NUMBER SYSTEMS.ppt

LESSON OUTLINE
• NUMBER SYSTEMS
• NUMBER NOTATIONS
• ARITHMETIC
• BASE CONVERSIONS
• SIGNED NUMBER REPRESENTATION
• CODES
• DECIMAL CODES
• GRAY CODE
• ASCII CODE
Number Systems and Codes 2

NUMBER SYSTEMS
THE DECIMAL (REAL), BINARY, OCTAL,
HEXADECIMAL NUMBER SYSTEMS ARE USED TO
REPRESENT INFORMATION IN DIGITAL SYSTEMS.
ANY NUMBER SYSTEM CONSISTS OF A SET OF
DIGITS AND A SET OF OPERATORS (+, , , ).

RADIX OR BASE
Decimal (base 10) 0 1 2 3 4 5 6 7 8 9
Binary (base 2) 0 1
Octal (base 8) 0 1 2 3 4 5 6 7
Hexadecimal (base 16) 0 1 2 3 4 5 6 7 8 9 A B C D E F
The radix or base of the number system
denotes the number of digits used in the
system.

Decimal Binary Octal Hexadecimal
00 0000 00 0
01 0001 01 1
02 0010 02 2
03 0011 03 3
04 0100 04 4
05 0101 05 5
06 0110 06 6
07 0111 07 7
08 1000 10 8
09 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F

POSITIONAL NOTATION
IT IS CONVENIENT TO REPRESENT A NUMBER
USING POSITIONAL NOTATION. A POSITIONAL
NOTATION IS WRITTEN AS A SEQUENCE OF
DIGITS WITH A RADIX POINT SEPARATING THE
INTEGER AND FRACTIONAL PART.
WHERE R IS THE RADIX, N IS THE NUMBER OF
DIGITS OF THE INTEGER PART, AND M IS THE
NUMBER DIGITS OF THE FRACTIONAL PART.
 r
m
n
n
r a
a
a
a
a
a
a
N 




 
 2
1
0
1
2
1 .

POLYNOMIAL NOTATION
A NUMBER CAN BE EXPLICITLY REPRESENTED IN POLYNOMIAL
NOTATION.
WHERE RP IS A WEIGHTED POSITION AND P IS THE POSITION
OF A DIGIT.
m
m
n
n
n
n
r r
a
r
a
r
a
r
a
r
a
r
a
r
a
N 








 














 
 2
2
1
1
0
0
1
1
2
2
1
1

EXAMPLES
IN BINARY NUMBER SYSTEM
IN OCTAL NUMBER SYSTEM
IN HEXADECIMAL NUMBER SYSTEM
  3
2
1
0
1
2
3
4
2 2
1
2
1
2
0
2
0
2
1
2
0
2
1
2
1
011
.
11010 


















  3
2
1
0
1
2
8 8
4
8
2
8
1
8
3
8
7
8
6
124
.
673 














  1
0
1
2
16 16
16
6
16
0
16
3
.
306 







 D
D

ARITHMETIC
(101101)2 +(11101)2
:
1111 1
+
101101
11101
1001010
Addition:
In binary number system,

ADDITION
(6254)8+(5173)8 : 1 1
+
6254
5173
13447
In octal number system,
(9F1B)16 +(4A36)16 : 1 1
+
9F1B
4A36
D951
In hexadecimal number system,

SUBTRACTION
(101101)2 -(11011)2
:
10 10
-
101101
11011
10010

SUBTRACTION
(9F1B)16 -(4A36)16 : 16
-
9F1B
4A36
54E5
In octal number system,
In hexadecimal number system,
(6254)8 -(5173)8 : 8
-
6254
5173
1061

MULTIPLICATION
(1101)2  (1001)2 :

1101
1001
1101
0000
0000
1101
1110101

DIVISION
(1110111)2 (1001)2 : 1101
1001 1110111
1001
1011
1001
1011
1001
10

BASE CONVERSIONS
CONVERT (100111010)2 TO BASE 8
 
 8
0
1
2
0
1
1
1
2
0
1
2
3
4
5
6
7
8
2
472
8
2
8
7
8
4
8
2
8
1
8
2
8
4
8
4
2
0
2
1
2
0
2
1
2
1
2
1
2
0
2
0
2
1
100111010



































   
 8
2
2
472
2
7
4
010
111
100
100111010




BASE CONVERSION
 
         
 10
10
10
10
10
10
0
1
2
3
4
5
6
7
8
2
314
2
8
16
32
256
2
0
2
1
2
0
2
1
2
1
2
1
2
0
2
0
2
1
100111010

























BASE CONVERSION
 
 16
0
1
2
0
0
1
1
2
0
1
2
3
4
5
6
7
8
2
13
16
16
3
16
1
16
2
16
8
16
1
16
2
16
1
2
0
2
1
2
0
2
1
2
1
2
1
2
0
2
0
2
1
100111010
A
A



































   
 16
2
2
13
3
1
1010
0011
0001
100111010
A
A 



BASE CONVERSION FROM BASE 8
• CONVERT (372)8 TO BASE 2
 
 2
8
11111010
010
111
011
2
7
3
372



 
     
 10
10
10
10
0
1
2
8
250
2
56
192
8
2
8
7
8
3
372










   
 16
2
8 1010
1111
372
FA
A
F 



BASE CONVERSION FROM BASE 16
• CONVERT (9F2)16 TO BASE 2
19
 
 2
16
10
1001111100
0010
1111
1001
2
9
2
9


 F
F
 
   8
2
16
4762
)
2
010
6
110
7
111
4
100
(
0010
1111
1001
2
9
2
9



 F
F
 
     
 10
10
10
10
0
1
2
16
2546
2
240
2304
16
2
16
16
9
2
9









 F
F

BINOMIAL EXPANSION
(SERIES SUBSTITUTION)
TO CONVERT A NUMBER IN BASE R TO BASE P.
• REPRESENT THE NUMBER IN BASE P IN BINOMIAL SERIES.
• CHANGE THE RADIX OR BASE OF EACH TERM TO BASE P.
• SIMPLIFY.

CONVERT BASE 10 TO BASE R
THEREFORE (174)10 = (256)8
8 1 7 4 6 LSB
8 2 1 5
8 2 2 MSB
0 0

CONVERT (0.275)10 TO BASE 8
THEREFORE (0.275)10 = (0.21463)8
8  0.275  2.200 MSD
8  0.200  1.600
8  0.600  4.800
8  0.800  6.400
8  0.400  3.200 LSD

CONVERT (0.68475)10 TO BASE 2
THEREFORE (0.68475)10 = (0.10101)2
2  0.68475  1. 3695 MSD
2  0.3695  0.7390
2  0.7390  1.4780
2  0.4780  0.9560
2  0.9560  1.9120 LSD

SIGNED NUMBER
REPRESENTATION
THERE ARE 3 SYSTEMS TO REPRESENT SIGNED NUMBERS IN
BINARY NUMBER SYSTEM:
• SIGNED-MAGNITUDE
• 1'S COMPLEMENT
• 2'S COMPLEMENT

SIGNED-MAGNITUDE SYSTEM
IN SIGNED-MAGNITUDE SYSTEMS, THE MOST SIGNIFICANT
BIT REPRESENTS THE NUMBER'S SIGN, WHILE THE
REMAINING BITS REPRESENT ITS ABSOLUTE VALUE AS AN
UNSIGNED BINARY MAGNITUDE.
• IF THE SIGN BIT IS A 0, THE NUMBER IS POSITIVE.
• IF THE SIGN BIT IS A 1, THE NUMBER IS NEGATIVE.

SIGNED-MAGNITUDE SYSTEM

1'S COMPLEMENT SYSTEM
• A 1'S COMPLEMENT SYSTEM REPRESENTS THE
POSITIVE NUMBERS THE SAME WAY AS IN THE
SIGNED-MAGNITUDE SYSTEM. THE ONLY
DIFFERENCE IS NEGATIVE NUMBER
REPRESENTATIONS.
• LET BE N ANY POSITIVE INTEGER NUMBER AND
BE A NEGATIVE 1'S COMPLEMENT INTEGER OF N.
IF THE NUMBER LENGTH IS N BITS, THEN
__
N
.
)
1
2
( N
N n




EXAMPLE OF 1'S COMPLEMENT
FOR EXAMPLE IN A 4-BIT SYSTEM, 0101 REPRESENTS +5 AND
1010 REPRESENTS 5
 
1010
0101
1111
0101
)
0001
10000
(
0101
0001
24









• A 2'S COMPLEMENT SYSTEM IS SIMILAR TO 1'S
COMPLEMENT SYSTEM, EXCEPT THAT THERE IS ONLY ONE
REPRESENTATION FOR ZERO.
• LET BE N ANY POSITIVE INTEGER NUMBER AND
BE A NEGATIVE 2'S COMPLEMENT INTEGER OF N. IF THE
NUMBER LENGTH IS N BITS, THEN
__
N
.
2 N
N n



EXAMPLE OF 2'S COMPLEMENT
FOR EXAMPLE IN A 4-BIT SYSTEM, 0101 REPRESENTS +5 AND
1011 REPRESENTS 5
CS 3402--Digital Logic
1011
0101
10000
0101
24





ADDITION AND SUBTRACTION IN
SIGNED AND MAGNITUDE
(a) 5
+2
0101
+0010
7 0111
(b) -5
-2
1101
+1010
-7 1111
(c) 5
-2
0101
+1010
3 0011
(d) -5
+2
1101
+0010
-3 1011

(a) 5
+2
0101
+0010
7 0111
(b) -5
-2
1010
+1101
-7 1 0111
1
1000
(c) 5
-2
0101
+1101
3 1 0010
1
0011
(d) -5
+2
1010
+0010
-3 1100
1’S COMPLEMENT

(a) 5
+2
0101
+0010
7 0111
(b) -5
-2
1011
+1110
-7 1 1001
(c) 5
-2
0101
+1110
3 1 0011
(d) -5
+2
1011
+0010
-3 1101
2’S COMPLEMENT

OVERFLOW CONDITIONS
CARRY-IN  CARRY-OUT
0111 1000
5 0101 -5 1011
+3 +0011 -4 +1100
-8 1000 7 10111
CARRY-IN = CARRY-OUT
0000 1110
+5 0101 -2 1110
+2 +0010 -6 +1010
7 0111 -8 11000
Number Systems and Codes
36

HEXADECIMAL SYSTEM
(9F1B)16 -(4A36)16 : 16
9F1B
-
4A36
54E5
(9F1B)16 +(4A36)16 : 1 1
9F1B
+
4A36
E951
Addition
Subtraction

CODES
• DECIMAL CODES
• GRAY CODE
• ASCII CODE

IN HISTORY: CODE BREAKERS
• ALAN TURING, WHO CRACKED NAZI CODE TO WIN WORLD
WAR II. TURING'S ELECTRO-MECHANICAL MACHINE, A
FORERUNNER OF MODERN COMPUTERS, UNRAVELED THE
ENIGMA CODE USED BY NAZI GERMANY AND HELPED GIVE
THE ALLIES AN ADVANTAGE IN THE NAVAL STRUGGLE FOR
CONTROL OF THE ATLANTIC.
39

CODES
• WHAT ARE THE USES OF CODES?
SIMPLY PUT, CODING IS USED FOR COMMUNICATING WITH
COMPUTERS. PEOPLE USE CODING TO GIVE COMPUTERS AND
OTHER MACHINES INSTRUCTIONS ON WHAT ACTIONS TO
PERFORM. FURTHER, WE USE IT TO PROGRAM THE WEBSITES,
APPS, AND OTHER TECHNOLOGIES WE INTERACT WITH EVERY
DAY.
WHAT ARE SOME CODES WE USE EVERYDAY??
EX. BRAILLE, MORSE CODE, BARD CODES, POST CODES,
TELEPHONE NUMBERS

DECIMAL CODES
Decimal Digit BCD Excess-3 2421
8421
0 0000 0011 0000
1 0001 0100 0001
2 0010 0101 0010
3 0011 0110 0011
4 0100 0111 0100
5 0101 1000 1011
6 0110 1001 1100
7 0111 1010 1101
8 1000 1011 1110
9 1001 1100 1111

BCD CODE
BCD was commonly used for displaying
alpha-numeric in the past but in modern-
day BCD is still used with real-time clocks
or RTC chips to keep track of wall-clock
time and it's becoming more common for
embedded microprocessors to include an
RTC.

GRAY CODE
Decimal Equivalent Binary Code Gray Code
0 0000 0000
1 0001 0001
2 0010 0011
3 0011 0010
4 0100 0110
5 0101 0111
6 0110 0101
7 0111 0100
8 1000 1100
9 1001 1101
10 1010 1111
11 1011 1110
12 1100 1010
13 1101 1011
14 1110 1001
15 1111 1000

GRAY CODE
Gray codes are widely used to prevent
spurious output from electromechanical
switches and to facilitate error correction in
digital communications such as digital
terrestrial television and some cable TV
systems.

ASCII CODE
• ASCII: AMERICAN STANDARD CODE FOR
INFORMATION INTERCHANGE.
• USED TO REPRESENT CHARACTERS AND
TEXTUAL INFORMATION
• EACH CHARACTER IS REPRESENTED WITH 1
BYTE
• UPPER AND LOWER CASE LETTERS: A..Z AND A..Z
• DECIMAL DIGITS -- 0,1,…,9
• PUNCTUATION CHARACTERS -- ; , . :
• SPECIAL CHARACTERS --$ & @ / {
• CONTROL CHARACTERS -- CARRIAGE RETURN (CR) ,
LINE FEED (LF), BEEP

ASCII CODE

END TASK
• TASK (ASSIGNMENT):
USING THE ASCII CODE CONVERSION TABLE, DECODE THE
FOLLOWING DIGITS SET:
067 101 098 117 032 084 101 099 104 110 111 108 111
103 105 099 097 108 032 085 110 105 118 101 114 115
105 116 121.
WHAT IS THE TEXT EQUIVALENT? WHAT IS THE BINARY VALUE
OF THIS CHARACTER SET?

NUMBER SYSTEMS.ppt

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