6. Numbers. The line of numbers: Important subsets of IR:

Size: px

Start display at page:

Download "6. Numbers. The line of numbers: Important subsets of IR:"

Garey McLaughlin
5 years ago
Views:

1 6. Nubes We do not give n xiotic definition of the el nubes hee. Intuitive ening: Ech point on the (infinite) line of nubes coesponds to el nube, i.e., n eleent of IR. The line of nubes: Ipotnt subsets of IR: IN the set of ll ntul nubes (positive integes), does not contin the IN : IN {} the set of ll non-negtive integes Z the set of ll integes {... ; ; ; ; ;... } Q I the set of ll tionl nubes (epesentble s fctions of integes p/q, whee q ) We hve: IN Z Q IR. I Rek: Evey tionl nube cn be epesented s decil nube with its expnsion fte the decil dot eithe coing to n end o becoing peiodic. 7

2 Exples: /4.5 / (peiodic) /6.6 (ultitely peiodic) Exple fo tnsfotion in the othe diection: (note the diffeent nottions: decil dot in nglosxon counties, co in Geny) Itionl nubes e el nubes tht e not tionl, i.e., cnnot be expessed s fction of integes. Thei decil expnsion becoes neve peiodic. Exples: Aithetic opetions on IR: Addition Opetion sybol: b exists fo evey, b IR. cn be seen s function with two guents: b is in pefix nottion (, b). 8

3 Rules fo dding nubes: b b (couttivity) ( b) c (b c) (ssocitivity) ( is the neutl eleent of ddition) Fo evey, thee is nube such tht ( ) We hve lwys: ( ). Subtction cn be deived fo ddition: b ( b). Multipliction Opetion sybol: (often oitted!) (soeties lso * insted of ). b exists fo evey, b IR. Rules fo ultipliction: b b ( b) c (b c) ( is the neutl eleent of ultipliction) Rule cobining ddition nd ultipliction: (b c) b c (distibutivity) Note: By convention, binds stonge thn Fo evey, thee is nube / such tht /. We hve lwys: /(/). Othe nottions fo / :,. 9

4 / is clled the invese of. Division cn be deived fo ultipliction: : b /b Anothe nottion fo : b is b. : b is not defined fo b. The powe of nube A powe with positive intege exponent is defined s n iteted ultipliction: Exple: is clled the bsis, 3 the exponent. By definition, fo ll. Fo n >, we define s the powe with negtive exponent n : n /( n ) ( ( n ) ). Exple: The oot of nube Fo evey positive el nube nd evey positive intege n thee exists positive el nube x which fulfills the eqution x n. This (unique) x is clled the n-th oot of. Two nottions fo x: Fo odd integes n nd negtive we cn extend this definition by /n ( ) /n. 3

5 Fo even n, the n-th oot of negtive nube is not defined in IR. To ovecoe this estiction, it is possible to extend the set of el nubes IR: The so-clled iginy unit i is defined which fulfills i i. IR is extended to the set C I of coplex nubes. Ech coplex nube hs the fo b i with, b IR. It is possible to clculte with coplex nubes in the se wy s with el nubes. Visuliztion s points in the plne (with elvlued coodintes, b). Bck to the el nubes: The opetion "n-th oot of..." does invet the powe opetion. Attention: We hve (by definition) but:! Hee, x denotes the bsolute vlue of x: x x if x nd x x othewise., b : the distnce between nd b. 3

6 In the context of sque oots, the solution foul fo qudtic equtions ("pq foul") is often useful tool: Fo the eqution x px q, the solutions (if they exist) e: Condition fo the existence of the solution(s): Fo contol puposes, Viet's theoe cn be useful: The two solutions fulfill x x p nd x x q. The powe of el nubes with tionl exponent: The powe k/n is defined s. (By using liits of seies of tionl nubes fo the intoduction of liits see lte, the definition of powe cn lso be extended to itionl exponents.) 3

7 Rules fo powes: s s : s s ( ) s s b ( b) Becuse the powe opetion n is not couttive, thee e two diffeent evese opetions: You cn sech fo bsis o you cn sech fo n exponent. The fist cse leds to the oot, the second cse to the logith. Definition: Let, b > be el nubes. The (unique) solution of b x is x logb (logith of to the bse b). Often the so-clled ntul logith is used, which uses the Eule nube e s its bse: ln loge. Othe fequent cses: biny logith (bse ); decil logith (bse ). In genel, we hve: logb ln / ln b. Rules fo logiths (hold fo bity bse): log(x y) log x log y log (x / y) log x log y log (x y ) y log x 33

