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Unformatted text preview: PREFACE TO THE THIRD EDITION 'i'he most signiﬁcant Change in this third edition is the incinsioh ofa new (starred)
Chapter 17 on planetary motion, in which calculus is employed for a substantial
physics problem. in preparation for this, the old Appendix to Chapter 4 has been repiaced by
three Appendices: the ﬁrst two COver vectors and conic sections, while poiar coor
dinates are now deferred untii the third Appendix, which also discusses the polar
coordinate equations of the conic sections. Moreover, the Appendix to Chapter 1?.
has been extended to treat vector operations on vectorvaiued curves. Another iarge change is merely a rearrangement of old material: “The Cos
mopolitan integral," previously a second Appendix to Chapter 23, is now an
Appendix to the chapter on “Integration in Elementary Terms" (previously Chap
ter i8, now Chapter 19}; moreover, those problems from that chapter which used
the matcriai from that Appendix now appear as problems in the newly placed
Appendix, A few other changes and renumbcring of Problems result from corrections, and
elimination of incorrect problems. I was both startled and somewhat dismayed when i reailzed that after altow
ing l3 years to elapse between the ﬁrst and second editions of the book, I have
ailowed another 14 years to elapse before this third edition. During this time I
seem to have accumuiated a notsosliort list of corrections, but no longer have
the originai communications, and therefore cannot properiy thank the various in
dividuals involved (who by now have probably lost interest anyway). 1 have had
time to make only a few changes to the Suggested Reading, which after ail these
years probably requires a complete revision; this wiil have to wait until the next
edition, which I hope to make in a more timely fashion. MICHAEL SPIVAK PREFACE TO THE FOURTH EDETION Promises, promises! In the preface to the third edition I noted that it was 13 years
herween the first and second editions, and then another [4 years before the third,
expressing the hope that the next edition would appear sooner. ‘r‘l'eil, here it is
another 14 years later before the fourth, and presumably final, edition. I Ait'hough smnil changes have been made to some material, especialiy in Chap
ters 5 and 20, this edition tlilfers mainly in the introduction of additional problems,
a complete update ofthe Suggested Reading, and the correction ofnttmcrotts er—
rors. These have been brought to my attention over the years by, among; others,
Nils von Barth; Philip Loewen; Fernando hchias; Lance hIilter, who provided a
long list, particularly for the answer hook; and Michael hIaltenfort, who provided
an amazingly extensive list of misprints, errors, and criticisms. ' Most of all, however, E am indebted to my friend Ted Shifrin, who has been
using the book for the text in his renowned course at the University of Georgia
for all these years, and who prodded and helped the to Iinaily make this needed
revision I musz also thank the students inhis course this last academic year, who
served as guinea pigs for the new edition, resuhing, in particuiar, in the current
proofin Prohiem 820 for the Rising Sun Lemma, far simpler than Reisz’s original
proof, or even the proof in [38} of the Suggested Reading, which itself has now
been updated considerably, again with great help from Ted. MICHAEL RPIVAK. CONTENTS PREFACE vi PART 1 Prologue 1 Basic Properties of Numbers 3
2 Numbers of Various Sons 21 PART II Foundations 3 Functions 39 Appendix Ordered Pain‘ 54
4 Graphs 56
Appendix 1" Vectors 75
AppetidixZ, The ConicSecrious 80
Appendix 3. Polar Cam'rﬁuarer 84
Limits 90
Continuous Functicns l 15
Three Hard Theorems 322 
Least Upper Bounds E33
Appendix. Uniﬁrm Contbmiyl 144 oo‘JGm PART III Derivatives and Integrals 9 Derivatives 149
10 Difl'crcmialion 168
1] Signiﬁcance of the Derivative E88
Appendix: Corwaxilwmd Concnviti 219
12 Inverse Functions 231)
Appendix. Paranwlric Repmmmlian omeveS 244
13 integrals 253
Appendix. Riemarrrt Sum: 28?.
