number systems and codes. zpositional number systems zarithmetic operations zcodes
TRANSCRIPT
NUMBER SYSTEMS AND CODES
NUMBER SYSTEMS AND CODES
POSITIONAL NUMBER SYSTEMSARITHMETIC OPERATIONSCODES
ROMAN NUMBER SYSTEM
I + III = IV
ROMAN NUMBER SYSTEM
L V I : L IV = ?
POSITIONAL NUMBER SYSTEMS
101 1011021011.12
131011002213
POSITIONAL NUMBER SYSTEMS
GENERAL FORM OF NUMBER:
BASE OR RADIX (r)RADIX POINTVALUE:
npp ddddddd 210121 .
1p
ni
iirdD
BINARY NUMBERS
DIGITAL SYSTEMS: HIGH OR LOW1 OR 0BINARY RADIX (BASE TWO)
npp bbbbbbb 210121 .
1
2p
ni
iibB
HEXADECIMAL, OCTAL NUMBERS
HEXADECIMAL - RADIX 160,1,2,…,9,A,B,C,D,E,F,10,11...
OCTAL - RADIX 80,1,2,…,7,8,10,11...
CONVERSIONS
BINARY TO OCTAL:
801
0123012
012345
822
428284
2021202)202021(
202120202021
42010100100010
CONVERSIONS
BINARY TO HEXADECIMAL (HEX):
1622 1010110111011010 DA
CONVERSIONS
RADIX-r TO DECIMAL:
DECIMAL TO RADIX-r
021
1
0
))((( drrdrd
rdD
pp
p
i
ii
121
0
))((( drrdrdQr
dQrD
pp
EXAMPLE: RADIX-16 TO DECIMAL
D5916=13·162 + 5·16 + 9
D5916 =341710
EXAMPLE: DECIMAL TO RADIX-16
3417:16=213 remainder 9 (LSD):16=13 remainder 5
:16=0 remainder 13 (MSD)
341710=D5916
NUMBER SYSTEMS AND CODES
POSITIONAL NUMBER SYSTEMSARITHMETIC OPERATIONSCODES
ADDITION OF NONDECIMAL NUMBERS
SAME TECHNIQUE, DIFFERENT TABLES
17
+ 15
32
10001
1111
100000
11111
X
Y
X+Y
CARRY
NEGATIVE NUMBERS
SIGNED-MAGNITUDE REPRESENTATION:
000001012=+510 100001012=-510
011111112=+12710 111111112=-12710
000000002=+010 100000002=-010
COMPLICATED ADDER/SUBTRACTOR
NEGATIVE NUMBERS
COMPLEMENT NUMBER SYSTEMS SYSTEM-DEPENDENT DEFINITION DIRECT ADDITION AND SUBTRACTION
RADIX-COMPLEMENT
FOR AN n-DIGIT NUMBER D:
EXAMPLE FOR r=10, n=2, D=23:
DISCARD EXTRA HIGH-ORDER DIGITS
DrD n
01007723
77231023 2
DD
D
RADIX-COMPLEMENT
rn IS AN n+1 DIGIT NUMBERrn-1 IS AN n DIGIT NUMBER:
1
1
1)1(
rm
mmmr
DrDn
n
RADIX-COMPLEMENT
FOR D=dn-1dn-2…d0 WE HAVE:
WHERE:
1
1
1))1((
021
021
ddd
dddmmm
DrD
nn
nn
n
drd )1(
EXAMPLE: TEN’S-COMPLEMENT
871018709
1129099991)1290)110((
1))1((
;1290
;4
;10
4
DrD
D
n
r
n
EXAMPLE: TEN’S-COMPLEMENT
01000000854306145694
8543061854305
;145694
;6
;10
DD
D
D
n
r
TWO’S-COMPLEMENT REPRESENTATION
BINARY NUMBERS
WEIGHT OF MSB IS -2p-1
npp bbbbbbb 210121 .
