1 Numbers. 1.1 Prologue

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1 1 Numers 1.1 Prologue Aout 500 yers go, group of philosophers nd students gthered nd founded society to seek Truth nd Knowledge ner the modern town of Crotone, on the cost of southern Itly. They clled themselves mthemtikoi, fter the Greek word μάθημα (mthem), which mens lerning, study, science. The mthemtikoi nd their disciples were Greek immigrnts to the Itlin peninsul; their little colony ws prt of Mgn Greci, ptchwork of Greek colonies on the Itlin cost. The nme of their leder ws Pythgors. The mthemtikoi were secretive unch; we don t know much out their lives nd customs. But some of their discoveries hve survived over the millenni, kept live y people who pprecited the euty nd utility of their work. Now it is our turn. The Pythgorens elieved numers hve divine, lmost mysticl qulities. In our more prcticl time, we think numer is something you punch into clcultor. Mthemticlly speking, wht is numer? For instnce, wht do you sy if someone sks you, Wht is 5? There re mny possile nswers. Five is how mny fingers I hve on one hnd. Five is wht follows four. And five represents fiveness, the strct property tht ll sets of five ojects hve in common. The lst definition seems to e circulr: How cn we tell tht two sets hve the sme numer of elements without counting? And to count, you lredy need to know one, two, three, four, five, etc. However, it turns out tht there is simple method of compring the numers of elements in two sets without counting. If we cn estlish one-to-one correspondence etween the two sets, such tht ech element of the first set is pired up with exctly one element of the second set nd vice-vers, then we cn e sure the two sets hve the sme numer of elements. For exmple, if you hve severl coins on the tle nd you cn touch different coin with ech finger of one hnd, nd no coins remin untouched, then you know there re s mny coins s there re fingers on your hnd. Another exmple: suppose there re severl students nd severl ckpcks on the plyground. Ask ech student to pick up one ckpck. If there re not enough ckpcks for ll the students, the set of students hs more elements, nd if there re some ckpcks left on the ground, the set of ckpcks hs more elements. If every student gets one ckpck, nd there re no ckpcks left, then the numer of students nd the numer of ckpcks re the sme.

2 CHAPTER 1 ~ NUMBERS It would e nice to crete one specil set of five different ojects, so tht we could compre other sets to it. Suppose the world is populted only y sets, s in pure set theory. How cn we devise five different ojects in such world? Tke the empty cll it 1. Now tke the set cll it 0. Now tke set of one element: { } set tht contins 0 nd 1 s elements: {, { }} cll it. (We cnnot just tke {, } ecuse sets re not llowed to hve duplicte elements.), { },{, { }} is 3. And so on. An interesting wy to get something out of { } nothing! 1. Positionl Numer Systems Once we hve figured out, more or less, wht whole numer is, we need to decide how to represent different numers. Look t 34, for exmple. This comintion of two digits represents crucil invention of humn civiliztion, positionl numer system. Without it we would still e writing 34 s XXXIV or something. Wheres our system mkes it esy to see: 34 is three times ten, plus 4. Ten is mgic numer, the se of our positionl system. We proly picked ten simply ecuse humns hve ten fingers. If you need to tell strnger who doesn t spek your lnguge tht you re willing to swp your ipod for 34 pples, it is convenient to gesture three times with oth hnds open nd then once more with four outstretched fingers. So the importnce of ten is universl cross lnguges nd cultures. Our system hs ten digits, 0 through 9; it is clled the deciml or se-10 system. If you hve numer d... n dd1d0, then tht numer is equl to deciml digits n 1 n n d 10 + d d 10 + d 10 + d. For exmple, n 347 = Now imgine wht would hppen if we humns hd only three fingers. Three would ecome our mgic numer! In our deciml system, 10 comes fter 9. But in the se-3 system, 10 comes fter : we hve 0, 1,, 10, 11, 1, then 0, 1,, nd then (1 se 3) would men 3 + 1= 710 (7 se 10). 14 would e written s 11 ( 113 = = 1410), nd insted of 34 we would write ( 101 = = 34 ). 3 10

3 CHAPTER 1 ~ NUMBERS 3 The se-3 system would hve its dvntges. There would e fewer digits to lern, nd it would e esier to lern how to count: one, two, ten, eleven, twelve, twenty, nd so on. And you could live to celerte your 100th irthdy (our ge 9), 1000th irthdy (our ge 7), nd even 10000th irthdy (our ge 81). On the other hnd, your phone numer, insted of 10 digits, would need 0 or 1 hrd to rememer. There re mny pplets (little progrms emedded into we pges) on the Internet tht illustrte inry nd hex representtion of numers nd let you ply with numer conversions. We cn do rithmetic on numers written in ny se in the sme mnner s we do rithmetic in se 10. Exmple 1 Clculte Solution Exmple Clculte (= 3 10 ) = 10 3 write 0, crry (= 10 ) = 3 write, crry (= 3 10 ) = 10 3 write 0, crry (= 3 10 ) = 10 3 write 0, crry 1. Write 1.

