Computer Number Systems

Transcription

1 Computer Number Systems Thorne, Edition 2 : Section 1.3, Appendix I (Irvine, Edition VI : Section 1.3) SYSC3006 1

2 Starting from What We Already Know Decimal Numbers Based Number Systems : 1. Base defines set of symbols 2. Number = string of symbols 3. Interpretation rules Digits numbered right to left, start at 0 eg. 4-digit number d 3 d 2 d 1 d 0 Value of digit depends on symbol and position within number Value of number is sum of value of digits Decimal Numbers : Base-10 with symbols = { 0, 1, 2, 3,, 9} 4-digit example: d 3 d 2 d 1 d 0 = 1436 = d d d d = Note : 1. Addition/Subtraction : We carry and borrow 10 s 2. Multiplication/Division by 10 : Shift right/left SYSC3006 2

3 Applying Same Methods to Binary Numbers Binary Numbers : Base-2 with symbols = { 0, 1} 4-digit example: d 3 d 2 d 1 d 0 = 1010b = d d d d = = = = 10d Note : 1. Addition/Subtraction : We carry and borrow 2 s 2. Multiplication/Division by 2 : Shift right/left SYSC3006 3

4 Applying Same Methods to Binary Numbers Addition Subtraction 111b 1010b 101b 1001b + 10b + 111b - 10b - 111b Multiplication by Powers-of-2 Division by Powers-of-2 By 2 1 : 100b * 10b = By 2 1 : 100b / 10b = By 2 2 : 11b * 100b = By 2 2 : 1100b / 100b = By 2 3 : 100b * 1000b = By 2 3 : b / 1000b = SYSC3006 4

5 4-bit Binary Numbers you should know Binary Decimal Binary Decimal SYSC3006 5

6 Range of Representation Within a computer, all numbers are represented in a finite-width Registers are finite width, depending on processor Memory cells are 8 or 16 or 32 or 64-width Range : n-bit values can represent (at most) 2 n different counting numbers If start representation at 0, then largest value represented is 2 n 1 Previous page : 4-bit numbers Maximum number of values : 2 4 = 16 Range : (2 4 1) Observation : Binary representation often requires lots of bits Εxample: = (10 bits!) Converting binary decimal is not so convenient SYSC3006 6

7 Hexadecimal Numbers : Base 16 Why Base-16? 16 = 2 4 Will provide us with a shortcut for working with large binary number analogous to acronyms (eg. CU, UBC, MIT) Hexadecimal Numbers : Base-16 with symbols = { 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F} 4-digit example: d 3 d 2 d 1 d 0 = 12AFh = d d d d = A F 16 0 = = = 4783d Addition/Subtraction : We carry and borrow 16 s Multiplication/Division by 16 : Shift right/left SYSC3006 7

8 Applying Same Methods to Hexadecimal Numbers Addition Subtraction 181h 1510h 1E1h 1001h + 89h + E11h - 1Fh - 111h Multiplication by Powers-of-16 Division by Powers-of-16 By 16 1 : 20h * 10h = By 16 1 : 105h / 10h = By 16 2 : 3 * 100h = By 16 2 : 10E0h / 100h = SYSC3006 8

9 Hexadecimal and Binary Numbers : 2 4 = 16 Converting Binary to Hexadecimal 16-bit: (= 35, ) group in 4 s replace with hex 8 A 6 E bit: (= ) group in 4 s replace with hex Converting Hex Binary : Replace each hex digit by 4-digit binary rep. Example : 134h = b SYSC3006 9

10 Common Pitfalls for Beginners 1. What do I mean when I say Ten hex? 2. Are these two the same thing? b and A2h 3. What are the widths of the following numbers? b 2h 22h 1010h 1010b 4. Memory is always composed of bytes One byte = 8 bits Contents of a byte = 8 binary digits b b = 2 hex digits = (2*4 bits) 00h FFh = 0 255d SYSC

11 Representing Negative Numbers For unsigned numbers (counting, whole) : All n bits are used for the magnitude of the number Example : 4-bit number : 0000b. 1111b For signed numbers (negative) : One bit must be used for the sign : Most significant bit = sign bit 0 = positive, 1 = negative n-1 bits are used for the magnitude of the number. Encoding the magnitude portion : 2 typical approaches 1. Signed magnitude encoding : A natural progression from unsigned numbers, but No good for math 2. Two s complement encoding Hard to understand, but Math works SYSC

