Theory and Problems of
ADVANCED CALCULUS
Second Edition
ROBERT WREDE, Ph.D.
MURRAY R. SPIEGEL, Ph.D.
Former Professor and Chairman of Mathematics Rensselaer Polytechnic Institute Hartford Graduate Center
Schaum’s Outline Series
McGRAW-HILL New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto
distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher.
0-07-139834-1
The material in this eBook also appears in the print version of this title: 0-07-137567-8
All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps.
McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in cor- porate training programs. For more information, please contact George Hoare, Special Sales, at george_hoare@mcgraw- hill.com or (212) 904-4069.
TERMS OF USE
This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms.
THE WORK IS PROVIDED “AS IS”. McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WAR- RANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PAR- TICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any dam- ages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work.
Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, con- sequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise.
DOI: 10.1036/0071398341
iii
the reason for the longevity of Professor Spiegel’s advanced calculus. His collection of solved and unsolved problems remains a part of this second edition.
Advanced calculus is not a single theory. However, the various sub-theories, including vector analysis, infinite series, and special functions, have in common a dependency on the fundamental notions of the calculus. An important objective of this second edition has been to modernize terminology and concepts, so that the interrelationships become clearer. For exam- ple, in keeping with present usage fuctions of a real variable are automatically single valued;
differentials are defined as linear functions, and the universal character of vector notation and theory are given greater emphasis. Further explanations have been included and, on occasion, the appropriate terminology to support them.
The order of chapters is modestly rearranged to provide what may be a more logical structure.
A brief introduction is provided for most chapters. Occasionally, a historical note is included; however, for the most part the purpose of the introductions is to orient the reader to the content of the chapters.
I thank the staff of McGraw-Hill. Former editor, Glenn Mott, suggested that I take on the project. Peter McCurdy guided me in the process. Barbara Gilson, Jennifer Chong, and Elizabeth Shannon made valuable contributions to the finished product. Joanne Slike and Maureen Walker accomplished the very difficult task of combining the old with the new and, in the process, corrected my errors. The reviewer, Glenn Ledder, was especially helpful in the choice of material and with comments on various topics.
ROBERTC. WREDE
Copyright 2002, 1963 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
v
CHAPTER 1 NUMBERS 1
Sets. Real numbers. Decimal representation of real numbers. Geometric representation of real numbers. Operations with real numbers. Inequal- ities. Absolute value of real numbers. Exponents and roots. Logarithms.
Axiomatic foundations of the real number system. Point sets, intervals.
Countability. Neighborhoods. Limit points. Bounds. Bolzano- Weierstrass theorem. Algebraic and transcendental numbers. The com- plex number system. Polar form of complex numbers. Mathematical induction.
CHAPTER 2 SEQUENCES 23
Definition of a sequence. Limit of a sequence. Theorems on limits of sequences. Infinity. Bounded, monotonic sequences. Least upper bound and greatest lower bound of a sequence. Limit superior, limit inferior.
Nested intervals. Cauchy’s convergence criterion. Infinite series.
CHAPTER 3 FUNCTIONS, LIMITS, AND CONTINUITY 39
Functions. Graph of a function. Bounded functions. Montonic func- tions. Inverse functions. Principal values. Maxima and minima. Types of functions. Transcendental functions. Limits of functions. Right- and left-hand limits. Theorems on limits. Infinity. Special limits. Continuity.
Right- and left-hand continuity. Continuity in an interval. Theorems on continuity. Piecewise continuity. Uniform continuity.
CHAPTER 4 DERIVATIVES 65
The concept and definition of a derivative. Right- and left-hand deriva- tives. Differentiability in an interval. Piecewise differentiability. Differ- entials. The differentiation of composite functions. Implicit differentiation. Rules for differentiation. Derivatives of elementary func- tions. Higher order derivatives. Mean value theorems. L’Hospital’s rules. Applications.
Copyright 2002, 1963 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
CHAPTER 5 INTEGRALS 90
Introduction of the definite integral. Measure zero. Properties of definite integrals. Mean value theorems for integrals. Connecting integral and differential calculus. The fundamental theorem of the calculus. General- ization of the limits of integration. Change of variable of integration.
Integrals of elementary functions. Special methods of integration.
Improper integrals. Numerical methods for evaluating definite integrals.
Applications. Arc length. Area. Volumes of revolution.
CHAPTER 6 PARTIAL DERIVATIVES 116
Functions of two or more variables. Three-dimensional rectangular coordinate systems. Neighborhoods. Regions. Limits. Iterated limits.
Continuity. Uniform continuity. Partial derivatives. Higher order par- tial derivatives. Differentials. Theorems on differentials. Differentiation of composite functions. Euler’s theorem on homogeneous functions.
Implicit functions. Jacobians. Partial derivatives using Jacobians. The- orems on Jacobians. Transformation. Curvilinear coordinates. Mean value theorems.
CHAPTER 7 VECTORS 150
Vectors. Geometric properties. Algebraic properties of vectors. Linear independence and linear dependence of a set of vectors. Unit vectors.
Rectangular (orthogonal unit) vectors. Components of a vector. Dot or scalar product. Cross or vector product. Triple products. Axiomatic approach to vector analysis. Vector functions. Limits, continuity, and derivatives of vector functions. Geometric interpretation of a vector derivative. Gradient, divergence, and curl. Formulas involving r. Vec- tor interpretation of Jacobians, Orthogonal curvilinear coordinates.
Gradient, divergence, curl, and Laplacian in orthogonal curvilinear coordinates. Special curvilinear coordinates.
CHAPTER 8 APPLICATIONS OF PARTIAL DERIVATIVES 183
Applications to geometry. Directional derivatives. Differentiation under the integral sign. Integration under the integral sign. Maxima and minima. Method of Lagrange multipliers for maxima and minima.
Applications to errors.
CHAPTER 9 MULTIPLE INTEGRALS 207
Double integrals. Iterated integrals. Triple integrals. Transformations
of multiple integrals. The differential element of area in polar
coordinates, differential elements of area in cylindrical and spherical
coordinates.
CHAPTER 10 LINE INTEGRALS, SURFACE INTEGRALS, AND
INTEGRAL THEOREMS 229
Line integrals. Evaluation of line integrals for plane curves. Properties of line integrals expressed for plane curves. Simple closed curves, simply and multiply connected regions. Green’s theorem in the plane. Condi- tions for a line integral to be independent of the path. Surface integrals.