4 4 CHAPTER 1 ~ NUMBERS Solution (= 6 10 ) = 11 5 write 1, crry (= 5 10 ) = 10 5 write 0, crry (= 9 10 ) = 14 5 write 4, crry (= 3 10 ) = 3 5 write 3. The next step, of course, is to pretend tht we don t hve ny fingers t ll, just two hnds. Then we would hve only two digits, 0 nd 1. This se- system is clled the inry system. In the inry system, non-negtive whole numers re represented like this: Bse 10: Binry: It is very convenient to use inry representtion in computers, ecuse ech it in computer s memory cn represent only two vlues, 0 nd 1. In some sense the inry system my e more nturl thn our se-10 system, ecuse is the smllest possile se, while 10 is n ritrry se. However, inry numers re hrd to work with for humns, ecuse even smll numers hve too mny digits. For exmple, = Computer progrmmers use compromise: system tht is esy to convert to inry yet esy to red for humns (t lest for progrmmers). There re two such systems: octl nd hexdeciml. Nowdys, the hexdeciml system is more populr. It uses se 16. The 16 digits in the hexdeciml system re denoted y 0, 1,, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, nd F. The digit A stnds for 10, B for 11, etc.; F stnds for 15. Ech of these digits cn e represented y four-digit inry numer, s follows:

5 CHAPTER 1 ~ NUMBERS A 1010 B 1011 C 1100 D 1101 E 1110 F 1111 To convert hex (hexdeciml) numer into inry, just replce ech hex digit with its 4-digit inry representtion. To convert inry numer to hex, split the string of inry digits into quds, strting from the right, nd replce ech qud with its hex equivlent. Exmple 3 B6F8 16 = = 76EA 16 Experienced computer progrmmers rememer the it ptterns for ll 16 hex digits nd cn do these conversions very quickly. To convert inry numer into se 10 mnully, it is fster to first convert it into hex, then hex into se 10. Exmple = hex 1D = = 466 The sme is true for converting deciml into inry: first convert the numer into hex, then hex into inry. (Figure 1-1) gives n exmple of inry rithmetic. To dd two inry numers, write them one underneth the other with the unit digits ligned. Add the unit digits.

6 6 CHAPTER 1 ~ NUMBERS If the result is, sutrct nd set the crry it to 1. Write the result. Add the s digits nd the crry it. If the result is or 3, sutrct nd set the crry it to 1. Write the result. And so on Crry it Figure 1-1. Binry ddition Computer processors perform these opertions t lightning speed. Exercises 1. Look up Pythgors s iogrphy on the Internet. Where did he study mthemtics?. Find ll two-digit numers (se 10) such tht the numer is equl to twice the sum of its digits. Find ll two-digit numers such tht the numer is equl to 7 times the sum of its digits. 3. Tke ny numer nd sutrct from it the sum of ll its digits (se 10). Show tht tht difference is lwys evenly divisile y In Sudoku puzzle you need to fill 9-y-9 grid with numers in such wy tht ech row, ech column, nd ech of the nine 3-y-3 squres contins ll the numers from 1 to 9. Tke ny completed Sudoku grid nd comine the first three columns into one column of 3-digit numers. Do the sme for the next three columns nd the lst three columns. Show tht the sum of the numers in ll three new columns is the sme. Wht is the vlue of tht sum? 5. Write nd 4 10 in se Write 11 3 nd 00 3 in se Write in se 10.

7 CHAPTER 1 ~ NUMBERS 7 8. Write 43 5 in se Clculte nd without converting the numers into deciml (se-10) representtion. 10. How cn you quickly tell whether numer written in se 3 is even (evenly divisile y )? Hint: see Question Tke 3-y-3 mgic squre: It contins ll the numers from 1 to 9, nd the sum of the numers in ech row, ech column, nd ech of the two digonls is the sme. Sutrct 1 from ech numer. Convert ech numer into two-digit se-3 numer (dding leding zero where necessry). In the resulting grid, ech row hs 0, 1, s first digits nd s second digits, in some order. The sme for columns. All nine comintions of two digits re different. Such n rrngement is known s Greco-Romn squre. (The originl design used the Greek letters α, β, γ insted of 0, 1, s the first digits nd the Ltin letters A, B, C s second digits. Ech row nd ech column would hve ll three Greek letters nd ll three Ltin letters, nd ll nine comintions would e different.) Construct 5-y-5 Greco-Romn squre nd then 5-y-5 mgic squre. 1. Wht is the lrgest integer tht hs 7 digits in its inry representtion? How out 15 digits in its inry representtion? 13. Convert hex 90AB into inry. 14. Convert inry into hex. 15. Wht re the results of the following opertions, written in se-? () ()

8 8 CHAPTER 1 ~ NUMBERS 16. Wht re the results of the following opertions, written in hex? () () Rtionl nd Irrtionl Numers Pythgorens nd philosophers efore them clerly understood positive integers nd frctions, nd how numers cn e used to mesure lengths. If you estlish unit length ( yrdstick ), you cn mesure pproximtely the length of stright line segment in yrds. If you mrk smller equl segments (like inches) on your yrdstick, you cn mesure lengths more ccurtely, in yrds nd inches. The Pythgorens were not mesuring nything, of course; they were interested in the theory. They knew how to divide given segment into n equl prts, using prllel lines: 1 n Theoreticlly, you cn otin segment of length 1, s smll s you wish, for ny n n, so you cn mesure the length of ny segment with ny desired precision. Pythgors elieved tht the length of ny segment cn e expressed s rtionl numer, rtio of two whole numers. It ws just mtter of time efore one of his students sked: Wht is the length of the digonl of squre whose side is 1? Wht rtio of whole numers represents this length? (Legend hs it tht the student didn t end well: fellow Pythgorens drowned him in fountin for this heresy. ) If this numer is d, we must hve d = =, ccording to Pythgors s own theorem. In our modern nottion, d is written s. After futile ttempts to show tht d = is rtionl numer, one of the students proly proved tht it is not! His or her (the Society dmitted women on equl terms) proof most likely ws geometric. It proly used the method of proof clled proof y contrdiction, in which we ssume tht the opposite to wht we need to prove is true nd then show tht it leds to nonsense. Something like this. Suppose the side of squre is 1 = mx nd the digonl d is nx, where m nd n re integers nd x is smll segment, common mesure of 1 nd d. The rtio of the length of the digonl to the length

9 CHAPTER 1 ~ NUMBERS 9 digonl d nx n of the side would e = = =. This rtio is the sme for ny squre. side 1 mx m n Let s ssume tht m nd n re the smllest possile integers such tht d =. m However, we cn construct smller squre (Figure 1-) with the side ( n m) x, nd the digonl ( m n) x. O m n n-m n-m m-n m-n m n n = n m m Figure 1-. A geometric proof of irrtionlity of. m n In this squre, d =. ( m n) nd ( n m) re integers tht re smller thn n m n nd m, respectively. But we chose the smllest m nd n. So our hypothesis tht m nd n exist ws wrong in the first plce. Here is more modern, lgeric proof lso proof y contrdiction. Suppose there exists rtionl numer n m such tht n n = =. Let s reduce this m m frction s fr s possile until m nd n hve no common fctors. n = m, so n must e n even numer. (If n were odd, n would e odd, too.) Therefore, n= k, for some positive integer k. Then n = 4k 4k = m k = m. So m must e even, too. So oth m nd n re even, nd is their common fctor. This contrdicts our ssumption tht m nd n hve no common fctors. We hve to conclude tht is not rtionl numer.