8 The ode eltion on IR Evey two el nubes, b cn be odeed: Eithe < b, o b, o > b. b ens < b o b. We hve: < b c < b c (nlogously fo ), fo c > : < b c < b c but fo c < : < b c > b c Bounded intevls An open, bounded intevl (, b) is the set of ll el nubes x which e popely between nd b, i.e., which fulfill < x < b. Attention! The se nottion s fo odeed pis is used, but the ening is diffeent. If < b, (, b) is n infinite set. (, b) 34

9 In closed intevl [, b], the end points e included: [, b] { x IR x b }. [, b] An intevl closed on the ight-hnd side: (, b] An intevl closed on the left-hnd side: [, b) Unbounded intevls (, ) { x IR < x }. 35

10 (, ) [, ) nlogously fo intevls unbounded to the left: (, ) (, ] The neighbouhood of nube Let ε > be positive el nube. The intevl ( b ε, b ε ) is clled the ε-neighbouhood of the nube b. We hve ( b ε, b ε ) { x IR x b < ε }. Tht ens: The neighbouhood contins ll nubes fo which the distnce to b is slle thn the given theshold ε. 36

11 Bounds An uppe bound of set M of el nubes is nube with > x fo ll x M. Anlogously: lowe bound (exchnge > by < ). A set of nubes is clled bounded if thee exists n uppe bound nd lowe bound fo it. If set hs n uppe bound, it hs infinitely ny uppe bounds. We e inteested in the sllest one: The sllest uppe bound of set M IR is clled the supeu of M, denoted sup M. Anlogously: The lgest lowe bound of set M IR is clled the infiu of M, denoted inf M. Exples: inf {; ; 3; 4}, sup {; ; 3; 4} 4, 37

12 Nube systes Question: How to epesent nubes? We concentte on positive integes hee. The dditionl digits in the hexdecil syste: A, B, C, D 3, E 4, F 5. Tnsfotion fo one nube syste to the othe: Specil cse (esy): fo biny to hexdecil Evey 4 biny digits coespond diectly to hexdecil digit Exple: C 6 38

13 fo bity syste to decil: Hone schee Input: zn zn... z to bse b stt with hn zn clculte fo k n, n,..., : Output: z h hk hk * b zk Exple: Input: biny nube (n 4, b ) Stt: hn h3 z3 k n 3: h h3 * z * k : h h * z * 5 k : h h * z *5 z fo decil to bity: Invese Hone schee stt with h z ( input) clculte fo k,, 3,... : zk hk od b, hk hk div b (od: est when dividing by b, div: integl pt fo dividing by b) Output: zn zn... z to bse b 39

14 Exple: Input: decil nube 34, tnsfo in teny syste (b 3) Stt: h 34 k : z h od 3 34 od 3, h h div 3 34 div 3 k : z h od 3 od 3, h h div 3 div 3 3 k 3: z h od 3 3 od 3, h3 h div 3 3 div 3, k 4: z3 h3 od 3 od 3, h4 h3 div 3 div 3 (Stop) z Rek: Abity el nubes cn lso be epesented using n bity intege b > s bse. Digits fte the dot e intepeted s coefficients of b n (n,, 3,...). Exple:. (bse b) / /4 /8 7/

15 7. Vectos We will wok with eleents fo the set The eleents e n-tuples of el nubes, we cll the vectos. To distinguish vecto-vlued vibles fo vibles stnding fo single nubes, often n owed lette ( ) o pinting in diffeent font is used. Two wys to wite down vecto: ow vecto, e.g., (; 5; ) colun vecto To distinguish el nubes fo vectos, we cll the lso scls: M n IR n vecto (fo n ; 3 geoeticlly: epesenttion by ow ; diected entity ) IR scl ( undiected entity ) 4

16 3 L,,... e clled coponents of the vecto (lso: coodintes) specil cses:, cn be epesented s plne: ech vecto coesponds to point in the plne. Often vecto is epesented s n ow pointing fo the oigin to this point. 3-diensionl spce. IR n is clled n n-diensionl vecto spce. 4

17 Exple of vecto in highe-diensionl vecto spce IR n (n > 3) : The ge-clss vecto of popultion (e.g., of foest stnd) - yes old 5 h - yes old -3 yes old h 7 h 3-4 yes old 8 h

18 44 Equlity of vectos: Two vectos e equl iff ll thei coesponding coponents e equl. L M M ^ ^ b b b b b n n n b n K^ Addition of vectos: Definition of the su of two vectos in IR n n n n n b b b b b b M M M Popeties of the ddition of vectos: b b couttivity ) ( ) ( c b c b ssocitivity, v neutl eleent whee is the zeo vecto:

19 M IR n Geoeticl intepettion of vecto ddition: The ows of both vectos e plced one fte the othe, nd the oigin is connected with the new end point. (in physics: "pllelog of foces") 45

20 The su in the cse of ge-clss vectos: ggegtion of two foest stnds into one ge-clss stuctue of the totl e 46

21 Fo ll vectos fo IR n, thee exists exctly one vecto which fulfills ( )?. invese (negtive) eleent ( ) Diffeence of vectos: b ( b) (s in the cse of el nubes) b b 47

22 Geoeticl intepettion of the diffeence of vectos: invesion of the diection we get thus the "connecting vecto" of the endpoints of both vectos. 48

23 49 Multipliction of vecto with scl ( inne poduct, vecto poduct!) IR, IR n n n M M : IR n Exple: 6 5) ( geoeticl ening: expnsion, esp. copession of by the fcto 3 The diection is inveted, if the fcto is <.