The divergence theorem. Stoke’s theorem.
CHAPTER 11 INFINITE SERIES 265
Definitions of infinite series and their convergence and divergence. Fun- damental facts concerning infinite series. Special series. Tests for con- vergence and divergence of series of constants. Theorems on absolutely convergent series. Infinite sequences and series of functions, uniform convergence. Special tests for uniform convergence of series. Theorems on uniformly convergent series. Power series. Theorems on power series.
Operations with power series. Expansion of functions in power series.
Taylor’s theorem. Some important power series. Special topics. Taylor’s theorem (for two variables).
CHAPTER 12 IMPROPER INTEGRALS 306
Definition of an improper integral. Improper integrals of the first kind (unbounded intervals). Convergence or divergence of improper integrals of the first kind. Special improper integers of the first kind.
Convergence tests for improper integrals of the first kind. Improper integrals of the second kind. Cauchy principal value. Special improper integrals of the second kind. Convergence tests for improper integrals of the second kind. Improper integrals of the third kind. Improper integrals containing a parameter, uniform convergence. Special tests for uniform convergence of integrals. Theorems on uniformly conver- gent integrals. Evaluation of definite integrals. Laplace transforms.
Linearity. Convergence. Application. Improper multiple integrals.
CHAPTER 13 FOURIER SERIES 336
Periodic functions. Fourier series. Orthogonality conditions for the sine and cosine functions. Dirichlet conditions. Odd and even functions.
Half range Fourier sine or cosine series. Parseval’s identity. Differentia-
tion and integration of Fourier series. Complex notation for Fourier
series. Boundary-value problems. Orthogonal functions.
CHAPTER 14 FOURIER INTEGRALS 363
The Fourier integral. Equivalent forms of Fourier’s integral theorem.
Fourier transforms.
CHAPTER 15 GAMMA AND BETA FUNCTIONS 375
The gamma function. Table of values and graph of the gamma function.
The beta function. Dirichlet integrals.
CHAPTER 16 FUNCTIONS OF A COMPLEX VARIABLE 392
Functions. Limits and continuity. Derivatives. Cauchy-Riemann equa- tions. Integrals. Cauchy’s theorem. Cauchy’s integral formulas. Taylor’s series. Singular points. Poles. Laurent’s series. Branches and branch points. Residues. Residue theorem. Evaluation of definite integrals.
INDEX 425
1
Numbers
Mathematics has its own language with numbers as the alphabet. The language is given structure with the aid of connective symbols, rules of operation, and a rigorous mode of thought (logic). These concepts, which previously were explored in elementary mathematics courses such as geometry, algebra, and calculus, are reviewed in the following paragraphs.
SETS
Fundamental in mathematics is the concept of aset,class, or collectionof objects having specified characteristics. For example, we speak of the set of all university professors, the set of all letters A;B;C;D;. . .;Z of the English alphabet, and so on. The individual objects of the set are called membersorelements. Any part of a set is called asubsetof the given set, e.g.,A,B,Cis a subset of A;B;C;D;. . .;Z. The set consisting of no elements is called theempty setornull set.
REAL NUMBERS
The following types of numbers are already familiar to the student:
1. Natural numbers1;2;3;4;. . .; also calledpositive integers, are used in counting members of a set. The symbols varied with the times, e.g., the Romans used I, II, III, IV, . . . Thesumaþb andproductaborabof any two natural numbersaandbis also a natural number. This is often expressed by saying that the set of natural numbers is closed under the operations of additionandmultiplication, or satisfies theclosure propertywith respect to these operations.
2. Negative integers and zerodenoted by1;2;3;. . .and 0, respectively, arose to permit solu- tions of equations such asxþb¼a, whereaandbare any natural numbers. This leads to the operation ofsubtraction, or inverse of addition, and we writex¼ab.
The set of positive and negative integers and zero is called the set ofintegers.
3. Rational numbersorfractionssuch as23,54, . . . arose to permit solutions of equations such as bx¼afor all integersaandb, whereb6¼0. This leads to the operation ofdivision, orinverse of multiplication, and we writex¼a=borabwhereais thenumeratorandbthedenominator.
The set of integers is a subset of the rational numbers, since integers correspond to rational numbers whereb¼1.
4. Irrational numberssuch as ffiffiffi p2
andare numbers which are not rational, i.e., they cannot be expressed asa=b(called thequotientofaandb), whereaandbare integers andb6¼0.
The set of rational and irrational numbers is called the set ofreal numbers.
Copyright 2002, 1963 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
DECIMAL REPRESENTATION OF REAL NUMBERS
Any real number can be expressed in decimal form, e.g., 17=10¼1:7, 9=100¼0:09, 1=6¼0:16666. . .. In the case of a rational number the decimal exapnsion either terminates, or if it does not terminate, one or a group of digits in the expansion will ultimately repeat, as for example, in
1
7¼0:142857 142857 142. . .. In the case of an irrational number such as ffiffiffi p2
¼1:41423. . . or ¼3:14159. . .no such repetition can occur. We can always consider a decimal expansion as unending, e.g., 1.375 is the same as 1.37500000 . . . or 1.3749999 . . . . To indicate recurring decimals we some- times place dots over the repeating cycle of digits, e.g.,17¼0:11_44_22_88_55_77,_ 196 ¼3:166._
The decimal system uses the ten digits 0;1;2;. . .;9. (These symbols were the gift of the Hindus.
They were in use in India by 600A.D.and then in ensuing centuries were transmitted to the western world by Arab traders.) It is possible to design number systems with fewer or more digits, e.g. thebinary systemuses only two digits 0 and 1 (see Problems 32 and 33).
GEOMETRIC REPRESENTATION OF REAL NUMBERS
The geometric representation of real numbers as points on a line called thereal axis, as in the figure below, is also well known to the student. For each real number there corresponds one and only one point on the line and conversely, i.e., there is aone-to-one(see Fig. 1-1)correspondencebetween the set of real numbers and the set of points on the line. Because of this we often use point and number interchangeably.
(The interchangeability of point and number is by no means self-evident; in fact, axioms supporting the relation of geometry and numbers are necessary. The Cantor–Dedekind Theorem is fundamental.) The set of real numbers to the right of 0 is called the set ofpositive numbers; the set to the left of 0 is the set ofnegative numbers, while 0 itself is neither positive nor negative.
(Both the horizontal position of the line and the placement of positive and negative numbers to the right and left, respectively, are conventions.)