10 10 CHAPTER 1 ~ NUMBERS The discovery of irrtionl numers eventully rought us to where we re in mthemtics. You might e wondering: Which re there more of, rtionl numers or irrtionl numers? Does this question even mke sense? After ll, there is n infinite numer of rtionl numers nd n infinite numer of irrtionl numers (see Question <...> in the exercises). How cn one infinity e igger or smller thn nother? It turns out there is wy to compre. The theory ws developed y Georg Cntor ( ), Germn mthemticin orn in Russi. In Cntor s set theory, infinite sets cn e compred in mnner similr to the wy we compred finite sets in the prologue to this chpter. Given two sets, A nd B, we cn try to estlish one-to-one correspondence etween the elements of A nd B. If such correspondence exists, we conclude tht the sets hve n equl numer of elements, or, s mthemticins sy, hve the sme crdinlity. For exmple, the set of positive integers hs the sme crdinlity s the set of negtive integers, ecuse we cn estlish one-to-one correspondence etween these sets: for exmple, n corresponds to -n. But then strnge things egin to hppen. We re used to the fct tht if we tke finite set nd choose some (ut not ll) of its elements, the resulting suset hs fewer elements thn the whole set. Not so for infinite sets! An infinite set cn hve the sme crdinlity s its smller suset. For exmple, it is esy to estlish one-toone correspondence etween the set of ll positive integers nd its suset of ll even positive integers (see Question <...> in the exercises). Therefore, these sets hve the sme crdinlity. Cntor identified the whole clss of sets tht hve the sme crdinlity s the set of positive integers. He clled such sets countle, ecuse estlishing one-to-one correspondence etween the elements of A nd the set of positive integers is nlogous to counting (or enumerting) the elements of A: we cn rrnge ll the elements of countle set into one infinite sequence {,,...,...}. 1 n If one-to-one correspondence etween sets A nd B does not exist, we cn try to estlish one-to-one correspondence etween A nd some suset of B. If such correspondence exists, we conclude tht B hs more elements thn A, tht is, B hs greter crdinlity thn A. This wy we cn compre crdinlities. Cntor proved tht there re infinities of different orders of mgnitude, tht is, different crdinlities (see Question <...> in the exercises). The crdinlity of countle sets is the smllest possile crdinlity, ecuse every infinite set hs countle suset. Eventully Cntor proved tht the set of rtionl numers is countle (see

11 CHAPTER 1 ~ NUMBERS 11 Question <...> in the exercises) nd, little lter, tht the set of irrtionl numers is not countle. So there re more irrtionl numers thn rtionl numers. Exercises 1. Prove tht 3 is irrtionl.. Show tht 1+ is irrtionl. Hint: multiply this numer y itself. 3. Show tht + 3 is irrtionl. 4. Is the sum of two rtionl numers lwys rtionl numer? 5. Is the verge of two rtionl numers lwys rtionl numer? 6. Is the sum of two irrtionl numers lwys n irrtionl numer? 7. Wht cn you sy out the sum of rtionl numer nd n irrtionl numer? As we hve seen (Question ), it cn e irrtionl. Is it lwys irrtionl? Justify your nswer. 8. Show tht there re no significnt gps in rtionl numers, tht is, ny intervl, no mtter how smll, contins t lest one rtionl numer. 9. Look up Georg Cntor s iogrphy nd portrit on the Internet. Write down few fcts out his life nd work. 10. A 10-inch squre cke hs 30 risins in it. Prove tht you cn cut out piece of the cke tht fits inside 6-inch round plte nd hs t lest two risins in it. 11. Give n exmple of n infinite countle set of irrtionl numers. 1. Explin why there re t lest s mny irrtionl numers s rtionl numers. 13. Show tht the set of ll integers, including positive, negtive, nd zero, is countle. Hint: find wy to rrnge ll integers into one infinite sequence.

12 1 CHAPTER 1 ~ NUMBERS 14. Show tht the set of ll frctions with positive numertor nd denomintor is countle. 15. Consider the set of ll infinite sequences of 0s nd 1s. Suppose this set is countle. Then we cn rrnge ll such sequences in n infinite list, ligned one elow the other. The result will e n infinite tle of 0s nd 1s, something like this: The digits on the digonl of the tle form sequence of 0s nd 1s. Becuse our list contins ll possile sequences, this digonl sequence is somewhere in our list. Wht out its complement sequence, in which ll 0s re replced with 1s nd 1s with 0s? Cn it e in our list? Why or why not? Wht does this construction prove? 1.4 The Numer Line If numers re used to mesure distnces, it is only nturl to represent them s points on stright line tht extends to infinity in oth directions: - -1 This model is clled the numer line By convention the numer line is drwn horizontlly, with n rrow on the right tht indictes the positive direction. One specil point on the numer line, clled the origin, represents 0. The points to the right of 0 represent positive numers, nd the points to the left of 0 represent negtive numers. The segment from 0 to 1 represents the unit length. The segment etween ny two consecutive integers hs the sme unit length.