£4 The Fundamental Theorem of Calculus 285 xiii (Ibrahim: 15 The Trigonomen‘ic Functions 303
‘5‘l6 Jr is Irrational 324
”:17 i’ianctary Motion 330
$8 The Logaz‘ilhm anti Exponential Functions 339
19 Integration in Eéamcmary Terms 363
Appwul‘i't The Comopoiilmr Inkgm! 402 _ 1 PARTIV Inﬁnite Sequences and Inﬁnite Series ' G AL C I L l S 20 Approximation by Polynomia} Functions 4H
*2! e is Transcendental 442 22 Inﬁnite Sequences 452 23 Inﬁnizc Series 475 '24 Uniform Cnnvcx‘g‘emc and Power Series 499 25 Complex Numbers 526 26 Complex Functions 541 2'? Compiex Power Series 555 PART V Epilogu e 28 Fields 58?
29 Construction nfthc Real Numbers 588
30 Uniqueness ol‘lhc Rea? Numbers 601 Suggcxled Reading 609 Aimee): {(0 5313.7wa pmMcuuj 619
Glosmfr qfé'wnbols 665 hedex 669 To be smartions that
you are ignorant i: a great step
to knowledge. BENJAMIN DISRAELI CHA?TER BASiC PROPERTIES OF NUMBERS The title of this chapter expresses in a few worth the mathematical knowledge
required to read this book. in fact, this short chapter is simply an cxplanation of
what is meant by tin: “basic properties of numbers,” ail of whichmaddition and
muitiplicat‘ion, subtraction and divasion, solutions ol‘cquations and inequalities,
factoring and other algebraic manipulations—are aircaciy Familiar to us: chcrr
titclcss, this chapter is not a review Despite the fnn‘tiliarity of the subject, the
survey we are about to nntlcrtalte wiil probably seem quite novel; it (locsjnot aim
to present an extended review of old material, but to condense this knowledge
into a Few simple and obvious propertics'ofnumbers. Some may eve1% item 100
obvious to mention, but a surprising number ol‘tiivcrsc and important Riots turn
out to be. consequences ofthc ones we shail emphasize. Ol‘thc tweivo properties which we shail study in this chapter, the first him: are
concerned with the fttndamcntai operations of addition and tnuitiplication. For
the moment we consider only addition: this operation is performed on a pair
ol‘ nttnihcmwthe sum (l t~ b exists for any two given numimrs (t and 2‘) {which
may possibly be the some number, of course} it might seem reasonabic to regard
addition as an operation which can be performed on scvcrai numbers at once, and
consider the sum at + —t— (1,. of J] numbers (It. . . . ,a,. as ti basic concept. It is
more convenient, however, to consider addition of pairs of numbers only, And to
cieﬁnc other sums in terms oi‘sttms of this type. For the Sum of three numbers
a, 12, and c, this may be tlont: in two dili‘orcnt ways. Ont: can hast add it and r,
obtaining b + c, and than atltl {t to this llLI111i3ttl‘, obtnining l! + (t’: l a); or one can
ﬁrst add (i and b, and than aclcl the sum (1 +1: to t, obtaining {a + b) + 5. Of
comic, the two compound sums obtained are equal, and this fact is the my ﬁrst property we shall iist:
(Pl) if a, I), and 0 art: any l‘ttm'lhm‘s, than n +(b +6) = (a to) +12 3‘1“: statcmcnt of this property cical‘ly renders a soparale concept of (111‘. sum 0!
three numbers superfluous; we simply agree that a + b + t denotes the number
a +(btc) = {a +b) ~t~ct Addition of Four numbers requires similar, though sliglttiy
more involved, considerations, The symbol a t b + c + d is deﬁned to moan (i) ((alb)+c}+d.
or {9.) (rt + (b + 6)) + d.
or {3) a 4 {(b + c) + d),
or 4) a + (b+ {c+d)).
) (a ~i~bi + (Mind). AA infogtte This definition is unambiguous since these numbers are all equal, Fortunately, this
fact need not be listed separately, since it foilows from the property Pi already
listed. For example, we know from Pl that ((,+b)+e=u+(b+cl. and it follows immediately that (i) and (2) are equaL The equality of (2) and (3)
is a direct consequenee of Pi, although this may not be apparent at first sight
(one must let I; t 0 play the role of I: in Pi, and d the role of c). The equalities
(3) m (4) = (5) are also simple to prove. it is probably obvious that an appeal to Pl \viil aiso suffice to prove the equality
of the i4 possible ways ol‘sunun‘tng live numbers, but it may not be so clear how we
can reasonably arrange a proofti'tat this is so without actnaily listing these 14 sums.