Between any two rational numbers (or irrational numbers) on the line there are infinitely many rational (and irrational) numbers. This leads us to call the set of rational (or irrational) numbers an everywhere denseset.
OPERATIONS WITH REAL NUMBERS
Ifa,b,cbelong to the setRof real numbers, then:
1. aþbandabbelong toR Closure law
2. aþb¼bþa Commutative law of addition
3. aþ ðbþcÞ ¼ ðaþbÞ þc Associative law of addition
4. ab¼ba Commutative law of multiplication
5. aðbcÞ ¼ ðabÞc Associative law of multiplication 6. aðbþcÞ ¼abþac Distributive law
7. aþ0¼0þa¼a, 1a¼a1¼a
0 is called theidentity with respect to addition, 1 is called theidentity with respect to multi- plication.
_5 _4 _3 _2 _1 0 1 2 3 4 5
1 2 4
_3
_p √2 e p
Fig. 1-1
8. For anyathere is a numberxinRsuch thatxþa¼0.
xis called theinverse of a with respect to additionand is denoted bya.
9. For anya6¼0 there is a numberxinRsuch that ax¼1.
xis called theinverse of a with respect to multiplicationand is denoted bya1or 1=a.
Convention: For convenience, operations called subtraction and division are defined by ab¼aþ ðbÞandab¼ab1, respectively.
These enable us to operate according to the usual rules of algebra. In general any set, such asR, whose members satisfy the above is called afield.
INEQUALITIES
Ifabis a nonnegative number, we say thataisgreater than or equal to borbisless than or equal to a, and write, respectively,aAborb%a. If there is no possibility thata¼b, we writea>borb<a.
Geometrically, a>b if the point on the real axis corresponding to a lies to the right of the point corresponding tob.
EXAMPLES. 3<5 or 5>3;2<1 or1>2;x@3 means thatxis a real number which may be 3 or less than 3.
Ifa,b;andcare any given real numbers, then:
1. Eithera>b,a¼bora<b Law of trichotomy 2. Ifa>bandb>c, thena>c Law of transitivity 3. Ifa>b, then aþc>bþc
4. Ifa>bandc>0, thenac>bc 5. Ifa>bandc<0, thenac<bc
ABSOLUTE VALUE OF REAL NUMBERS
The absolute value of a real numbera, denoted byjaj, is defined asaifa>0,aifa<0, and 0 if a¼0.
EXAMPLES. j 5j ¼5,j þ2j ¼2,j 34j ¼34,j ffiffiffi p2
j ¼ ffiffiffi p2
,j0j ¼0.
1. jabj ¼ jajjbj orjabc. . .mj ¼ jajjbjjcj. . .jmj
2. jaþbj@jaj þ jbj orjaþbþcþ þmj@jaj þ jbj þ jcj þ jmj 3. jabjAjaj jbj
The distance between any two points (real numbers)aandbon the real axis isjabj ¼ jbaj.
EXPONENTS AND ROOTS
The productaa. . .aof a real numberaby itself ptimes is denoted byap, wherepis called the exponentandais called thebase. The following rules hold:
1. apaq¼apþq 3. ðapÞr¼apr 2. ap
aq¼apq 4. a
b
p¼ap bp
These and extensions to any real numbers are possible so long as division by zero is excluded. In particular, by using 2, with p¼q and p¼0, respectively, we are lead to the definitions a0¼1, aq¼1=aq.
Ifap¼N, wherepis a positive integer, we callaapthrootofNwritten ffiffiffiffi
pN
p . There may be more than one realpth root ofN. For example, since 22¼4 andð2Þ2¼4, there are two real square roots of 4, namely 2 and2. For square roots it is customary to define ffiffiffiffi
pN
as positive, thus ffiffiffi p4
¼2 and then ffiffiffi
p4
¼ 2.
Ifpandqare positive integers, we defineap=q¼ ffiffiffiffiffi ap pq
.
LOGARITHMS
Ifap¼N,pis called thelogarithmofNto the basea, writtenp¼logaN. IfaandNare positive anda6¼1, there is only one real value forp. The following rules hold:
1. logaMN¼logaMþlogaN 2. logaM
N ¼logaMlogaN 3. logaMr¼rlogaM
In practice, two bases are used, basea¼10, and thenatural basea¼e¼2:71828. . .. The logarithmic systems associated with these bases are called commonand natural, respectively. The common loga- rithm system is signified by logN, i.e., the subscript 10 is not used. For natural logarithms the usual notation is lnN.
Common logarithms (base 10) traditionally have been used for computation. Their application replaces multiplication with addition and powers with multiplication. In the age of calculators and computers, this process is outmoded; however, common logarithms remain useful in theory and application. For example, the Richter scale used to measure the intensity of earthquakes is a logarith- mic scale. Natural logarithms were introduced to simplify formulas in calculus, and they remain effective for this purpose.
AXIOMATIC FOUNDATIONS OF THE REAL NUMBER SYSTEM
The number system can be built up logically, starting from a basic set ofaxiomsor ‘‘self-evident’’
truths, usually taken from experience, such as statements 1–9, Page 2.
If we assume as given the natural numbers and the operations of addition and multiplication (although it is possible to start even further back with the concept of sets), we find that statements 1 through 6, Page 2, withRas the set of natural numbers, hold, while 7 through 9 do not hold.
Taking 7 and 8 as additional requirements, we introduce the numbers1;2;3;. . .and 0. Then by taking 9 we introduce the rational numbers.
Operations with these newly obtained numbers can be defined by adopting axioms 1 through 6, whereRis now the set of integers. These lead toproofsof statements such asð2Þð3Þ ¼6,ð4Þ ¼4, ð0Þð5Þ ¼0, and so on, which are usually taken for granted in elementary mathematics.
We can also introduce the concept of order or inequality for integers, and from these inequalities for rational numbers. For example, ifa,b,c,d are positive integers, we definea=b>c=d if and only if ad>bc, with similar extensions to negative integers.
Once we have the set of rational numbers and the rules of inequality concerning them, we can order them geometrically as points on the real axis, as already indicated. We can then show that there are points on the line which do not represent rational numbers (such as ffiffiffi
p2
,, etc.). These irrational numbers can be defined in various ways, one of which uses the idea ofDedekind cuts(see Problem 1.34).
From this we can show that the usual rules of algebra apply to irrational numbers and that no further real numbers are possible.
POINT SETS, INTERVALS
A set of points (real numbers) located on the real axis is called aone-dimensional point set.
The set of pointsxsuch thata@x@bis called aclosed intervaland is denoted by½a;b. The set a<x<bis called anopen interval, denoted byða;bÞ. The setsa<x@banda@x<b, denoted by ða;band½a;bÞ, respectively, are calledhalf openorhalf closedintervals.
The symbolx, which can represent any number or point of a set, is called avariable. The given numbersaorbare calledconstants.
Letters were introduced to construct algebraic formulas around 1600. Not long thereafter, the philosopher-mathematician Rene Descartes suggested that the letters at the end of the alphabet be used to represent variables and those at the beginning to represent constants. This was such a good idea that it remains the custom.
EXAMPLE. The set of allxsuch thatjxj<4, i.e.,4<x<4, is represented byð4;4Þ, an open interval.
The setx>acan also be represented bya<x<1. Such a set is called aninfiniteorunbounded interval. Similarly,1<x<1represents all real numbersx.
COUNTABILITY
A set is calledcountableordenumerableif its elements can be placed in 1-1 correspondence with the natural numbers.
EXAMPLE. The even natural numbers 2;4;6;8;. . .is a countable set because of the 1-1 correspondence shown.
Given set Natural numbers
2 4 6 8 . . . l l l l 1 2 3 4 . . .
A set isinfiniteif it can be placed in 1-1 correspondence with a subset of itself. An infinite set which is countable is calledcountable infinite.
The set of rational numbers is countable infinite, while the set of irrational numbers or all real numbers is non-countably infinite (see Problems 1.17 through 1.20).
The number of elements in a set is called itscardinal number. A set which is countably infinite is assigned the cardinal numberFo(the Hebrew letteraleph-null). The set of real numbers (or any sets which can be placed into 1-1 correspondence with this set) is given the cardinal numberC, called the cardinality of the continuuum.
NEIGHBORHOODS
The set of all pointsxsuch thatjxaj< where >0, is called aneighborhoodof the pointa.
The set of all points x such that 0<jxaj< in which x¼a is excluded, is called a deleted neighborhoodofaor an open ball of radiusabouta.
LIMIT POINTS
Alimit point,point of accumulation, orcluster pointof a set of numbers is a numberlsuch that every deletedneighborhood oflcontains members of the set; that is, no matter how small the radius of a ball aboutlthere are points of the set within it. In other words for any >0, however small, we can always find a member xof the set which is not equal to l but which is such that jxlj< . By considering smaller and smaller values ofwe see that there must be infinitely many such values ofx.
A finite set cannot have a limit point. An infinite set may or may not have a limit point. Thus the natural numbers have no limit point while the set of rational numbers has infinitely many limit points.
A set containing all its limit points is called aclosed set. The set of rational numbers is not a closed set since, for example, the limit point ffiffiffi
p2
is not a member of the set (Problem 1.5). However, the set of all real numbersxsuch that 0@x@1 is a closed set.
BOUNDS
If for all numbersxof a set there is a numberMsuch thatx@M, the set isbounded aboveandMis called anupper bound. Similarly ifxAm, the set isbounded belowandmis called alower bound. If for allxwe havem@x@M, the set is calledbounded.
IfMis a number such that no member of the set is greater thanMbut there is at least one member which exceedsMfor every >0, thenMis called theleast upper bound(l.u.b.) of the set. Similarly if no member of the set is smaller thanmm but at least one member is smaller thanmm þfor every >0, thenmm is called thegreatest lower bound(g.l.b.) of the set.
BOLZANO–WEIERSTRASS THEOREM
The Bolzano–Weierstrass theorem states that every bounded infinite set has at least one limit point.
A proof of this is given in Problem 2.23, Chapter 2.
ALGEBRAIC AND TRANSCENDENTAL NUMBERS A numberxwhich is a solution to thepolynomial equation
a0xnþa1xn1þa2xn2þ þan1xþan¼0 ð1Þ wherea06¼0,a1;a2;. . .;anare integers andnis a positive integer, called thedegreeof the equation, is called an algebraic number. A number which cannot be expressed as a solution of any polynomial equation with integer coefficients is called atranscendental number.
EXAMPLES. 23and ffiffiffi p2
which are solutions of 3x2¼0 andx22¼0, respectively, are algebraic numbers.
The numbers and ecan be shown to be transcendental numbers. Mathematicians have yet to determine whether some numbers such aseoreþare algebraic or not.
The set of algebraic numbers is a countably infinite set (see Problem 1.23), but the set of transcen- dental numbers is non-countably infinite.
THE COMPLEX NUMBER SYSTEM
Equations such as x2þ1¼0 have no solution within the real number system. Because these equations were found to have a meaningful place in the mathematical structures being built, various mathematicians of the late nineteenth and early twentieth centuries developed an extended system of numbers in which there were solutions. The new system became known as thecomplex number system.
It includes the real number system as a subset.
We can consider a complex number as having the formaþbi, whereaandbare real numbers called therealandimaginary parts, andi¼ ffiffiffiffiffiffiffi
p1
is called theimaginary unit. Two complex numbersaþbi andcþdiareequalif and only ifa¼candb¼d. We can consider real numbers as a subset of the set of complex numbers withb¼0. The complex number 0þ0icorresponds to the real number 0.
Theabsolute valueormodulusofaþbiis defined asjaþbij ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2þb2
p . Thecomplex conjugateof aþbiis defined asabi. The complex conjugate of the complex numberzis often indicated byzzorz. The set of complex numbers obeys rules 1 through 9 of Page 2, and thus constitutes a field. In performing operations with complex numbers, we can operate as in the algebra of real numbers, replac- ingi2 by1 when it occurs. Inequalities for complex numbers are not defined.
From the point of view of an axiomatic foundation of complex numbers, it is desirable to treat a complex number as an ordered pairða;bÞof real numbersaandbsubject to certain operational rules which turn out to be equivalent to those above. For example, we defineða;bÞ þ ðc;dÞ ¼ ðaþc;bþdÞ, ða;bÞðc;dÞ ¼ ðacbd;adþbcÞ, mða;bÞ ¼ ðma;mbÞ, and so on. We then find that ða;bÞ ¼að1;0Þ þ bð0;1Þand we associate this withaþbi, whereiis the symbol forð0;1Þ.
POLAR FORM OF COMPLEX NUMBERS
If real scales are chosen on two mutually perpendicular axesX0OXandY0OY(thexandyaxes) as in Fig. 1-2 below, we can locate any point in the plane determined by these lines by the ordered pair of numbersðx;yÞcalledrectangular coordinatesof the point. Examples of the location of such points are indicated byP,Q,R,S, andT in Fig. 1-2.
Since a complex numberxþiycan be considered as an ordered pairðx;yÞ, we can represent such numbers by points in anxyplane called thecomplex planeorArgand diagram. Referring to Fig. 1-3 above we see thatx¼cos,y¼sinwhere¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2þy2
p ¼ jxþiyjand, called theamplitudeor argument, is the angle which lineOPmakes with the positivexaxisOX. It follows that
z¼xþiy¼ðcosþisinÞ ð2Þ
called thepolar formof the complex number, whereandare calledpolar coordintes. It is sometimes convenient to write cisinstead of cosþisin.
If z1¼x1þiyi¼1ðcos1þisin1Þ and z2¼x2þiy2¼2ðcos2þisin2Þ and by using the addition formulas for sine and cosine, we can show that
z1z2¼12fcosð1þ2Þ þisinð1þ2Þg ð3Þ z1
z2¼1
2
fcosð12Þ þisinð12Þg ð4Þ zn¼ fðcosþisinÞgn¼nðcosnþisinnÞ ð5Þ wherenis any real number. Equation (5) is sometimes calledDe Moivre’s theorem. We can use this to determine roots of complex numbers. For example, ifnis a positive integer,
z1=n¼ fðcosþisinÞg1=n ð6Þ
¼1=n cos þ2k n
þisin þ2k n
k¼0;1;2;3;. . .;n1
X¢_4 _3 _2 _1 1 2 3 4 X
Y4
Y¢
3 2 1
_1 _2 _3
O Q(_3, 3)
S(2, _2) P(3, 4)
T(2.5, 0)
R(_2.5, _1.5)
Fig. 1-2
X′ O X
Y
Y′
ρ φ
y
x
P(x, y)
Fig. 1-3
from which it follows that there are in generalndifferent values ofz1=n. Later (Chap. 11) we will show thatei¼cosþisinwheree¼2:71828. . .. This is calledEuler’s formula.
MATHEMATICAL INDUCTION
The principle of mathematical induction is an important property of the positive integers. It is especially useful in proving statements involving all positive integers when it is known for example that the statements are valid forn¼1;2;3 but it issuspectedorconjecturedthat they hold for all positive integers. The method of proof consists of the following steps:
1. Prove the statement forn¼1 (or some other positive integer).
2. Assume the statement true forn¼k; wherek is any positive integer.
3. From the assumption in 2 prove that the statement must be true forn¼kþ1. This is part of the proof establishing the induction and may be difficult or impossible.
4. Since the statement is true forn¼1 [from step 1] it must [from step 3] be true forn¼1þ1¼2 and from this forn¼2þ1¼3, and so on, and so must be true for all positive integers. (This assumption, which provides the link for the truth of a statement for a finite number of cases to the truth of that statement for the infinite set, is called ‘‘The Axiom of Mathematical Induc- tion.’’)
Solved Problems
OPERATIONS WITH NUMBERS
1.1. Ifx¼4,y¼15,z¼ 3,p¼23,q¼ 16, andr¼34, evaluate (a) xþ ðyþzÞ, (b) ðxþyÞ þz, (c) pðqrÞ, (d) ðpqÞr, (e) xðpþqÞ
(a) xþ ðyþzÞ ¼4þ ½15þ ð3Þ ¼4þ12¼16 (b) ðxþyÞ þz¼ ð4þ15Þ þ ð3Þ ¼193¼16
The fact that (a) and (b) are equal illustrates theassociative law of addition.
(c) pðqrÞ ¼23fð16Þð34Þg ¼ ð23Þð243Þ ¼ ð23Þð18Þ ¼ 242 ¼ 121 (d) ðpqÞr¼ fð23Þð16Þgð34Þ ¼ ð182Þð34Þ ¼ ð19Þð34Þ ¼ 363 ¼ 121
The fact that (c) and (d) are equal illustrates theassociative law of multiplication.
(e) xðpþqÞ ¼4ð2316Þ ¼4ð4616Þ ¼4ð36Þ ¼126 ¼2
Another method: xðpþqÞ ¼xpþxq¼ ð4Þð23Þ þ ð4Þð16Þ ¼8346¼8323¼63¼2 using the distributive law.
1.2. Explain why we do not consider (a) 00 (b) 10as numbers.
(a) If we definea=bas that number (if it exists) such thatbx¼a, then 0=0 is that number xsuch that 0x¼0. However, this is true for all numbers. Since there is no unique number which 0/0 can represent, we consider it undefined.
(b) As in (a), if we define 1/0 as that numberx(if it exists) such that 0x¼1, we conclude that there is no such number.
Because of these facts we must look upon division by zero as meaningless.
1.3. Simplifyx25xþ6 x22x3. x25xþ6
x22x3¼ðx3Þðx2Þ ðx3Þðxþ1Þ¼x2
xþ1provided that the cancelled factorðx3Þis not zero, i.e.,x6¼3.
Forx¼3 the given fraction is undefined.
RATIONAL AND IRRATIONAL NUMBERS 1.4. Prove that the square of any odd integer is odd.
Any odd integer has the form 2mþ1. Sinceð2mþ1Þ2¼4m2þ4mþ1 is 1 more than the even integer 4m2þ4m¼2ð2m2þ2mÞ, the result follows.
1.5. Prove that there is no rational number whose square is 2.
Letp=qbe a rational number whose square is 2, where we assume thatp=qis in lowest terms, i.e.,pandq have no common integer factors except1 (we sometimes call such integersrelatively prime).
Thenðp=qÞ2¼2,p2¼2q2andp2is even. From Problem 1.4,pis even since ifpwere odd,p2would be odd. Thusp¼2m:
Substitutingp¼2minp2¼2q2 yieldsq2¼2m2, so thatq2 is even andqis even.
Thus p and q have the common factor 2, contradicting the original assumption that they had no common factors other than1. By virtue of this contradiction there can be no rational number whose square is 2.
1.6. Show how to find rational numbers whose squares can be arbitrarily close to 2.
We restrict ourselves to positive rational numbers. Sinceð1Þ2¼1 andð2Þ2¼4, we are led to choose rational numbers between 1 and 2, e.g., 1:1;1:2;1:3;. . .;1:9.
Since ð1:4Þ2¼1:96 and ð1:5Þ2¼2:25, we consider rational numbers between 1.4 and 1.5, e.g., 1:41;1:42;. . .;1:49:
Continuing in this manner we can obtain closer and closer rational approximations, e.g.ð1:414213562Þ2 is less than 2 whileð1:414213563Þ2is greater than 2.
1.7. Given the equation a0xnþa1xn1þ þan¼0, where a0;a1;. . .;an are integers and a0 and an6¼0. Show that if the equation is to have a rational rootp=q, thenpmust divide anand q must dividea0exactly.
Sincep=qis a root we have, on substituting in the given equation and multiplying byqn, the result a0pnþa1pn1qþa2pn2q2þ þan1pqn1þanqn¼0 ð1Þ or dividing byp,
a0pn1þa1pn2qþ þan1qn1¼ anqn
p ð2Þ
Since the left side of (2) is an integer, the right side must also be an integer. Then sincepandqare relatively prime,pdoes not divideqnexactly and so must dividean.
In a similar manner, by transposing the first term of (1) and dividing byq, we can show thatqmust dividea0.
1.8. Prove that ffiffiffi p2
þ ffiffiffi p3
cannot be a rational number.
Ifx¼ ffiffiffi p2
þ ffiffiffi p3
, thenx2¼5þ2 ffiffiffi p6
,x25¼2 ffiffiffi p6
and squaring,x410x2þ1¼0. The only possible rational roots of this equation areffiffiffi 1 by Problem 1.7, and these do not satisfy the equation. It follows that
2 p þ ffiffiffi
p3
, which satisfies the equation, cannot be a rational number.
1.9. Prove that between any two rational numbers there is another rational number.
The set of rational numbers is closed under the operations of addition and division (non-zero denominator). Therefore,aþb
2 is rational. The next step is to guarantee that this value is betweena andb. To this purpose, assumea<b. (The proof would proceed similarly under the assumptionb<a.) Then 2a<aþb, thusa<aþb
2 andaþb<2b, thereforeaþb 2 <b.
INEQUALITIES
1.10. For what values ofxisxþ3ð2xÞA4x?
xþ3ð2xÞA4xwhenxþ63xA4x, 62xA4x, 64A2xx, 2Ax, i.e.x@2.
1.11. For what values ofxisx23x2<102x?
The required inequality holds when
x23x210þ2x<0; x2x12<0 or ðx4Þðxþ3Þ<0 This last inequality holds only in the following cases.
Case 1:x4>0andxþ3<0, i.e.,x>4 andx<3. This isimpossible, sincexcannot be both greater than 4 and less than3.
Case 2:x4<0 andxþ3>0, i.e. x<4 and x>3. This is possible when3<x<4. Thus the inequality holds for the set of allxsuch that3<x<4.
1.12. IfaA0 andbA0, prove that12ðaþbÞA ffiffiffiffiffi pab
.
The statement is self-evident in the following cases (1) a¼b, and (2) either or both ofaandbzero.
For bothaandbpositive anda6¼b, the proof is by contradiction.
Assume to the contrary of the supposition that12ðaþbÞ< ffiffiffiffiffi ab p
then14ða2þ2abþb2Þ<ab.
That is,a22abþb2¼ ðabÞ2<0. Since the left member of this equation is a square, it cannot be less than zero, as is indicated. Having reached this contradiction, we may conclude that our assumption is incorrect and that the original assertion is true.
1.13. Ifa1;a2;. . .;an andb1;b2;. . .;bn are any real numbers, proveSchwarz’s inequality ða1b1þa2b2þ þanbnÞ2@ða21þa22þ þa2nÞðb21þb22þ þb2nÞ For all real numbers, we have
ða1þb1Þ2þ ða2þb2Þ2þ þ ðanþbnÞ2A0 Expanding and collecting terms yields
A22þ2CþB2A0 ð1Þ
where
A2¼a21þa22þ þa2n; B2¼b21þb22þ þb2n; C¼a1b1þa2b2þ þanbn ð2Þ The left member of (1) is a quadratic form in . Since it never is negative, its discriminant, 4C24A2B2, cannot be positive. Thus
C2A2B20 or C2A2B2 This is the inequality that was to be proved.
1.14. Prove that1 2þ1
4þ1
8þ þ 1
2n1<1 for all positive integersn>1.
Sn¼1 2þ1
4þ1
8þ þ 1 2n1 Let
1
2Sn¼ 1 4þ1
8þ þ 1 2n1þ1
2n Then
1 2Sn¼1
21
2n: ThusSn¼1 1
2n1<1 for alln: Subtracting,
EXPONENTS, ROOTS, AND LOGARITHMS 1.15. Evaluate each of the following:
ðaÞ 3438 314 ¼34þ8
314 ¼34þ814¼32¼ 1 32¼1
9
ðbÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5106Þð4102Þ
8105 s
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 54
8 106102 105 s
¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2:5109
p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 251010
p ¼5105or 0:00005
ðcÞ log2=3 278
¼x: Then 23x
¼278 ¼ 323
¼ 23 3
or x¼ 3
ðdÞ ðlogabÞðlogbaÞ ¼u: Then logab¼x;logba¼yassuminga;b>0 anda;b6¼1:
Thenax¼b,by¼aandu¼xy.
SinceðaxÞy¼axy¼by¼awe haveaxy¼a1orxy¼1 the required value.
1.16. IfM>0,N>0; anda>0 buta6¼1, prove that logaM
N ¼logaMlogaN.
Let logaM¼x, logaN¼y. Thenax¼M,ay¼Nand so M
N¼ax
ay¼axy or loga
M
N¼xy¼logaMlogaN
COUNTABILITY
1.17. Prove that the set of all rational numbers between 0 and 1 inclusive is countable.
Write all fractions with denominator 2, then 3;. . .considering equivalent fractions such as12;24;36;. . .no more than once. Then the 1-1 correspondence with the natural numbers can be accomplished as follows:
Rational numbers Natural numbers
0 1 12 13 23 14 34 15 25 . . . l l l l l l l l l 1 2 3 4 5 6 7 8 9 . . .
Thus the set of all rational numbers between 0 and 1 inclusive is countable and has cardinal numberFo
(see Page 5).
1.18. IfAandBare two countable sets, prove that the set consisting of all elements fromAorB(or both) is also countable.
SinceAis countable, there is a 1-1 correspondence between elements ofAand the natural numbers so that we can denote these elements bya1;a2;a3;. . ..
Similarly, we can denote the elements ofBbyb1;b2;b3;. . ..
Case 1: Suppose elements ofAare all distinct from elements ofB. Then the set consisting of elements from AorBis countable, since we can establish the following 1-1 correspondence.
AorB
Natural numbers
a1 b1 a2 b2 a3 b3 . . . l l l l l l 1 2 3 4 5 6 . . .
Case 2: If some elements ofAandBare the same, we count them only once as in Problem 1.17. Then the set of elements belonging toAorB(or both) is countable.
The set consisting of all elements which belong toAorB(or both) is often called theunionofAandB, denoted byA[BorAþB.
The set consisting of all elements which are contained in bothAand Bis called theintersectionofAand B, denoted byA\BorAB. IfAandBare countable, so isA\B.
The set consisting of all elements inAbutnotinBis writtenAB. If we letBBbe the set of elements which are not inB, we can also writeAB¼ABB. IfAandBare countable, so isAB.
1.19. Prove that the set of all positive rational numbers is countable.
Consider all rational numbersx>1. With each such rational number we can associate one and only one rational number 1=xinð0;1Þ, i.e., there is aone-to-one correspondencebetween all rational numbers>1 and all rational numbers inð0;1Þ. Since these last are countable by Problem 1.17, it follows that the set of all rational numbers>1 is also countable.
From Problem 1.18 it then follows that the set consisting of all positive rational numbers is countable, since this is composed of the two countable sets of rationals between 0 and 1 and those greater than or equal to 1.
From this we can show that the set of all rational numbers is countable (see Problem 1.59).
1.20. Prove that the set of all real numbers in½0;1is non-countable.
Every real number in½0;1has a decimal expansion:a1a2a3. . .wherea1;a2;. . .are any of the digits 0;1;2;. . .;9.
We assume that numbers whose decimal expansions terminate such as 0.7324 are written 0:73240000. . . and that this is the same as 0:73239999. . ..
If all real numbers in½0;1 are countable we can place them in 1-1 correspondence with the natural numbers as in the following list:
1 2 3 ...
$
$
$
0:a11a12a13a14. . . 0:a21a22a23a24. . . 0:a31a32a33a34. . .
...
We now form a number
0:b1b2b3b4. . .
whereb16¼a11;b26¼a22;b36¼a33;b46¼a44;. . .and where allb’s beyond some position are not all 9’s.
This number, which is in½0;1is different from all numbers in the above list and is thus not in the list, contradicting the assumption that all numbers in½0;1were included.
Because of this contradiction it follows that the real numbers in½0;1cannot be placed in 1-1 corre- spondence with the natural numbers, i.e., the set of real numbers in½0;1is non-countable.
LIMIT POINTS, BOUNDS, BOLZANO–WEIERSTRASS THEOREM
1.21. (a) Prove that the infinite sets of numbers 1;12;13;14;. . . is bounded. (b) Determine the least upper bound (l.u.b.) and greatest lower bound (g.l.b.) of the set. (c) Prove that 0 is a limit point of the set. (d) Is the set a closed set? (e) How does this set illustrate the Bolzano–Weierstrass theorem?
(a) Since all members of the set are less than 2 and greater than1 (for example), the set is bounded; 2 is an upper bound,1 is a lower bound.
We can find smaller upper bounds (e.g.,32) and larger lower bounds (e.g.,12).
(b) Since no member of the set is greater than 1 and since there is at least one member of the set (namely 1) which exceeds 1for every positive number, we see that 1 is the l.u.b. of the set.
Since no member of the set is less than 0 and since there is at least one member of the set which is less than 0þfor every positive(we can always choose for this purpose the number 1=nwherenis a positive integer greater than 1=), we see that 0 is the g.l.b. of the set.
(c) Letxbe any member of the set. Since we can always find a numberxsuch that 0<jxj< for any positive number(e.g. we can always pickxto be the number 1=nwherenis a positive integer greater than 1=), we see that 0 is a limit point of the set. To put this another way, we see that any deleted neighborhood of 0 always includes members of the set, no matter how small we take >0.
(d) The set is not a closed set since the limit point 0 does not belong to the given set.
(e) Since the set is bounded and infinite it must, by the Bolzano–Weierstrass theorem, have at least one limit point. We have found this to be the case, so that the theorem is illustrated.
ALGEBRAIC AND TRANSCENDENTAL NUMBERS 1.22. Prove that ffiffiffi
32 p þ ffiffiffi
p3
is an algebraic number.
Letx¼ ffiffiffi
32 p þ ffiffiffi
p3
. Thenx ffiffiffi p3
¼ ffiffiffi
32 p
. Cubing both sides and simplifying, we findx3þ9x2¼ 3 ffiffiffi
p3
ðx2þ1Þ. Then squaring both sides and simplifying we findx69x44x3þ27x2þ36x23¼0.
Since this is a polynomial equation with integral coefficients it follows that ffiffiffi
32 p þ ffiffiffi
3 p
, which is a solution, is an algebraic number.
1.23. Prove that the set of all algebraic numbers is a countable set.
Algebraic numbers are solutions to polynomial equations of the form a0xnþa1xn1þ þan¼0 wherea0;a1;. . .;anare integers.
LetP¼ ja0j þ ja1j þ þ janj þn. For any given value ofPthere are only a finite number of possible polynomial equations and thus only a finite number of possible algebraic numbers.
Write all algebraic numbers corresponding toP¼1;2;3;4;. . .avoiding repetitions. Thus, all algebraic numbers can be placed into 1-1 correspondence with the natural numbers and so are countable.
COMPLEX NUMBERS
1.24. Perform the indicated operations.
(a) ð42iÞ þ ð6þ5iÞ ¼42i6þ5i¼46þ ð2þ5Þi¼ 2þ3i (b) ð7þ3iÞ ð24iÞ ¼ 7þ3i2þ4i¼ 9þ7i
(c) ð32iÞð1þ3iÞ ¼3ð1þ3iÞ 2ið1þ3iÞ ¼3þ9i2i6i2¼3þ9i2iþ6¼9þ7i ðdÞ 5þ5i
43i ¼5þ5i 43i 4þ3i
4þ3i¼ð5þ5iÞð4þ3iÞ
169i2 ¼2015iþ20iþ15i2 16þ9
¼35þ5i
25 ¼5ð7þiÞ
25 ¼7
5 þ1 5i ðeÞ iþi2þi3þi4þi5
1þi ¼i1þ ði2ÞðiÞ þ ði2Þ2þ ði2Þ2i
1þi ¼i1iþ1þi 1þi
¼ i 1þi1i
1i¼ii2 1i2¼iþ1
2 ¼1 2þ1
2i ðfÞ j34ijj4þ3ij ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð3Þ2þ ð4Þ2
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4Þ2þ ð3Þ2 q
¼ ð5Þð5Þ ¼25
ðgÞ 1 1þ3i 1
13i
¼ 13i
19i21þ3i 19i2
¼ 6i
10
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0Þ2þ 6
10
2
s
¼3 5
1.25. Ifz1andz2are two complex numbers, prove thatjz1z2j ¼ jz1jjz2j.
Letz1¼x1þiy1,z2¼x2þiy2. Then
jz1z2j ¼ jðx1þiy1Þðx2þiy2Þj ¼ jx1x2y1y2þiðx1y2þx2y1Þj
¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx1x2y1y2Þ2þ ðx1y2þx2y1Þ2 q
¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21x22þy21y22þx21y22þx22y21 q
¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx21þy21Þðx22þy22Þ
q ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x21þy2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x22þy22
q ¼ jx1þiy1jjx2þiy2j ¼ jz1jjz2j:
1.26. Solvex32x4¼0.
The possible rational roots using Problem 1.7 are1,2,4. By trial we findx¼2 is a root. Then the given equation can be written ðx2Þðx2þ2xþ2Þ ¼0. The solutions to the quadratic equation ax2þbxþc¼0 are x¼b ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b24ac p
2a . For a¼1, b¼2, c¼2 this gives x¼2 ffiffiffiffiffiffiffiffiffiffiffi 48 p
2 ¼
2 ffiffiffiffiffiffiffi p4
2 ¼22i
2 ¼ 1i.
The set of solutions is 2,1þi,1i.
POLAR FORM OF COMPLEX NUMBERS
1.27. Express in polar form (a) 3þ3i, (b) 1þ ffiffiffi p3
i, (c) 1, (d) 22 ffiffiffi p3
i. See Fig. 1-4.
(a) Amplitude¼458¼=4 radians. Modulus¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 32þ32
p ¼3 ffiffiffi
2 p
. Then 3þ3i¼ðcosþisinÞ ¼ 3 ffiffiffi
p2
ðcos=4þisin=4Þ ¼3 ffiffiffi p2
cis=4¼3 ffiffiffi p2
ei=4
(b) Amplitude¼1208¼2=3 radians. Modulus¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ2þ ð ffiffiffi
3 p Þ2 q
¼ ffiffiffi 4
p ¼2. Then1þ3 ffiffiffi 3 p
i¼ 2ðcos 2=3þisin 2=3Þ ¼2 cis 2=3¼2e2i=3
(c) Amplitude¼1808¼radians. Modulus¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ2þ ð0Þ2 q
¼1. Then1¼1ðcosþisinÞ ¼ cis¼ei
(d) Amplitude ¼2408¼4=3 radians. Modulus ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ2þ ð2 ffiffiffi
p3 Þ2
q ¼4. Then 22 ffiffiffi
p3
¼ 4ðcos 4=3þisin 4=3Þ ¼4 cis 4=3¼4e4i=3
45° 120° 180°
240°
3 3√2 3
_2√3
√3
(a) (b) (c) (d)
2
_1 _1
_2
4
Fig. 1-4
1.28. Evaluate (a) ð1þ ffiffiffi p3
iÞ10, (b) ð1þiÞ1=3. (a) By Problem 1.27(b) and De Moivre’s theorem,
ð1þ ffiffiffi 3
piÞ10¼ ½2ðcos 2=3þisin 2=3Þ10¼210ðcos 20=3þisin 20=3Þ
¼1024½cosð2=3þ6Þ þisinð2=3þ6Þ ¼1024ðcos 2=3þisin 2=3Þ
¼102412þ12 ffiffiffi 3 p i
¼ 512þ512 ffiffiffi 3 p
i (b) 1þi¼ ffiffiffi
2
p ðcos 1358þisin 1358Þ ¼ ffiffiffi 2
p ½cosð1358þk3608Þ þisinð1358þk3608Þ. Then
ð1þiÞ1=3¼ ð ffiffiffi 2
p Þ1=3 cos 1358þk3608 3
þisin 1358þk3608 3
The results fork¼0;1;2 are ffiffiffi2 p6
ðcos 458þisin 458Þ; ffiffiffi2
p6
ðcos 1658þisin 1658Þ; ffiffiffi2
p6
ðcos 2858þisin 2858Þ
The results fork¼3;4;5;6;7;. . .give repetitions of these. These complex roots are represented geometrically in the complex plane by pointsP1;P2;P3on the circle of Fig. 1-5.
MATHEMATICAL INDUCTION
1.29. Prove that 12þ22þ33þ42þ þn2¼16nðnþ1Þð2nþ1Þ.
The statement is true forn¼1 since 12¼16ð1Þð1þ1Þð21þ1Þ ¼1.
Assumethe statement true forn¼k. Then
12þ22þ32þ þk2¼16kðkþ1Þð2kþ1Þ Addingðkþ1Þ2to both sides,
12þ22þ32þ þk2þ ðkþ1Þ2¼16kðkþ1Þð2kþ1Þ þ ðkþ1Þ2¼ ðkþ1Þ½16kð2kþ1Þ þkþ1
¼16ðkþ1Þð2k2þ7kþ6Þ ¼16ðkþ1Þðkþ2Þð2kþ3Þ
which shows that the statement is true forn¼kþ1ifit is true forn¼k. But since it is true forn¼1, it follows that it is true forn¼1þ1¼2 and forn¼2þ1¼3;. . .;i.e., it is true for all positive integersn.
1.30. Prove thatxnyn hasxyas a factor for all positive integersn.
The statement is true forn¼1 sincex1y1¼xy.
Assumethe statement true forn¼k, i.e., assume thatxkykhasxyas a factor. Consider xkþ1ykþ1¼xkþ1xkyþxkyykþ1
¼xkðxyÞ þyðxkykÞ
The first term on the right hasxyas a factor, and the second term on the right also hasxyas a factor because of the above assumption.
Thusxkþ1ykþ1hasxyas a factorifxkykdoes.
Then sincex1y1hasxyas factor, it follows thatx2y2hasxyas a factor,x3y3hasxyas a factor, etc.
P2
P3
P1 165°
285°
45°
√26
Fig. 1-5