13 CHAPTER 1 ~ NUMBERS 13 Negtive numers secured their plce in mthemtics reltively lte. They were known in one form or nother since ntiquity, ut for long time mthemticins viewed them with suspicion. As opposed to nturl numers (positive integers * ), mthemticins regrded negtive numers s, well..., unnturl. But if you consider 1 s step in the right direction, it mkes sense to hve nme for step in the opposite direction: negtive 1. One positive step nd one negtive step cncel ech other, nd you end up where you strted. Numers represented y ll points on the numer line zero, positive nd negtive, rtionl nd irrtionl re clled rel numers. The letter (doule-lined R) or simply R is trditionlly used to represent the set of ll rel numers. Are rel numers relly rel? This is good philosophicl question. To egin with, there is no perfectly stright infinite line in the oservle world. Our numer line is n strction. There is no tool tht would llow us to mesure the length of the digonl of squre precisely, so tht length is n strction, too (nd there is no perfectly squre squre to egin with tht s n strction s well). Mthemtics dels with strctions. There is no clcultor or computer in the world tht cn give us the vlue of exctly, ecuse it would need n infinite sequence of digits to disply tht numer. We lwys settle for some pproximtion. Theoreticlly, we cn get s close to s we wnt, for exmple, But we cnnot get hold of the exct numer, in ny rel sense, eyond sying tht is numer such tht ( ) =. But suppose we did hve n idel unit squre, nd we could mesure the length of the digonl with perfect precision, nd we hd n idel compss, which could tke tht mesurement, nd we could plce the idel compss s leg onto the idel point 0 on the numer line nd drw n idel circle * Some uthors cll {1,, 3,...} nturl numers. This is the system we use. Others include 0 nd cll {0, 1,, 3,...} nturl numers. Some people cll ll integers, zero, positive, nd negtive whole numers. Others cll only {0, 1,, 3,...} whole numers. We will cll ll integers, zero, positive, nd negtive, integers.

14 14 CHAPTER 1 ~ NUMBERS suppose ll tht were possile. Still, how do we know tht the circle would intersect our numer line in two points, nd? Wht if our numer line hs holes in those plces? How do we know tht our numer line is continuous nd connected? We don t. We postulte this. We wnt to work with rel numers ecuse this leds to eutiful nd useful mthemtics, so we postulte tht they exist! The construction of rel numers does not strt from scrtch: we hve lredy done lot of work. We hve set of nturl numers, = {1,, 3,...}, with ddition nd multipliction. We hve dded to them zero nd negtive numers nd uilt integers, = {..., 1, 0,1,,...}. So we re now le to do sutrction. We hve defined the set of rtionl numers (often denoted ), s frctions with integer numertors nd denomintors. We know how to dd, sutrct, multiply, nd divide frctions (t lest we should know how...). We know tht these opertions on rtionl numers oey some simple lws (Figure 1-3). Given ny two rtionl numers, we cn lso tell whether one is greter thn the other. This reltion of totl order hs some nice properties (Figure 1-4). (It is clled totl order ecuse for ny two numers we cn tell which one is greter, s opposed to prtil order, in which only certin pirs of elements cn e compred, not necessrily ll.) Commuttive lw for ddition: + = + Associtive lw for ddition: ( + ) + c = +( + c) Commuttive lw for multipliction: = Associtive lw for multipliction: ( ) c = ( c) Distriutive lw: (+c) = + c Figure 1-3. The lws of rithmetic Reflexivity: Antisymmetry: Trnsitivity: if nd then = if nd c then c Figure 1-4. The properties of n order reltion

15 CHAPTER 1 ~ NUMBERS 15 Moreover, for rtionl numers, rithmetic is consistent with the ordering: if nd c d then + c + d if nd k 0 then k k Exmple 1 Wht is greter, 3 7 or 3 8? Solution We know tht if > then 1 < 1 (when we divide y lrger numer we get smller frction). In this exmple, the numertors re the sme nd 7 < 8, so 3 > Exmple Wht is greter, 3 5 or 4 7? Solution The lest common denomintor of these frctions is = 1 ; 4 = 0. 1 > 0, so 3 > 4. Or, equivlently, we cn just cross-multiply (ecuse oth denomintors 5 7 re positive): 37 > 45, so 3 > So rtionl numers hve ll these nice properties. We cn lso use rtionl numers to mesure ny quntity to ny desired precision. Wht else do we wnt? Why re we not stisfied? We wnt to get rid of the holes. Rtionl numers hve lots of holes etween them. For exmple, is not rtionl numer. The eqution

16 16 CHAPTER 1 ~ NUMBERS x = hs no rtionl solutions. We wnt to e le to work with such equtions. We wnt to hve set of numers tht hs ll the nice properties of rtionl numers, nd no holes. The notion of continuity of spce nd time is deeply ingrined in our intuition. Are we orn with it or do we get it from studying mthemtics nd physics? There re severl wys of constructing set of rel numers. Different constructions re equivlent nd led to the sme result. Here we will descrie the construction sed on sequences of nested closed intervls. Given two numers, nd, where >, we cn consider set of ll numers x such tht x. This set is clled closed intervl from to nd denoted [, ]. Now suppose 1 1. Then [, ] is inside [ 1, 1] : 1 1 We sy tht the closed intervl [, ] is nested in [ 1, 1]. Now let s tke [ 3, 3] nested in [, ]. And so on we get n infinite sequence of closed intervls in which ech next intervl is nested in the previous one: The postulte of the completeness of rel numers implies tht ny sequence of nested closed intervls hs non-empty intersection, tht is, there is t lest one rel numer tht elongs to every intervl in the sequence. This is just one of the equivlent forms of the completeness postulte. If the lengths of the nested intervls ecome smller nd smller nd pproch zero, then the intersection of the intervls must e exctly one numer, exctly one point on the numer line.