12 Signed Magnitude Encoding of Signed Numbers Magnitude encoded in remaining n 1 bits using binary number system (as for counting #s) Example : b = -1d magnitude = b = 1 sign bit = 1 negative number Problems 1. Two representations for 0 positive 0 (sign bit = 0) negative 0 (sign bit = 1) 2. Awkward arithmetic operations b b SYSC

13 Two s Complement Encoding of Signed Numbers Magnitude of negative numbers encoded with flip-and-add Magnitude of positive numbers encoded as normal binary number Definitions : complement of one bit: b = 1 b complement of 0 = 1 complement of 1 = 0 1s complement of an n-bit value: complement each bit 2 s complement of an n-bit value: complement each bit, and then add 1 (flip-and-add) Ignore any carry out of most significant bit SYSC

14 Two s Complement Encoding of Signed Numbers Example : Find the 8-bit 2 s complement representation of b Complement b Add b = FFh 2 s complement operation Encoding of negative numbers is weird! Example : Find the 8-bit 2 s complement representation of b = 01h That s it! SYSC

15 Two s Complement Encoding of Signed Numbers Example : Find the decimal value of this 2 s complement number FEh (sign bit = 1) Complement b b Add b = 2d FEh = -2d Example : Find the decimal value of this 2 s complement number 79h (sign bit = 0) b = = 121d 2 s complement operation SYSC

16 Range of Signed Representations Whether using signed magnitude or 2 s complement MSBit = sign bit magnitude = n-1 remaining bits Range : n-bit values still represent (at most) 2 n different counting numbers, but Use half of binary values for negative values (i.e. 2 n 1 values), the rest for positive (and 0) Range for negative and positive values : All negatives start with n n 1-1 (Total 2 n numbers) SYSC

17 Why 2 Complement for Signed Numbers? 1. Negating a negated number should give the original number ( x) = x complement add It works! 2. How many representations for 0? complement add ignore carry out of most significant bit SYSC

18 One Special Case with 2 Complement for Signed Numbers Consider 8-bit 2 s complement encodings: 2 8 = 256, 2 7 = 128 range: ( 2 n 1 ) (2 n 1 1) complement add SYSC

19 One Special Case with 2 Complement for Signed Numbers How to get encoding for 128? 1. Using math = subtract Using 2 s complement What if we negate 128? [ special case!! ] complement add SYSC

20 Character Encoding Representation of displayable characters: characters: { A, B,..., Y, Z } 26 2 = 52 (upper and lower case) decimal (10) digits: {0, 1,..., 8, 9 } punctuation:!,.? / : ; math symbols: + * = ( and / above) brackets: ( ) [ ] { } < > # $ % ^ & \ ~ blank space: 90+ symbols (??) Various encoding schemes have been used SYSC

21 (7-bit) ASCII character encoding ASCII = American Standard Code for Information Interchange 7-Bit ASCII Encoding 7 bits to encode each character (128 codes) often extended to 8-bit (byte) values by making most significant bit (msb) = 0 [in following: all codes are given as hex values] 00 1F non-displayable control char s 00h NULL 07h BELL 08h backspace 09h tab 0Ah line feed 0Ch form feed 0Dh carriage return others often serve special purposes in communication applications SYSC

22 Decimal ASCII Code Table SYSC

23 (7-bit) ASCII character encoding 30h 39h 30h 0 39h 9 decimal digit char s character 0 number 41h 5Ah 41h A Upper Case Letter char s 5Ah Z 61h 7Ah 61h a Lower Case Letter char s 7Ah z SYSC

24 (7-bit) ASCII character encoding Example : 306 is FUN! encoding: E 21 shorthand for binary! Other Character Encoding Schemes: IBM standardized an 8-bit scheme (256 char s) as defacto standard (PC s!) See Appendix I : overlaps with 7-bit ASCII for displayable char s Java: unicode 16-bit scheme (65,536 char s) multi-lingual character sets SYSC

25 Programmers : Be Aware Important Concept!!!!!! fixed-width binary values are used to represent information in computers the computer works with the binary representations (NOT the information!!) the same binary value can be used to represent different information! 8-bit example: unsigned: signed: bit ASCII: other? SYSC