CHAPTER 1 INTRODUCTION NUMBER SYSTEMS AND CONVERSION

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1 Fudmetls of Logic Desig Chp. CHAPTE /9 INTODUCTION NUMBE SYSTEMS AND CONVESION This chpter i the book icludes: Objectives Study Guide. Digitl Systems d Switchig Circuits. Number Systems d Coversio. Biry Arithmetic.4 epresettio of Negtive Numbers.5 Biry Codes Problems

2 Fudmetls of Logic Desig Chp. Objectives /9 Topics itroduced i this chpter: Differece betwee Alog d Digitl System Differece betwee Combitiol d Sequetil Circuits Biry umber d digitl systems Number systems d Coversio Add, Subtrct, Multiply, Divide Positive Biry Numbers s Complemet, s Complemet for Negtive biry umber BCD code, 6--- code, excess- code

3 Fudmetls of Logic Desig Chp.. Digitl Systems d Switchig Circuits /9 Digitl systems computtio, dt processig, cotrol, commuictio, mesuremet elible, Itegrtio Alog Cotiuous - Nturl Pheome (Pressure, Temperture, Speed ) - Difficulty i relizig, processig usig electroics Digitl Discrete - Biry Digit Sigl Processig s Bit uit - Esy i relizig, processig usig electroics - High performce due to Itegrted Circuit Techology

4 Fudmetls of Logic Desig Chp. Biry Digit? 4/9 Biry Two vlues(, ) Ech digit is clled s bit Good thigs i Biry Number Number represettio with oly two vlues (,) C be implemeted with simple electroics devices (ex: Voltge High(), Low() ; Switch O () Off() )

5 Fudmetls of Logic Desig Chp. Switchig Circuit 5/9 Combitiol Circuit : outputs deped o oly preset iputs, ot o pst iputs Sequetil Circuit: outputs deped o both preset iputs d pst iputs hve memory elemets Figure -: Switchig circuit

6 6/9 Fudmetls of Logic Desig Chp.. Number Systems d Coversio ). ( N F A Deciml: Biry: dix(bse): Octl-Deciml: Hex-Deciml:

7 7/9 Fudmetls of Logic Desig Chp.. Number Systems d Coversio ) ( N remider Q, N remider Q, Q 4 remider Q, Q Coversio of Deciml to Bse-

8 Fudmetls of Logic Desig Chp.. Number Systems d Coversio 8/9 Exmple: Deciml to Biry Coversio 5 6 rem. = = 6 rem. = = rem. = = 5 rem. = = rem. = = 4 rem. = = 5

9 9/9 Fudmetls of Logic Desig Chp.. Number Systems d Coversio m m m F ) (. F F m m F F m m F F m m ) (.5.65 F ) (.5.5 F ) (..5 F. 65. Coversio of deciml frctio to Bse- Exmple:

10 Fudmetls of Logic Desig Chp.. Number Systems d Coversio /9 Exmple: Covert.7 to biry.7 ().4 ().8 ().6 (). ().4 ().8 Process strts repetig here becuse.4 ws previously obtied.7.

11 Fudmetls of Logic Desig Chp.. Number Systems d Coversio /9 Exmple: Covert. 4 to bse Coversio of Biry to Hex rem. rem (5).5 7 ().75 7 (5).5 7 () D. 5C6 4 D 5 C

12 Fudmetls of Logic Desig Chp.. Number Systems d Coversio /9 Coversio of Biry to Octl, Hex-deciml ( ) = ( ) 8, octl () = ( ) 8, octl () = ( ) 6, Hexdeciml () = ( ) 6, Hexdeciml

13 Fudmetls of Logic Desig Chp.. Biry Arithmetic /9 Additio d crry to the ext colum Exmple: 4 crries

14 Fudmetls of Logic Desig Chp.. Biry Arithmetic 4/9 Subtrctio d borrow from the ext colum Exmple: (idictes borrow From the rd colum) borrows borrows