17 CHAPTER 1 ~ NUMBERS 17 Formlly we define which sequences of nested closed intervls with rtionl ends re equivlent to ech other (hve the sme intersection ). We cll the whole clss of equivlent sequences rel numer. We then define rithmetic opertions on the clsses of equivlent sequences tht extend opertions on rtionl numers. Exercises 1. Mrk [-3, ] on the numer line.. Which of the following frctions is greter, 7 5 or 10 7? 3. Using the lws of rithmetic nd the definition of, show tht = We know tht if nd k 0 then k k. Using this fct nd the trnsitivity property of the order reltion, prove lgericlly tht if 0 nd 0 c d then c d. 5. Recll tht rectngle with sides of lengths x nd y hs n re of xy nd demonstrte geometriclly the result in Question Give n exmple of numers such tht nd c d ut c > d. 7. Is it true tht 7 < < 10? Answer this question without clcultor, using 5 7 only rithmetic for integers nd frctions nd the definition of. 8. Which frction is closer to, 7 5 or 10? Answer this question without 7 using clcultor. Hint: squre the differences, collect the like terms on different sides, then squre gin. 9. Suppose you hve sequence of nested closed intervls [ n, n] tht lie to the right of 0 nd intersect t x > 0. How would you construct sequence of nested closed intervls tht intersect t 1 x?

18 18 CHAPTER 1 ~ NUMBERS 10. A set of numers x such tht < x< is clled n open intervl from to nd is denoted (, ). Give n exmple of sequence of nested open intervls whose intersection is empty. 11. We know tht flls somewhere etween 1 nd, tht is, elongs to [1, ]. Let s tke tht intervl nd divide it into 10 equl prts. Wht is their length? will fll into one of these suintervls. If we count suintervls from left to right, strting from 0, wht is the numer of the suintervl tht contins? Wht re its endpoints, expressed s simple frctions? 1. In Question 11 we divided [1, ] into 10 suintervls of equl lengths nd determined which one of these 10 suintervls contins. Now let s tke tht suintervl nd divide it into 10 equl prts. will fll into one of these smller suintervls. Let s divide it into 10 equl prts. And so on... If we continue this process, we get n infinite sequence of nested closed intervls. If [1, ] is the first intervl in the sequence, wht is the length of the fourth intervl? n-th intervl? Wht is the intersection of this sequence of nested closed intervls? 13. In Question 1 we defined sequence of nested closed intervls round. Now get the pproximte vlue of on your clcultor. Wht re the ends of the fourth intervl, expressed s simple frctions? Wht is the numer of the fourth intervl in the sequence within the third intervl, if we count from left to right, strting from 0? 14. In Questions we divided [1, ] into 10 equl prts, then gin into 10 equl prts, nd so on, nd otined the digits of fter the deciml point. If insted of 10 we lwys divided the intervl in hlf, we would otin inry digits. Wht re the first three inry digits of fter the inry point? 1.5 Regions on the Numer Line The distnce from x to 0 on the numer line is clled the solute vlue of x nd denoted x. When x 0, x = x; when x < 0, x = x. The solute vlue of 0 is 0; the solute vlue of ny other numer is positive. The numers x nd x re symmetricl with respect to 0

19 CHAPTER 1 ~ NUMBERS 19 x x so x = x. -x 0 x x represents the distnce from x to. Exmple = 4.5; =. Exmple = = is positive ecuse 5 7 >. 5 For ny rel numers x nd y, xy = x y. In prticulr, for ny x, positive or negtive, negtive (y the definition of the x = x only when x 0. In generl, for ny rel numer x, x x opertion). So, = x. = x. x, if it is defined, is never Exmple 3 Show, without clcultor, tht 10 <

