Bit Juggling. Representing Information. representations. - Some other bits. - Representing information using bits - Number. Chapter
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1 Representng Informaton Bt Jugglng - Representng nformaton usng bts - Number representatons - Some other bts Chapter REMINDER: Problem Set #1 s now posted and s due next Wednesday L3 Encodng Informaton 1
2 Motvatons Computers Process Informaton Informaton s measured n bts By vrtue of contanng only swtches and wres dgtal computer technologes use a bnary representaton of bts How do we use/nterpret bts We need standards of representatons for Letters Numbers Colors/pxels Musc Etc. Today Last Week Last Tme It sn t a dream; the semester really has started. L3 Encodng Informaton 2
3 Encodng Encodng descrbes the process of assgnng representatons to nformaton Choosng an approprate and effcent encodng s a real engneerng challenge (and an art) Impacts desgn at many levels - Mechansm (devces, # of components used) - Effcency (bts used) - Relablty (nose) - Securty (encrypton) L3 Encodng Informaton 3
4 Fxed-Length Encodngs If all choces are equally lkely (or we have no reason to expect otherwse), then a fxed-length code s often used. Such a code should use at least enough bts to represent the nformaton content. ex. Decmal dgts 1 = {,1,2,3,4,5,6,7,8,9} 4-bt BCD (bnary code decmal) log 2 (1/1) = < 4bts ex. ~84 Englsh characters = {A-Z (26), a-z (26), -9 (1), punctuaton (8), math (9), fnancal (5)} 7-bt ASCII (Amercan Standard Code for Informaton Interchange) log 2 (84/ 1) = < 7bts L3 Encodng Informaton 4
5 Uncode ASCII s based towards western languages. Englsh n partcular. There are, n fact, many more than 256 characters n common use: â, m, ö, ñ, è,, 揗, 敇, 횝, カ, ℵ, ℷ, ж, క, ค Uncode s a worldwde standard that supports all languages, specal characters, classc, and arcane Several encodng varants 16-bt (UTF-8) ASCII equv range: xxxxxxx Lower 11-bts of 16-bt Uncode 1 1 y y y y x 1 xxxxxx 16-bt Uncode z z z z 1 z y y y y x 1 xxxxxx www 1 wwz z z z 1 z y y y y x 1 xxxxxx L3 Encodng Informaton 5
6 Encodng Postve Integers It s straghtforward to encode postve ntegers as a sequence of bts. Each bt s assgned a weght. Ordered from rght to left, these weghts are ncreasng powers of 2. The value of an n-bt number encoded n ths fashon s gven by the followng formula: v = n = 1 2 b = = = = = = L3 Encodng Informaton 6
7 Some Bt Trcks - You are gong to have to get accustomed to workng n bnary. Specfcally for Comp 411, but t wll be helpful throughout your career as a computer scentst. - Here are some helpful gudes 1. Memorze the frst 1 powers of 2 2 = = = = = = = = = = 512 L3 Encodng Informaton 7
8 More Trcks wth Bts - You are gong to have to get accustomed to workng n bnary. Specfcally for Comp 411, but t wll be helpful throughout your career as a computer scentst. - Here are some helpful gudes 2. Memorze the prefxes for powers of 2 that are multples of = Klo (124) 2 2 = Mega (124*124) 2 3 = Gga (124*124*124) 2 4 = Tera (124*124*124*124) 2 5 = Peta (124*124*124 *124*124) 2 6 = Exa (124*124*124*124*124*124) L3 Encodng Informaton 8
9 Even More Trcks wth Bts - You are gong to have to get accustomed to workng n bnary. Specfcally for Comp 411, but t wll be helpful throughout your career as a computer scentst. - Here are some helpful gudes When you convert a bnary number to decmal, frst break t down nto clusters of 1 bts. 4. Then compute the value of the leftmost remanng bts (1) fnd the approprate prefx (GIGA) (Often ths s suffcent) 5. Compute the value of and add n each remanng 1-bt cluster L3 Encodng Informaton 9
10 Other Helpful Clusters Oftentmes we wll fnd t convenent to cluster groups of bts together for a more compact representaton. The clusterng of 3 bts s called Octal. Octal s not that common today. v Seems natural to me! = n 1 = 8d 372 Octal - base *8 = + 2*8 1 = *8 2 = *8 3 = = 2 1 L3 Encodng Informaton 1
11 One Last Clusterng Clusters of 4 bts are used most frequently. Ths representaton s called hexadecmal. The hexadecmal dgts nclude -9, and A-F, and each dgt poston represents a power of 16. v = n 1 = 16d x7d Hexadecmal - base d = a b c d e f *16 = + 13*16 1 = *16 2 = L3 Encodng Informaton 11
12 Sgned-Number Representatons There are also schemes for representng sgned ntegers wth bts. One obvous method s to encode the sgn of the nteger usng one bt. Conventonally, the most sgnfcant bt s used for the sgn. Ths encodng for sgned ntegers s called the SIGNED MAGNITUDE representaton. Anythng werd? v = 1 S n = 2 2b S Even though ths approach seems straghtforward, t s not used that frequently n practce (wth one mportant excepton). L3 Encodng Informaton 12
13 2 s Complement Integers N bts -2 2 N-1 N sgn bt Range: 2 N-1 to 2 N-1 1 bnary pont The 2 s complement representaton for sgned ntegers s the most commonly used sgned-nteger representaton. It s a smple modfcaton of unsgned ntegers where the most sgnfcant bt s consdered negatve. 8-bt 2 s complement example: v = 2 n 1 b n 1 + n = 2 2b = = = 42 L3 Encodng Informaton 13
14 Why 2 s Complement? If we use a two s complement representaton for sgned ntegers, the same bnary addton mod 2 n procedure wll work for addng postve and negatve numbers (don t need separate subtracton rules). The same procedure wll also handle unsgned numbers! When usng sgned magntude representatons, addng a negatve value really means to subtract a postve value. However, n 2 s complement, addng s addng regardless of sgn. In fact, you NEVER need to subtract when you use a 2 s complement representaton. Example: 55 1 = = = = = = L3 Encodng Informaton 14
15 2 s Complement Trcks - Negaton changng the sgn of a number - Frst complement every bt (.e. 1, 1) - Add 1 Example: 2 = 11, -2 = = Sgn-Extenson algnng dfferent szed 2 s complement ntegers - 16-bt verson of 42 = bt verson of -2 = L3 Encodng Informaton 15
16 CLASS EXERCISE Helpful Table of the 9 s complement for each dgt s-complement Arthmetc (You ll never need to borrow agan) Step 1) Wrte down 2 3-dgt numbers that you want to subtract Step 2) Form the 9 s-complement of each dgt n the second number (the subtrahend) Step 3) Add 1 to t (the subtrahend) Step 4) Add ths number to the frst Step 5) If your result was less than 1, form the 9 s complement agan and add 1 and remember your result s negatve else subtract 1 What dd you get? Why weren t you taught to subtract ths way? L3 Encodng Informaton 16
17 Fxed-Pont Numbers By movng the mplct locaton of the bnary pont, we can represent sgned fractons too. Ths has no effect on how operatons are performed, assumng that the operands are properly algned = = = OR = -42 * 2-4 = -42/16 = L3 Encodng Informaton 17
18 Repeated Bnary Fractons Not all fractons can be represented exactly usng a fnte representaton. You ve seen ths before n decmal notaton where the fracton 1/3 (among others) requres an nfnte number of dgts to represent (.3333 ). In Bnary, a great many fractons that you ve grown attached to requre an nfnte number of bts to represent exactly. EX: 1 / 1 =.1 1 = / 5 =.2 1 =.11 2 = L3 Encodng Informaton 18
19 Crazness There are many other ways we can cluster bts and use them to encode numbers. Here s a scheme n whch the clusters overlap, and the weght values are not unque What possble use could ths have? -2 * 2 = * 2 2 = * 2 4 = * 2 6 = Bt Sequence Weght L3 Encodng Informaton 19
20 Bas Notaton There s yet one more way to represent sgned ntegers, whch s surprsngly smple. It nvolves subtractng a fxed constant from a gven unsgned number. Ths representaton s called Bas Notaton. v n = = 1 2b Bas EX: (Bas = 127) Why? Monotoncty 6 * 1 = 6 13 * 16 = L3 Encodng Informaton 2
21 Floatng Pont Numbers Another way to represent numbers s to use a notaton smlar to Scentfc Notaton. Ths format can be used to represent numbers wth fractons (3.9 x 1-4 ), very small numbers (1.6 x 1-19 ), and large numbers (6.2 x 1 23 ). Ths notaton uses two felds to represent each number. The frst part represents a normalzed fracton (called the sgnfcand), and the second part represents the exponent (.e. the poston of the floatng bnary pont). Exponent Normalzed Fracton 2 Exponent Normalzed Fracton dynamc range bts of accuracy L3 Encodng Informaton 21
22 - Sngle precson format IEEE 754 Format S Exponent Sgnfcand Ths s effectvely a sgned magntude fxed-pont number wth a hdden 1. The 1 s hdden because t provdes no nformaton after the number s normalzed The exponent s represented n bas 127 notaton. Why? Double precson format v = -1 s x 1.Sgnfcand x 2 Exponent-127 S Exponent Sgnfcand v = -1 s x 1.Sgnfcand x 2 Exponent-123 L3 Encodng Informaton 22
23 Summary 1) Selectng the encodng of nformaton has mportant mplcatons on how ths nformaton can be processed, and how much space t requres. 2) Computer arthmetc s constraned by fnte representatons, ths has advantages (t allows for complement arthmetc) and dsadvantages (t allows for overflows, numbers too bg or small to be represented). 3) Bt patterns can be nterpreted n an endless number of ways, however mportant standards do exst - Two s complement - IEEE 754 floatng pont L3 Encodng Informaton 23
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