21
1 2)2(p
ni
ii
pp bbB
TWO’S-COMPLEMENT EXAMPLES
1710=000100012
complement bits:11101110 +1111011112 = -1710
010=000000002
complement bits: 11111111 +11000000002 = 010
ONES’-COMPLEMENT REPRESENTATION
1710=000100012
complement bits:111011102 = -1710
+010=000000002
complement bits:111111112 = -010
EXCESS REPRESENTATION
M (0 M < 2m) REPRESENTS M-BB - BIAS
DECIMAL EXCESS 2m-1
-2 00-1 01 0 10 1 11
TWO’S COMPLEMENT ADDITION
COUNT UP BY ADDING ONE IGNORING CARRIES BEYOND MSB:
DECIMAL TWO’S COMPLEMENT-2 10-1 11 0 00 1 01
TWO’S-COMPLEMENT ADDITION EXAMPLES
410 01002
+ 110 + 00012
510 01012
-210 11102
+ 510 + 01012
310 1 00112
-210 11102
+ -610 + 10102
-810 1 10002
OVERFLOW
410 01002
+ 510 + 01012
910 10012 = -710
-810 10002
+ -810 + 10002
-1610 1 00002=0
SIGNS OF ADDENDS SAME AND DIFFERENT FROM SIGN OF SUM
CARRY INTO AND OUT OF SIGN POSITION DIFFERENT
TWO’S COMPLEMENT SUBTRACTION
1 (cin)
410 01002
- 510 + 10102
-110 11112 = -110
1 (cin)
810 10002
+ -810 + 01112
010 100002=0
BIT-BY-BIT COMPLEMENT OF SUBTRAHEND, ADD WITH EXTRA CARRY-IN OF 1
GRAPHICAL VIEW
UNSIGNED BINARY MULTIPLICATION
SHIFT AND ADD11 1011 multiplicand
13 1101 multiplier33 1011
11 0000 shifted 143 1011 multiplicands
101110001111 product
UNSIGNED BINARY MULTIPLICATION
USE PARTIAL PRODUCT:5 101 multiplicand
6 110 multiplier30 000 partial product
000 shifted multiplicand 0000 partial product
101 shifted multiplicand01010 partial product101 shifted multiplicand
011110 product
TWO’S COMPLEMENT
Bn = -bn-12n-1 + bn-22n-2 + … + b0
SIGN EXTENSION
TWO’S COMPLEMENT MULTIPLICATION
2 010 multiplicand - 2 110 multiplier
- 4 0000 sign extended partial product0000 s. e. shifted multiplicand
00000 s. e. partial product0010 s. e. & s. multiplicand
000100 s. e. partial product1110 s.e. & s. negated multiplicand111100 product
BINARY DIVISION
GRAMMAR SCHOOL METHOD: SHIFT AND SUBTRACT
NUMBER SYSTEMS AND CODES
POSITIONAL NUMBER SYSTEMSARITHMETIC OPERATIONSCODES
CODE?
CODE n. 1: A SYSTEM OF SIGNALS OR SYMBOLS FOR COMMUNICATION
HOW DO WE CREATE A CODES?
COMPUTERS USE THE BINARY SYSTEM
CODE: A SET OF n-BIT STRINGSONE SUCH STRING IS A CODE WORD
BINARY CODES FOR DECIMAL NUMBERS
DECIMAL BCD 1-OUT-OF-100 0000 00000000011 0001 00000000102 0010 00000001003 0011 00000010004 0100 00000100005 0101 00001000006 0110 00010000007 0111 00100000008 1000 01000000009 1001 1000000000
UNUSED1010, … 0000000000, ...
GRAY CODE
CHARACTER CODES
MOST COMMON NON-NUMERIC DATA IS TEXT
ASCII - AMERICAN STANDARD CODE FOR INFORMATION INTERCHANGE
128 7-BIT STRINGS
CODES FOR ACTIONS, CONDITIONS, STATES
FOR n DIFFERENT ACTIONS, CONDITIONS, OR STATES b-BIT BINARY CODE
nb 2log
TRAFFIC LIGHT EXAMPLE
REMARKS
UNDERSTAND FIGURE 2-4 AND NEXT-TO-LAST PARAGRAPH OF 2.6 (PAGE 43)
WHAT IS WRONG WITH FIGURE 2-5 (PAGE 51)?