Such a procedure is feasible, but would soon cease to be iixve considered collections
oi'six, seven, or more numbers; it wouid be totally inadequate to prove the equality
of ali pos‘sibie sums of an arbitrary finite collection of numbers n}, .,,,(t.,. This
fact may be taken for granted, but for those who would like to‘ worry about the
prooﬁand it is worth worrying about once) a reasonable approach is outlined in
?roblem 24. Henceforth, we shall usually make a tacit appeal to the results of this
problem and write sums n; +   + m, with a hiithe disregard for the arrangement
of parentheses. The number 0 has one property so important that we list it next: (F2) If a is any number, then n+0w0lrtnrt. An important role is also piayed by 0 in the third property of our list:
(P3) For every number u, there is a number we such that a ,4. (Wu) m (me) i r. = 0. Property P2 ought to represent a distinguishing characteristic of the number 0,
and it is comforting to note that we are aiready in a position to prove this, Indeed,
ifa number x satisﬁes 0+.r=a for any one number n, then .r = (3 (and consequently this equation also iioids for all
numbers a). The proof of this assertion involves nothing more than subtracting a
from both sides of the equation, in other words, adding “(I to both sides; as the
following detailed proof shows, all three properties ill—P3 must be used to justify
this operation. if u «l ' = n,
then (—a) l (a +th w {on} t~ n m 0:
hence {(ma) + a) + N m 0‘.
hence
hence .7 , Basic Properties Qannberx 5 As we have just hinted, it is convenient to regard subtraction as an operation
derived from addition: we consider rt —— b to be an abbreviation for a +(—b). it
is then possible to ﬁnd the solution of certain simple equations by a series of steps
(eachjustiﬁed by Pi, P2, or P3) similar to the ones just presented For the equation
a + x w a. For example: ' if x + 3 = 5,
then (,r 4 3) i {—3} = 5 + (”3);
hence x+(3+ (m3))=5~w3 m2;
hence x + {l m— 2;
hence .r m 2. Naturally, such elaborate solutions are oi'interest oniy until you become eonvineed
that they can always be supplied. in praotiee, it is usually gust a waste tigi‘timc to
solve an equation by indicating so expliciily the reliance on properties Plst P2, and
P3 (or any of the further properties we shall list}, ‘7 Only one other property ofaddition remains to be listed. When considering the
sums of three numbers a, b, and e, oniy two sums were mentioned: {a + b) + c
and rt + {b + e). Actually, several other arrangements are obtained if the order of
a, b, and e is changed. That these sums are all equal depends on (P4) If a and I: are any numbers, then
a +1) =b+a. The statement (£134 is meant to emphasize that although the operation of ad»
dition of pairs of numbers might eoneeivabiy depend on the order oiT the two
numbers, in fact it does not, it is helpful to remember that not all operations are
so well behaved. For example, subtraction does not have this property: usualiy
a — b aé b we. in passing we might ask just when a — .5 does equal b ,. 1’s, and is
is amusing to discover how powerless we are if we rely only on properties E’l—Pd
to justify our manipulations, Algebra of the most elementary variety shows that
a — b = b — a only when a = b. Nevertheless, it is impossible to derive this fact
from properties Pl—Pél; it is instructive to examine the nienientary algebra care
fully and determine which stems) cannot be justiﬁed by leP4, We wili indeed
be able to justify all steps in detail when a few more properties are iisted. Oddly
enough, however, the crucial property involves multiplication. The basic properties of multiplication are fortunately so similar to those for ad
