbers 15
EXAMPLES 1.12 Examples of the rules of arithmetic
rule examples
1. a1+1b1=1b1+1a 21+131=131+121=152. ab1=1ba 21131=131121=163.
a1+1(b1+1c)1=1(a1+1b)1+1c !21+1(31+14)1=121+171=19, and
@(21+13)1+141=151+141=194. a(bc)1=1(ab)c
!211(3114)1=1211121=124, and
@(2113)1141=161141=1245. a(b1+1c)1=1ab1+1ac
!211(31+14)1=121171=114, and
@211(31+14)1=1(2113)1+1(2114)1=161+181=1142(31+14)1=1(2113)1+1(2114)1=161181=1142(3114)1=1211311211(4)1=161+181=12
A corollary to rule 5 is(a1+1b)(c1+1d)1=1a(c1+1d)1+1b(c1+1d)
(21+13)(41+15)1=12(41+15)1+13(41+15)1=1181+1271=145
Three rules dene the properties of zero and unity:
6. a1+101=101+1a1=1a (addition of zero)7. a1101=1011a1=10
(multiplication by zero)8. a1111=1111a1=1a (multiplication by
unity)
We have already seen that subtraction of a number is the same as
addition of its negative, and that division by a number is the same
as multiplication by its inverse.However, division by zero is not
dened; there is no number whose inverse is zero.For example, the
number 12a, for positive values of a, becomes arbitrarily large as
thevalue of a approaches zero; we say that 12a tends to innity as a
tends to zero:
Although innity is represented by the symbol , it is not a
number. If it were a number then, by the laws of algebra, the
equations 1201=1 and 2201=1 would imply11=12.
The modulus of a real number a is dened as the positive square
root of a2;
(read as mod a). It is the magnitude of the number, equal to +a
if a ispositive, and equal to a if a is negative:
(1.13)
For example, and .| | =3 3| |3 3=
| |aa a
a a=
+ >
10 and to the left if x110 and below if y1
-
2.3 Factorization and simplication of expressions 35
The inverse operation of factorization is usually called
expansion or multiplyingout.
EXAMPLE 2.6 Factorize:
(i) 2xy2114x2y1+16xyThe expression 2xy is a common factor.
Therefore
2xy2114x2y1+16xy1=1(2xy)11(y)11(2xy)11(2x)1+1(2xy)11(3)
=12xy(y112x1+13)(ii) x2115x116
The aim is to express the quadratic function as the product of
two linearfunctions; that is, to nd numbers a and b such that
x2115x1161=1(x1+1a)(x1+1b)Expansion of the product gives
(x1+1a)(x1+1b)1=1x(x1+1b)1+1a(x1+1b)1=1x21+1bx1+1ax1+1aband,
therefore,
x2115x1161=1x21+1(a1+1b)x1+1abFor this equation to be true for
all values of x it is necessary that the coefcient of each power of
x be the same on both sides of the equal sign: a1+1b1=15 andab1=16.
The two numbers whose sum is 5 and whose product is 6 are a1=16and
b1=11. Therefore
x2115x1161=1(x116)(x1+11)(iii) x2119
Let x21191=1(x1+1a)(x1+1b)1=1x21+1(a1+1b)x1+1ab. In this case
there is no term linearin x: a1+1b1=10, so that b1=1a and
ab1=1a21=19. Therefore and thefactorization is
x21191=1(x1+13)(x113)This is an example of the general form
x211a21=1(x1+1a)(x11a).
(iv) x4115x21+14The quartic in x is a quadratic in disguise.
Replacement of x2 by y, followed byfactorization gives
y2115y1+141=1( y111)( y114)
a = = 9 3
-
36 Chapter 2 Algebraic functions
Therefore
x4115x21+141=1(x2111)(x2114)Both the quadratic factors have the
form x211a21=1(x1+1a)(x11a) discussed incase (iii) above:
x21111=1(x1+11)(x111) and x21141=1(x1+12)(x112)Therefore,
x4115x21+141=1(x1+11)(x111)(x1+12)(x112)0 Exercises 916
The expansion
(x1+1a)(x1+1b)1=1x21+1(a1+1b)x1+1ab (2.4)used in Examples 2.6,
has geometric interpretation as the area of a rectangle of
sides(x1+1a) and (x1+1b), as illustrated in Figure 2.3.3
Other important general forms are
(a1+1b)21=1a21+12ab1+1b2 square of side
(a1+1b)(a11b)21=1a2112ab1+1b2 square of side |a11b |
(2.5)(a1+1b)(a11b)1=1a211b2 difference of squares
The rst two equations of (2.5) can be combined by using the
symbol , meaning plusor minus:
(a11b)21=1a2112ab1+1b2 (2.6)in which either the upper symbol is
used on both sides of the equation or the lowersymbol is used on
both sides. Sometimes the symbol 3 is used in a similar way;
forexample, a131b1=1c represents the pair of equations a11b1=1+c
and a1+1b1=1c.