15 Fudmetls of Logic Desig Chp.. Biry Arithmetic 5/9 Subtrctio Exmple with Deciml 5 8 [ [ 5 8 ] ] colum colum ote borrow from colum [ [ ( ) ( 5) ] 8 ] ote borrow from colum [( ) [ [ ( ) 8 ) ] ] ] 87

16 Fudmetls of Logic Desig Chp.. Biry Arithmetic 6/9 Multiplictio Multiply: x () multiplicd () () 4 multiplier first prtil product secod prtil product sum of first two prtil products third prtil product sum fter ddig third prtil product fourth prtil product fil product (sum fter ddig fourth prtil prodoct)

17 Fudmetls of Logic Desig Chp.. Biry Arithmetic 7/9 Divisio The quotiet is with remider of.

18 Fudmetls of Logic Desig Chp..4 epresettio of Negtive Numbers 8/9 b b b Mgitude MSB () Usiged umber b b b b Sig deotes deotes + MSB Mgitude (b) Siged umber

19 Fudmetls of Logic Desig Chp..4 epresettio of Negtive Numbers 9/9 s complemet of positive iteger N N* N Tble -: Siged Biry Itegers (word legth = 4)

20 Fudmetls of Logic Desig Chp..4 epresettio of Negtive Numbers /9 s complemet of positive iteger N Exmple: N ( ) N N N N* N ( N) N == s complemet: s complemet + N N * d N ( ) N

21 Fudmetls of Logic Desig Chp..4 epresettio of Negtive Number /9 Additio of s complemet Numbers Cse 4 Cse 7 5 (correct swer) 6 Cse 5 wrog swer becuse of overflow (+ requires 5 bits icludig sig) 6 (correct swer) Cse () correct swer whe the crry from the sig bit is igored (this is ot overflow)

22 Fudmetls of Logic Desig Chp..4 epresettio of Negtive Numbers /9 Additio of s complemet Numbers Cse 5 4 Cse 6 7 () correct swer whe the lst crry is igored (this is ot overflow) 5 6 () wrog swer becuse of overflow (- requires 5 bits icludig sig)

23 Fudmetls of Logic Desig Chp..4 epresettio of Negtive Numbers /9 Additio of s complemet Numbers Cse Cse () (correct swer) (ed-roud crry) (correct swer, o overflow) Cse 5 4 () (ed-roud crry) (correct swer, o overflow)

24 Fudmetls of Logic Desig Chp..4 epresettio of Negtive Numbers 4/9 Additio of s complemet Numbers Cse () (ed-roud crry) (wrog swer becuse of overflow) Cse 4: A B (where B A) A B ( A) B ( B A) Cse 5: A B ( A B ) A B ( A) ( B) [ ( A B)]

25 Fudmetls of Logic Desig Chp..4 epresettio of Negtive Numbers 5/9 Additio of s complemet Numbers () ( ) ( ) (ed-roud crry) Additio of s complemet Numbers () ( 8) 9 (discrd lst crry)

26 Fudmetls of Logic Desig Chp..5 Biry Codes 6/ Deciml Digit Code (BCD) 6--- Code Excess- Code -out-of-5 Code Gry Code

27 Fudmetls of Logic Desig Chp..5 Biry Codes 7/ Code: N w w w w N 6 8 ASCII Code S t r t

28 Fudmetls of Logic Desig Chp..5 Biry Codes 8/9 Tble - ASCII code (icomplete)

ICS141: Discrete Mathematics for Computer Science I

ICS141: Discrete Mathematics for Computer Science I ICS4: Discrete Mthemtics for Computer Sciece I Dept. Iformtio & Computer Sci., J Stelovsky sed o slides y Dr. Bek d Dr. Still Origils y Dr. M. P. Frk d Dr. J.L. Gross Provided y McGrw-Hill 3- Quiz. gcd(84,96).