dition that little comment will be needed; both the meaning and the consequences
should be clear. (As in elementary algebra, the product of (i and b will be denoted
by a  b, or simply rib.) (P5) If (I, b, and c are any numbers, then
o{b‘c :{adﬂro (P6) if n is any number, then 'ofogue Moreover, i oi 0. (The assertion that ] 57$ 0 may seem a strange fact to list, but we have to
list it, because there is no way it could possibly be proved on the basis of the
other properties listcdmthcsc properties wouid all iioid if there were only one
number, namely, 0.) (P?) For every number :2 ye 0, there is a tiumbcr a“i such that (P8) If a and b are any numbers, than abzba. One detail which deserves emphasis is the appearance of the condition a 5x5 0
in P? This condition is quite necessary; since 05; = 0 {or ail nttrn‘hcrs b, there is no
numberﬂ" satisfying 9  0""1 m 1. 'fhis restriction has an important consequence
for division. just as subtraction was deﬁned in terms of addition, so division is
deﬁned in terms of multipiication: ihc symbol (1].!) mcans a . 1;". Since 0"' is
meaningless, 12/0 is also moaninglcss—division by O is always nndcﬁned. Property P7 has two important consequences. if a  b n a t c, it does not
ticccssai‘iiy follow that b w c; for if a = 0, then both a  b and a t c are (3, no matter
what I; and c are. However, if a gé 0, then i: = c; this can he deduced From P7 as
follows: If ab=¢iicanda;&0,
thorn a"  (ab) =a“i {aC):
hence (0"t a)b:{n'"' a)'c;
hence I . b m l 'c; hence b =c. It. is aiso a consequence of PT" that if a b = 0, then either a m 0 or b = 0. In fact. if ovmeandayEO,
then ﬂm2~{ab)m0;
hence (ffll >rt) b = 0;
hence l .‘i = 0;
hence it = 0. (It may happen that rt = 0 and b m 0 art: both true; this possihiiity is not oxcludcd
when we say "either (. = {J or b m 0”; in mathematics "or” is always used in the
sensc of “one or the other, or both“) This iattcr consequence of P”? is constantly used in the solution of equations.
Suppose, for example, that a number x is known to satisfy {x  l)(x —2}=G. Then it follows that either it w l m 0 or x — 2 = (J; hcncc x m i or x m 2. I . Basic Propertm quitmhcrs '7 On the basis of the eight properties listed so for it is still possible to prov: vciy
little. Listing the next property, which combines the operations of addition and
multiplication, will alter this situation drastically. (P9) If a, b, and c are any numbers, then
a{bi—c)=a b~ivac.
(Notice that the equation (in +c) > n =1; o + c a is also true, by Pig.) As an example of the usefulness of P9 wonwiil 110w determine just whcn'a m b m ii—o: 'ii
it If a ~b=‘f‘b«~a, ;
than «innit:tb—aiibwwbwa); ‘
hence a=b+bwo: ‘
hence a+a=(b+bal+n=b+b.
Consequently (l'(i+l)mbt(§+1}.
and therefore a = I). A second use of F9 is the justiﬁcation of the assertion n . O = 0 which we have
already made, and even used in a proof on page 6 {can you ﬁnd where?) This
{act was not listed as one of the basic properties, even though no proofwzis ocht‘cd
when it was ﬁrst mentioned. With lePS alone a proof was not possiblchsince the
number 0 appears only in PE and P3, which concern addition, while the {assertion
in question involves multipiication. With P9 tho proof is simpic, though perhaps
not obvious: We have a»0+av0=a(0+0) ,
mail; l. as we have already noted, this immediatciy implies {by adding ~(a {l) to both
sides) that a  D m 0. A series of further consequences of P9 may help explain the somewhat myste
rious rule that the product of two negative numbers is positive. To begin with,
we will establish the more easily acceptable assertion that (—a) . l: = ~«{a >12). To
prove this, note that (—a)b+nvb=[{—a}+a]b
:04:
:0 it follows immediately (by adding ~(rt ~ b) to both sides) that (—(1)  ii = —(o b).
Now note that (ma)  (“bi + [—(a bll w—w (Ma) . (—1)) + {—0) ' b
m (—a)  {(—12} i~ b] (—a) 0 0. ll 1! ‘rot'ogue Consequently, adding (a  b} to both sides, we obtain
(~a) . (Mb) : a  b. The fact that the ptoduct oi“ two negative numbers is positive is thus a consequence
of Pl—P9. in other words, g'fwe want PI to P9 to be true, the rate]??? the [tiniest :11“!th
negative tztttttbt'tzt t'sﬂrced upon us. The various consequences of P9 examined so far, although interesting and im—
portant, do not really indicate the signiﬁcance: of P9; after ail, we could have listed
Each of these properties separately. Actually, P9 is the justiﬁcation for almost all
algebraic manipulations. For example, aithough we have shown how to solve the equation
(x m DU #2) =0, we can hardly expect to be presented with an equation in this form, We are more
likely to he canfronted with the equation x2 m 3t + 2 = 0. ‘
The “faum’i‘eazion” x3 « 3x + 2 = (x w 1)(x m 2) is reaily a triple use of ?9: (xWi)t{x—'.Z)=x(x2}+{—i)'{x'2}
waxx“rtm2)+(—1)x+t—§)t~2)
mt2+x[(w2}+{w1)]+2
ms‘2—3x—tw2. A ﬁnai illustration ofthc importance of F9 is the fact. that this property is acttiaily
used every time one multiplies arabic numerals. For example, the calculation 13
x24 ""53
26 312
is a concise arrangement for the Foliowing equations: l324=i3(2~10+£i}
$13.2,10+§3.4.
m26'10‘i‘52. (Note that moving 26 to the tail in the above calculation is the same as writing
26  i0.) The multiplication 13 4 w 52 uses P9 also: i34:(I~iO+3)4
=ilO4+34
writ10+l2
=4tiUt~l10+2
={4+l}10+2
$5,104,;
m52. I. Basic Properties efﬁﬁunbert 9 The properties i’lwI’Q have descriptive names which are not essentiai to remetw
her, but which are often convenient for reference. We will take. this opportunity to
iist properties Pl~P9 together and indicate the names by which they are commonly
designated. (Pl) (Associative iaw for addition) a t (b + c) e (a + b) +3'c. (P2) (Existence of an additive a + 0 = O + a = a.
identity) ‘ ' (P3) (Existence of additive invcises 33 a t {~03 = (—tt) ut a =lt}. {P4} (Commutative law for addition): rt + b m I: + ct. % (P5) {Associative law for multiplica— a  (h c) = {a  b) . c.
tiou) (P6) (Existence of a muilipiicative n  i = l a = u: l ¢ 0.
identity) (i’?) (Existence of multiphcalive a .9"! w it” iCt m 1. for (1 ¢ ti.
inverses) (1‘8) (Commutative Law for muiti a J; :1; a,
plication) (P9) (Distributive iaw) a  (b + c') = (i  l) +(t c. The three basic properties of numbers which remain to he iistctl are concerned
with inequalities. Aitltough inequalities occur rarely in elementary mathematics,
they play a prominent role in calculus. The two notions of inequality; (2 «r. I)
(a is 3359 than 1;) and (t > b {a is greater than b}, are intimately related: (I < b
means the same as b > a (thus 1 e: 3 and 3 > 1 art: mereiy two ways of writing
the same assertion). The numbers a satisfying a :> 0 are called positive, while
those numbers ri' satisfying a < 0 are caiied negative. While positivity can thus
he deﬁned in terms of <, it is possible to reverse the procedure: a < b can he
defined to mean that b w a is positive. In fact, it is convenient to consider the
colicction of ail positive numhcrs, denoted by P, as the basic concept, and state
aii properties in terms of P: (P10) (Trichotoniy law) For every number a, one and only one of the
foilowing hoids: ti) a m 0,
(it) ﬂ is in the coliection P,
(iii) (z is in the collection P. (P1 i) (Closure under addition) 11” a and b are in P, then (1 +1) is in P. (F12) (Closure under muitiplication) If a and b are in P, then a . l: is
in P. Hologue These three properties should he Complententecl wiih the following deﬁnitions: a>b if a—bisinP;
a<b if h>a; 33b if a>boramb;
:95}; if a<b0ra=b.* Note, in parricular, that a > 0 if and only if a is in P. All the familiar facts about inequalities, however elementaty they may seem, are
consequences of i’lU—PlZ. lFor example, if a and l: are any two numbers, then
precisely one of the following holds: {5} a — b = 0,
(ii) a — b is in the collection F,
(iii) wk: ~— b) = b m a is in the collection P. Using the deﬁnitionsjust made, it follows that precisely one oft...
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