Unit 2: Binary Numbering Systems

Transcription

1 Unt 2: Bnary Numberng Systems Defntons Number bases Numercal representatons. Integer fxed pont. Bnary 2 s complement BCD Addton-subtracton Alphanumercal representatons Basc Bblography Any book on dgtal electroncs, for nstance: Dgtal Fundamentals. (Chapter 2) T.L. Floyd Pearsons Prentce Hall 2 1

2 Defntons Space of a representaton: number of bts to store a data (numercal or character) Byte (8 bts) Word (n bts, generally 16, 32, 64) Range of representaton: Maxmum and mnmum value that can be represented n a numberng system wth fxed number of dgts Resoluton of the representaton: Dfference between a number and the next one n the representaton Code length: number of elements that can be represented wth a n-bt representaton (example: for pure bnary wth n bts the code length s 2 n ) 3 Numberng bases (I) Bases 2, 8, 10 y 16 Bnary Octal Decmal Hexadecmal (base 2) (base 8) (base 10) (base 16) 0 0 (000) 0 (0000) 0 (0000) A (1010) 1 1 (001) 1 (0001) 1 (0001) B (1011) 2 (010) 2 (0010) 2 (0010) C (1100) 3 (011) 3 (0011) 3 (0011) D (1101) 4 (100) 4 (0100) 4 (0100) E (1110) 5 (101) 5 (0101) 5 (0101) F (1111) 6 (110) 6 (0110) 6 (0110) 7 (111) 7 (0111) 7 (0111) 8 (1000) 8 (1000) 9 (1001) 9 (1001) 4 2

3 Numberng bases (II) P 7 P 6 P 5 P 4 P 3 P 2 P 1 P 0 The poston of each bt represent ts weght Unts:10 0 Tens: 10 1 Hundreds: 10 2 Thousands: 10 3 Tens of Thousands: 10 4 To compute the decmal value: value Examples: n = 1 = 0 Bnary number Decmal value: x base 1x x x x x2 0 = 21 Hexadecmal number :78A Decmal value: 7x x x16 0 = Fxed pont representatons Pure Bnary n=8 bts x 7 x 6 x 5 x 4 x 3 x 2 x 1 x 0 n-1 0 Base 2 postonal system for ntegers Weghts are: Decmal value wth n bts: Range: [0, 2 n 1] P = 2 Value = n 1 = 0 2 x Resoluton = 1 6 3

4 Fxed pont representatons 2 s Complement (2C) Postve numbers: start wth 0, represented n pure bnary Negatve numbers: start wth 1, represented n 2C Then the MSB (Most Sgnfcant Bt) ndcates the sgn, but for operatons all n-bts are treated alke To represent a negatve number: - A = 2C of A. Operatons to obtan C2: _ Obtan 1C (1 s complement) of A : A = n 2 A (equvalent to replace 0 1) Add 1: A _ + 1 To obtan the decmal value (n bts): Value Range: [-2 n-1, -1, 0, (2 n-1-1)] n x f x n-1 = 0 = = 0 Value (2C ( number )) f x n-1 = 1 Resoluton = 1 7 Addton-Subtracton n 2 s complement Man reason to use 2C s that addton and subtracton operatons are smplfed: Operate wthout takng nto account the sgn of the operands Fnal carry s gnored. To subtract just add the 2C of the number: A B = A + 2C(B) Overflow occurs f: A 0 y B 0 and A + B < 0 A < 0 y B < 0 and A + B 0 Example: A=0111 and B=0101 : -A=1001 and -B=1011 A + B = = 1100 y C f = 0 : overflow A - B = A + (-B) = = 0010 y C f = 1 -A + B = = 1110 y C f = 0 -A - B = (-A) + (-B) = = 0100 y C f = 1 : overflow 8 4

5 Fxed pont representatons BCD: Bnary Coded Decmal Used to represent decmal dgts n bnary; Four bts represent one decmal dgt: Decmal dgt BCD Decmal dgt BCD To represent decmal numbers wth more dgts just group BCD packages Example: Addton n BCD Addton 10 5

6 Addton n hexadecmal 11 Alphanumerc representatons (I) Represent each character (7, A, j, =, *,.) by a group of bts. Examples 6 bts (2 6 =64 characters): Feldata and BCDIC 7 bts (2 7 =128 characters): ASCII 8 bts (2 6 =256 characters): extended ASCII and EBCDIC 16 bts (2 16 =65536 characters): UNICODE 12 6

7 Alphanumerc representatons (II) Phrases are represented groupng characters. Optons: Fxed length strng P E P E A N T O N I O R O S A Varable length strng Delmter character * P E P E * A N T O N I O * R O S A Explct length 4 P E P E 7 A N T O N I O 4 R O S A 13 Alphanumerc representatons (III) ASCII code 14 7