20 0 CHAPTER 1 ~ NUMBERS Solution Recll tht for ny rel numers nd, = ( )( + ). You cn verify this formul y opening the prentheses nd using the distriutive lw: nd cncel out, nd remins. So if we multiply 10 7 y 10 + we will get 7 rid of. This trick is clled multiplying y the conjugte expression = ( ) = = = Therefore, + =. + >, so < = < = Q.E.D. To e honest, we just wnted you to do little lger review... Recll from the previous section tht the set of numers x such tht x is clled closed intervl from to nd denoted [, ]. An open intervl is set of numers x such tht < x< ; it is denoted (, ). < x or x< re clled hlf-open (or, interchngely, hlf-closed) intervls. < x is denoted (, ]; x< is denoted [, ). Figure 1-5 summrizes the nottion for closed, open, nd hlf-open intervls nd shows conventionl wy to mrk them on the numer line. x Closed intervl [, ] < x < Open intervl (, ) x < Hlf-open intervl [, ) [ ( [ ] ) ) < x Hlf-open intervl (, ] ( ] Figure 1-5. Intervls on the numer line

21 CHAPTER 1 ~ NUMBERS 1 Some textooks use smll solid circles to mrk included endpoints nd hollow circles to mrk excluded endpoints of n intervl, insted of squre rckets nd prentheses, respectively: Exmple 4 Mrk on the numer line the set of ll x such tht x Solution These re ll the points on the numer line whose distnce from 0.5 does not exceed 1.5: - Exmple 5 [ -1 ] Mrk on the numer line the set of ll x such tht 1 1 x. Solution Since 1 x is positive, x must e positive, too. Since 1 1, we must hve 1 x x (when you flip over positive frctions, the inequlity chnges direction). Comining x > 0 nd x 1, we get (0, 1]. On the numer line this cn e shown s: -1 ( ] 0 1

22 CHAPTER 1 ~ NUMBERS So fr we hve tlked out finite (or ounded) intervls. A set of numers x such tht x is lso clled n intervl, ut this intervl is infinite (or unounded). It is denoted: [, ). The symol is clled infinity; it is not numer, just nottion for the direction to the right on the numer line, so is never included in ny intervl. Still, such n intervl is clled closed, ecuse it includes its only endpoint. Similrly, the set of numers x is denoted (, ]. Agin, is just nottion for the direction to the left on the numer line. x> nd x< re infinite open intervls. They re denoted (, ) nd (, ), respectively. Infinite intervls re summrized in Figure 1-6. Exmple 5 x Closed intervl [, ) x > Open intervl (, ) x Closed intervl (-, ] x < Open intervl (-, ) (-, ) [ ( Figure 1-6. Infinite intervls involve or Mrk on the numer line the set of ll numers x such tht x 1 >. Solution This region includes ll x whose distnce from 1 is greter thn. The region consists of two infinite intervls: x < 1 nd x > 3, or (, 1) nd (3, ) : ] ) - ) -1 (

23 CHAPTER 1 ~ NUMBERS 3 In the ove exmple, the set of solutions consists of two intervls. It is etter to write the nswer using their union. You re proly fmilir with the concept of the union of two sets. Given two sets, A nd B, the set of ll elements tht elong to either A or B (or to oth) is clled the union of A nd B nd denoted A B. For exmple, the union of A = { 1,, 3} nd B = {, 4, 7} is A B { 1,, 3, 4, 7} =. So, the solution of the inequlity x 1 > cn e written s (, 1) (3, ). This form of writing n nswer tht involves multiple intervls is much etter thn sying x > 3 nd x < 1. The ltter is misleding: it sounds like we wnt oth conditions to e met simultneously. When you tke the union of two sets, the order doesn t mtter: A B= B A. But it is customry to list the intervls connected y from left to right. Tht is why we wrote (, 1) (3, ) nd not (3, ) (, 1). You re proly lso fmilir with the concept of the intersection of two sets. Given two sets, A nd B, the set of ll elements tht elong to oth A nd B is clled the intersection of A nd B nd denoted A B. For exmple, the intersection of A = { 1,, 3, 4} nd B = {, 4, 7} is A B {, 4} =. Luckily, the intersection of two intervls is lwys either empty or single point or n intervl (see Question <...> in the exercises). Tht is why we don t see in nswers. A set tht contins single element x is denoted {x}. A set with severl elements, x, y, z, is denoted {x, y, z}. A single numer is mrked on the numer line s smll filled circle:

24 4 CHAPTER 1 ~ NUMBERS Exmple 6 Solve the inequlity line. 4 ( x 3) ( x ) ( x 1) 0 + nd mrk its solution on the numer Solution To solve n inequlity with one vrile x mens to descrie the set of ll x tht stisfy the inequlity. The solution set is usully descried s union of intervls 4 nd/or individul numers. In the given expression, ( x 3) ( x+ ) = 0 when x = 3 4 or x =. For ll other x, ( x 3) ( x+ ) > 0, ecuse the exponents nd 4 re 4 even numers, nd ( x 3) ( x+ ) = ( x 3)( x+ ). We cn divide oth sides of the inequlity y this positive fctor (recll tht for positive k, k k if nd only if ). Wht remins is ( x 1) 0. To the two vlues of x tht we hve lredy found, x = 3 nd x =, we need to dd ll the solutions of ( x 1) 0, which is x 1, tht is, [1, ). Note tht x = 3 elongs to this intervl, so there is no need to list it seprtely. The nswer is { } [1, ). ( Exercises In Questions 1-7 mrk the given intervl on the numer line. 1. [-,.5]. (0, 4) 3. (-1, 3] 4. [1, ) 5. [-1.5, )

25 CHAPTER 1 ~ NUMBERS 5 6. (-, -1] 7. (-, ) In Questions 8-15 mrk the given set of numers on the numer line. 8. {3} 9. {-, 3} 10. {1} [, 3] 11. (1.5,.5] {3, 3.5} 1. (, 1] [1, ) 13. (, 1] {0} [1, ) 14. (, 1) ( 1, ) 15. {1,, 3, 5, 8, 13, 1} In Questions 16-4 solve the given inequlity. Express the nswer s union of intervls nd sets of discrete numers. Mrk the nswer on the numer line. 16. x x x 4 x 0. x 3 < 1 1. x + 3. x ( x+ 1) 0

26 6 CHAPTER 1 ~ NUMBERS 3. 3 x ( x+ 1) ( x+ ) 0 4. x 1 + x Show tht the intersection of ny two intervls on the numer line is either empty, or one isolted point, or n intervl. 6. Which numer corresponds to the midpoint of [, ]? 7. Is it true tht for ny open intervl (, ) nd ny point x in it there exists n open intervl, centered t x, which lies inside (, )? In other words, is it true tht for ny open intervl (, ) nd ny point x in it there exists positive numer δ (perhps very smll, ut positive) such tht ( x δ, x+ δ ) lies inside (, )? If not, give n exmple of such,, nd x for which it is not true. If yes, explin how you cn find δ for ny given,, nd x. Note: the Greek letter δ (delt) is often used in mthemtics to represent the rdius of smll intervl or circle. 8. Prove tht ny closed intervl [, ] is complete spce, tht is, the intersection of ny nested sequence of closed intervls, ll of which lie within [, ], is not empty. 9. If A is set of rel numers, then the set of ll rel numers tht do not elong to A is clled its complement nd denoted A or A. Which of the following sttements re true for ny set? () A = A () A A = (c) A A= (empty set). Justify your nswers. 30. Mrk on the numer line [ 1, 0) (1, ] nd its complement. Now mrk on the numer line [ 1, 0), (1, ] nd their intersection. Is it true tht for ny two sets A nd B A B= A B? Justify your nswer. 31. Does n infinite intervl hve more points thn finite intervl? Give n exmple of one-to-one correspondence etween (0, 1) nd (1, ). 3. Give n exmple of one-to-one correspondence etween ( 1, 1) nd (, ). Hint: see Question 31.

27 1.6 Review Concepts, terms, methods, nd formuls introduced in this chpter: CHAPTER 1 ~ NUMBERS 7 One-to-one correspondence Positionl numer system Bse of numer system Binry system Hexdeciml system Rtionl numer Irrtionl numer Proof y contrdiction Crdinlity of set The numer line Rel numer, the set of ll rel numers Completeness of the numer line Lws of rithmetic: Commuttive lw for ddition: + = + Commuttive lw for multipliction: = Distriutive lw: (+c) = + c Totl order reltion Reflexivity: Antisymmetry: if nd then = Trnsitivity: if nd c then c If nd c d then + c + d If nd k 0 then k k x solute vlue of x x = x 0 xy = x y x Closed intervl [, ] < x < Open intervl (, ) x < Hlf-open intervl [, ) < x Hlf-open intervl (, ] Finite (ounded) nd infinite (unounded) intervl [ ( [ ( ] ) ) ]

28 8 CHAPTER 1 ~ NUMBERS x Closed intervl [, ) x > Open intervl (, ) x Closed intervl (-, ] x < Open intervl (-, ) (-, ) [ ( ] ) = ( )( + ) A B union of sets A nd B A B intersection of sets A nd B {x} set tht consists of only one element x {x, y, z} set with elements x, y, z

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