Factorization can be used to simplify algebraic fractions. For
example, in
xy xx xy
+
+
24 6
2
3 Euclid, The Elements, Book II, Propositions 4 and 7 are the
geometric equivalents of the rst two equations(2.5) for the squares
of (a1+1b) and (a11b).
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................................................................................................................................................................................................................................................................................
bx ab
x
2
ax
b
x
x a
Figure 2.3
-
2.4 Inverse functions 37
both numerator and denominator have the common factor x, and can
be divided bythis factor (when x110) without changing the value of
the fraction:
EXAMPLE 2.7 Simplication of fractions
(1)
(2)
(3)
0 Exercises 1722
2.4 Inverse functions
Given some function f and the equation y1=1f(x), it is usually
possible to dene, at leastfor some values of x and y, a function g
such that x1=1g(y). This new function is theinverse function of f
and is denoted by the symbol f 1 (not to be confused with
thereciprocal 12f ):
if y1=1f(x) then x1=1f 1( y) (2.7)
EXAMPLE 2.8 If y1=1f(x)1=12x1+13, nd x1=1f 1(y).To nd x in terms
of y,
(i) subtract 3 from both sides of the equation:
y1=12x1+1311y1131 =12x(ii) divide both sides by 2:
Therefore x1=1( y113)221=1f 1( y).In this example, y is a
single-valued function of x; that is, for each value of x there
exists just one value of y. Similarly, x is a single-valued
function of y.
0 Exercises 2325
EXAMPLE 2.9 If , express x in terms of y. y
ax bcx d
=
+
+
=
yx
32
a b
a ab b
a b a b
a b a b
a ba
2 2
2 22
+ +=
+
+ +=
( ) ( )
( ) ( )
( )( ++ b)
3 69 18
3 1 2
3 1 2
1 23 1 22
+
+=
+
+=
+
+
yx
y
x
yx
( )
( ) ( )
42
2 22
2xy
xy
xy
=
=
( )( )
xy xx xy
x y xx y
y xy
+
+=
+
+=
+
+
24 6
22 2 3
22 2 3
2 ( )( ) ( )
-
38 Chapter 2 Algebraic functions
To solve for x,
(i) multiply both sides of the equation by (cx1+1d):
(cx1+1d)y1=1ax1+1b(ii) expand the l.h.s.: cxy1+1dy1=1ax1+1b
(iii) subtract (ax1+1b) from both sides: cxy1+1dy11ax11b1=10(iv)
collect the terms in x1 and x0: (cy11a)x1+1(dy11b)1=10(v) subtract
(dy11b) from both sides: (cy11a)x1=1(dy11b)
(vi) divide both sides by (cy11a):We note that step (vi) is not
valid if (cy11a)1=10 because division by zero has nomeaning. Such
complications can normally be ignored.
This example demonstrates the type of algebraic manipulation
routinely used inthe solution of real problems.
0 Exercises 2629EXAMPLE 2.10 If y1=1f(x)1=1x21+11, express x in
terms of y.We have
y1=1x21+11, x21=1y111,y is a single-valued function of x, but x
is a double-valued function of y (except fory1=11); that is, for
each real value of y1>11 there exist two real values of x (if
y1
-
2.4 Inverse functions 39
In physical applications it is usually obvious from the context
which value is to bechosen. It is also seen, when x and y are real
numbers, that whereas y is dened for allvalues of x, 1
-
40 Chapter 2 Algebraic functions
EXAMPLE 2.12 The van der Waals equation
The equation of state for a slightly imperfect gas is
(2.8)
In this case, both T and p are easily expressed as explicit
functions of the other variables:
For V, equation (2.8) can be rearranged into
which is a cubic equation in V. It is possible to write down
explicit solutions of a cubic equation, but these are complicated
and seldom used. In this case, it is mostconvenient to regard
equation (2.8) as dening V as an implicit function of p, T, andn.
For any set of values of the independent variables and of the
constants, equation(2.8) can be solved numerically by an iterative
method such as the NewtonRaphsonmethod described in Chapter 20.
0 Exercises 32352.5 Polynomials
The general polynomial of degree n has the form
f(x)1=1a01+1a1x1+1a2x21+1-1+1anxn (2.9)where the coefcients
a0,1a1,1=,1an are constants, and n is a positive integer. Ifn1=10
the function is the constant a0. The polynomial is often written in
short-handnotation as
(2.10)
where the symbol represents summation. The notation tells us to
add together theterms aix
i in which the integer variable i takes in turn the values 0, 1,
2,1=1, n:
= a01+1a1x1+1a2x21+1-1+1anxn(remembering that x01=11 and
x11=1x).
i
n
ii
nna x a x a x a x a x
=
= + + + +0
00
11
22( ) ( ) ( ) ( )
f x a xi
n
ii( ) =
=
0
V n b
RTp
Vn a
pV
n abp
3 22 3
0 +
+ =
TnR
pn a
VV nb p
nRTV nb
n a
V= +
=
1 2
2
2
2( ),
pn a
VV nb nRT+
=
2
20( )
-
Only real coefcients are discussed here; the case of complex
coefcients is shownin Section 8.4 to involve no new principles.