24 5 We hve the following ules: k k b b v ) ( ) ( ) ( In the following, tes of the fo n i i i k i i k k R R, ), ( K e ipotnt. We spek of line cobintion of the vectos ;,, k i K the i e clled coefficients. Exple (in 3-diensionl spce): (hee witten s ow vectos fo convenience) }distibutive lws

25 The vecto is line cobintion of these fou vectos. In colun-vecto nottion, we clculte: The tivil line cobintion A line cobintion is clled tivil if ll coefficients,..., k e. It is clled nontivil if t lest one coefficient is not. A tivil line cobintion hs the zeo vecto s its esult. Cn the zeo vecto lso be the esult of nontivil line cobintion? An exple: 3 vectos in plne 5

26 We cn indeed constuct "cycle" of ultiples of these vectos which gives s its su the zeo vecto: This is nontivil line cobintion giving the zeo vecto! would be tivil. We sy:, b, c e linely dependent. 5

27 Definition: Line dependence / independence of vectos Given e k IN nd the vectos. These vectos e clled linely dependent, if thee exist el nubes,..., k, which e not ll equl to zeo, such tht. If the ltte eqution holds only if ll coefficients e, then the vectos e clled linely independent. One cn pove: Sevel vectos e linely dependent if nd only if one of the cn be epesented s line cobintion of the othes. Specil cses: IR : only sets with one eleent, { }, with e linely independent., is linely dependent both vectos e on line though the oigin. IR : { } IR 3 : {,, 3 } is linely dependent ll thee vectos e in plne going though the oigin of the coodinte syste. 53

28 How to test set of vectos fo line dependence Exple: Given e the thee vectos (; ; 3), (; ; ) nd ( ; ; ). Ae they linely dependent? Appoch: We hve to ssue 3 i i i Witten with colun vectos, this ens:. Fo ech coponent, we obtin n eqution, giving togethe the following syste of 3 line equtions: We cn solve this step by step fo the unknowns i. In this cse, we obtin quickly 3. So the syste cn only be fulfilled if ll coefficients e zeo, nd the 3 vectos hve been poven ls linely independent. 54

29 Exples fo tining: Linely dependent o independent? Decide youself! Rnk of set of vectos The nube of eleents of the xil linely independent subset of given set of vectos is clled the nk of the set of vectos. 55

30 56 The bsis of vecto spce IR n hs infinitely ny eleents. Is thee finite subset { } k K,,, such tht ll vectos fo IR n cn be epesented uniquely s line cobintion of the i? YES! Such set of vectos is clled bsis of IR n. Most siple exple of bsis:,,, M K M M e n e e, the stndd bsis of IR n. Thee e infinitely ny bses, which hve, howeve, ll the se nube of eleents (nely, n). This nube is clled the diension of the vecto spce. Exple:,, hs nk lin. indep. lin. dependent

31 57 If we eove, we obtin linely independent vecto syste:,, nk. If we dd now, e.g.,, we obtin bsis of IR 3, i.e., xil linely independent subset:,,, nk 3. lin. independent 3 is the diension of IR 3. If we dd n bity futhe eleent, e.g.,, the set becoes linely dependent: ) ( ) (.

32 The coodintes of vecto with espect to given bsis When n bity bsis is given, evey vecto cn be expessed uniquely s line cobintion of the eleents of this bsis (i.e., the coefficients e uniquely deteined). 58

33 Exple: 5 4 u 3 x u u u, u, 3 5 x u u 5 { u, u } bsis of IR (; ) e the coodintes of x. u w..t. {,u } 59

34 6 In the specil cse of the stndd bsis, we hve lwys: n n n n e e e M M K M M K The coponents,, n of vecto IR n e exctly the coodintes of with espect to the stndd bsis.

Class Summary. be functions and f( D) , we define the composition of f with g, denoted g f by

Class Summary. be functions and f( D) , we define the composition of f with g, denoted g f by Clss Summy.5 Eponentil Functions.6 Invese Functions nd Logithms A function f is ule tht ssigns to ech element D ectly one element, clled f( ), in. Fo emple : function not function Given functions f, g: