Scaling Up of E-Msr Codes Based Distributed Storage Systems with Fixed Number of Redundancy Nodes
NUMBER SYSTEMS AND CODES
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Transcript of NUMBER SYSTEMS AND CODES
NUMBER SYSTEMS AND
CODES
CS 3402--Digital Logic Number Systems and Codes 2
Outline
• Number systems– Number notations– Arithmetic– Base conversions – Signed number representation
• Codes– Decimal codes– Gray code– Error detection code– ASCII code
CS 3402--Digital Logic Number Systems and Codes 3
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 (+, , , ).
CS 3402--Digital Logic Number Systems and Codes 4
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.
CS 3402--Digital Logic Number Systems and Codes 5
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
CS 3402--Digital Logic Number Systems and Codes 6
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.
rmnnr aaaaaaaN 210121 .
CS 3402--Digital Logic Number Systems and Codes 7
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.
mm
nn
nnr rararararararaN
22
11
00
11
22
11
CS 3402--Digital Logic Number Systems and Codes 8
Examples
In binary number system
In octal number system
In hexadecimal number system
321012342 2121202021202121011.11010
3210128 848281838786124.673
101216 16166160163.306 DD
CS 3402--Digital Logic Number Systems and Codes 9
Arithmetic
(101101)2 +(11101)2 : 1111 1
+ 101101
11101
1001010
Addition:
In binary number system,
CS 3402--Digital Logic Number Systems and Codes 10
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,
CS 3402--Digital Logic Number Systems and Codes 11
Subtraction
(101101)2 -(11011)2 : 10 10
- 101101
11011
10010
In binary number system,
CS 3402--Digital Logic Number Systems and Codes 12
Subtraction
In octal number system,
In hexadecimal number system,
(6254)8 -(5173)8 : 8
- 6254
5173
1061
(9F1B)16 -(4A36)16 : 16
- 9F1B
4A36
54E5
CS 3402--Digital Logic Number Systems and Codes 13
Multiplication
(1101)2 (1001)2 :
1101
1001
1101
0000
0000
1101
1110101
In binary number system,
CS 3402--Digital Logic Number Systems and Codes 14
Division
(1110111)2 (1001)2 :
1101
1001 1110111
1001
1011
1001
1011
1001
10
In binary number system,
CS 3402--Digital Logic Number Systems and Codes 15
Base Conversions
Convert (100111010)2 to base 8
8
012
01112
0123456782
472
828784
8281828484
202120212121202021100111010
8
22
472274
010111100100111010
CS 3402--Digital Logic Number Systems and Codes 16
Base Conversion
Convert (100111010)2 to base 10
10
1010101010
0123456782
314
281632256
202120212121202021100111010
CS 3402--Digital Logic Number Systems and Codes 17
Base Conversion
Convert (100111010)2 to base 16
16
012
00112
0123456782
13
16163161
162168161162161
202120212121202021100111010
A
A
16
22
1331
101000110001100111010
AA
CS 3402--Digital Logic Number Systems and Codes 18
Base Conversion from base 8
• Convert (372)8 to base 2
• Convert (372)8 to base 10
• Convert (372)8 to base 16
2
8
11111010010111011
273372
10
101010
0128
250
256192
828783372
16
28 10101111372
FAAF
CS 3402--Digital Logic Number Systems and Codes 19
Base Conversion from base 16
• Convert (9F2)16 to base 2
• Convert (9F2)16 to base 8
• Convert (9F2)16 to base 10
2
16
101001111100001011111001
2929
FF
82
16
4762)2
0106
1107
1114
100(001011111001
2929
FF
10
101010
01216
2546
22402304
1621616929
FF
CS 3402--Digital Logic Number Systems and Codes 20
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.
CS 3402--Digital Logic Number Systems and Codes 21
Convert Base 10 to Base r
Convert (174)10 to base 8
Therefore (174)10 = (256)8
8 1 7 4 6 LSB
8 2 1 5
8 2 2 MSB
0 0
CS 3402--Digital Logic Number Systems and Codes 22
Convert Base 10 to Base r
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
CS 3402--Digital Logic Number Systems and Codes 23
Convert Base 10 to Base r
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
CS 3402--Digital Logic Number Systems and Codes 24
Signed Number Representation
There are 3 systems to represent signed numbers in binary number system:– Signed-magnitude– 1's complement– 2's complement
CS 3402--Digital Logic Number Systems and Codes 25
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.
CS 3402--Digital Logic Number Systems and Codes 26
Signed-magnitude system
CS 3402--Digital Logic Number Systems and Codes 27
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
.)12( NN n
CS 3402--Digital Logic Number Systems and Codes 28
Example of 1's Complement
For example in a 4-bit system, 0101 represents +5 and
1010 represents 5
1010
01011111
0101)000110000(0101000124
CS 3402--Digital Logic Number Systems and Codes 29
1's Complement system
CS 3402--Digital Logic Number Systems and Codes 30
2's Complement system
• 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 NN n
CS 3402--Digital Logic Number Systems and Codes 31
Example of 2's Complement
For example in a 4-bit system, 0101 represents +5 and
1011 represents 5
1011
010110000010124
CS 3402--Digital Logic Number Systems and Codes 32
2's Complement system
CS 3402--Digital Logic Number Systems and Codes 33
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
CS 3402--Digital Logic Number Systems and Codes 34
Addition and Subtraction in 1’s Complement
(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
CS 3402--Digital Logic Number Systems and Codes 35
Addition and Subtraction in2’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
CS 3402--Digital Logic Number Systems and Codes 36
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
CS 3402--Digital Logic Number Systems and Codes 37
Addition and Subtraction inHexadecimal System
(9F1B)16 -(4A36)16 : 16 9F1B
- 4A36 54E5
(9F1B)16 +(4A36)16 : 1 1 9F1B + 4A36
E951
Addition
Subtraction
CS 3402--Digital Logic Number Systems and Codes 38
Codes
• Decimal codes
• Gray code
• Error detection code
• ASCII code
CS 3402--Digital Logic Number Systems and Codes 39
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
CS 3402--Digital Logic Number Systems and Codes 40
Gray CodeDecimal 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
CS 3402--Digital Logic Number Systems and Codes 41
Error detection code Parity Bit (odd) Message
1 0000
0 0001
0 0010
1 0011
0 0100
1 0101
1 0110
0 0111
0 1000
1 1001
1 1010
0 1011
1 1100
0 1101
0 1110
1 1111
CS 3402--Digital Logic Number Systems and Codes 42
Error detection code Parity Bit (even) Message
0 0000
1 0001
1 0010
0 0011
1 0100
0 0101
0 0110
1 0111
1 1000
0 1001
0 1010
1 1011
0 1100
1 1101
1 1110
0 1111
CS 3402--Digital Logic Number Systems and Codes 43
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
CS 3402--Digital Logic Number Systems and Codes 44
Assignment 1
Page 74– 1.1: Only A+B and AB (a), (c), (f), and (g)– 1.2: Only A+B and AB (a), (c)– 1.3: Only A+B and AB (a), (c)– 1.4: (a), (c), (e)– 1.5: (a), (c), (e)– 1.6: (a), (e)– 1.7: (a), (b)– 1.8: (a), (b)– 1.10: (a), (c)– 1.11: (a), (c)– 1.12: (a), (c)– 1.13: (a), (b)