EXAMPLE 2.13 Write out in full:
(1)
(2)
(3)
0 Exercises 3639Degree n1=11: linear function
f(x)1=1a01+1a1x (2.11)This is the simplest type of function, and
is better known in the form
y1=1mx1+1c (2.12)The graph of the function is a straight line.
It has slope m, and intercepts the verticaly-axis (when x1=10) at
the point y1=1c, as shown in Figure 2.6.
If we take any two points on the line, with coordinates (x1, y1)
and (x2, y2), then
y11=1mx11+1cy21=1mx21+1c
( ) ( ) ( ) ( ) = + + = +=
x x x x x x xii
2 3 4 2 3 4
2
4
xn
x x x xx
x x xn
n
2 1 1 1 3 5 3 5
0
3
1 1 2 3 41
2 3 4
=+
= + + + = + + +
i x x x x x x x xi
i
= + + + = + +=
0 1 2 3 2 30 1 2 3 2 30
3
2.5 Polynomials 41
........ ........ ........ ........ ........ ........ ....
........ ........ ........ ........ ........ ........ ........
........ ........ ........ ........ ........ ........ ........
........ ........
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c/m
x
1
x
2
(x
1
, y
1
)
(x
2
, y
2
)
y
2
y
1
x
2
x
1
y
2
y
1
y
x
c
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Figure 2.6
-
42 Chapter 2 Algebraic functions
and
(2.13)
denes the constant slope. The line crosses the horizontal x-axis
at one point:
y1=10 when (2.14)This value of x is called the root of the
linear function. In general, the roots of apolynomial function are
those values of the variable for which the value of the functionis
zero; that is, the roots are the solutions of the polynomial
equation
f(x)1=10 (2.15)
EXAMPLE 2.14 Find the equation of the straight line that passes
through the points(1, 6) and (3, 2).
Let the line be y1=1mx1+1c. Then:at point (x1, y1)1=1(1, 6),
61=1m1+1cat point (x2, y2)1=1(3, 2), 21=13m1+1c
Solution of the pair of simultaneous equations (see Section 2.8)
gives m1=12 and c1=14. Therefore
y1=12x114The graph of the line is shown in Figure 2.7. The line
has slope m1=12, which meansthat the value of y increases twice as
fast as that of x. The line crosses the y-axis aty1=1c1=14, and
crosses the x-axis at x1=12.
0 Exercises 4043
x
cm
=
my y
x x=
2 1
2 1
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Figure 2.7
-
2.5 Polynomials 43
Degree n1=12: quadratic functionf(x)1=1a01+1a1x1+1a2x2
(2.16)
The quadratic function is usually written as
y1=1ax21+1bx1+1c (2.17)A typical graph is shown in Figure 2.1 in
Section 2.2. The shape of the curve is that ofa parabola. When the
constant a is positive, the function has a single minimum
value(turning point), and is symmetrical about a vertical line that
passes through the pointof minimum value. For the function in
Figure 2.1,
f(x)1=1x2112x113 (2.18)this minimum point has coordinates (x,
y)1=1(1, 4). The graph crosses the x-axis,when f(x)1=10, at the two
points x1=11 and x1=13. These are the roots of the
quadraticfunction, and they are the solutions of the quadratic
equation
x2112x1131=10 (2.19)In this example the roots are easily
obtained by factorization:
x2112x1131=1(x1+11)(x113)and the function is zero when either of
the linear factors is zero:
(the symbol means implies)Whilst it is possible to factorize a
variety of quadratic functions by trial and error,
as in Examples 2.6, the roots can always be found by
formula:4
ax21+1bx1+1c1=10when
(2.20)xb b ac
a=
2 42
x x
x x
x x2 2 3 0
1 0 1
3 0 3 =
+ = =
= =
when eitheror
4 A clay tablet (YBC 6967, Yale Babylonian Collection) of the
Old Babylonian Period (c. 18001600 BC)has inscribed on it in the
Sumerian cuneiform script the following problem (in modern
notation): giventhat xy1=160 and x11y1=17, nd x and y. The
prescription given for the (positive) solution corresponds to
and . The method and prescriptive approach is almost identical
to that used by Al-Khwarizmi two and a half millennia later. Modern
algebra became possible with thedevelopment of a general abstract
notation in the 15th to 17th centuries. One important step was
taken by FranoisVite (15401603). French lawyer, politician,
cryptoanalyst, and amateur mathematician, he made contributionsto
trigonometry and algebra. He is best remembered as the man who, in
his In artem analyticem isagoge(Introduction to the analytical art)
of 1591, introduced the systematic use of symbols (letters) into
the theory ofequations, distinguishing between constants and
variables.
y = + ( ) ( )7 2 60 7 22x = + +( ) ( )7 2 60 7 22
-
44 Chapter 2 Algebraic function
