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Chapter 3 Data Representation
Dr. Bernard Chen Ph.D.University of Central Arkansas
Spring 2009
Data Types The data types stored in digital computers
may be classified as being one of the following categories:
1. numbers used in arithmetic computations, 2. letters of the alphabet used in data processing,
and 3. other discrete symbols used for specific
purposes.
All types of data are represented in computers in binary-coded form.
Radix representation of numbers
• Radix or base: is the total number of symbols used to represent a value. A number system of radix r uses a string consisting of r distinct symbols to represent a value.
Radix representation of numbers Example: convert the following number
to the radix 10 format. 97654.35
The positions indicate the power of the radix.
Start from the decimal point right to left we get 0,1,2,3,4 for the whole numbers.
And from the decimal point left to right We get -1, -2 for the fractions= 9x104 + 7x103 + 6x102 + 5x101 + 4x100 +
3x10-1 + 5x10-2
Binary Numbers
Binary numbers are made of binary digits (bits): 0 and 1
Convert the following to decimal (1011)2 = 1x23 + 0x22 + 1x21 + 1x20 =
(11)10
Example
Use radix representation to convert the binary number (101.01) into decimal.
The position value is power of 2
1 0 1. 0 1
22 21 20 2-1 2-2
4 + 0 + 1 + 0 + 1/22 = 5.25 (101.01)2 (5.25)10
= 1 x 22 + 0 x 2 + 1 + 0 x 2-1 + 1 x 2-2
Binary Addition
1 1 1 1 0 1+ 1 0 1 1 1---------------------
0
1
0
1
1
1111
1 1 00
carries
Example Add (11110)2 to (10111)2
(111101)2 + (10111) 2 = (1010100)2carry
Binary Subtraction We can also perform subtraction (with borrows). Example: subtract (10111) from (1001101)
1 100 10 10 0 0 10
1 0 0 1 1 0 1- 1 0 1 1 1------------------------ 0 1 1 0 1 1 0
borrows
1+1=2
(1001101)2 - (10111)2 = (0110110)2
The Growth of Binary Numbers
n 2n
0 20=1
1 21=2
2 22=4
3 23=8
4 24=16
5 25=32
6 26=64
7 27=128
n 2n
8 28=256
9 29=512
10 210=1024
11 211=2048
12 212=4096
20 220=1M
30 230=1G
40 240=1T
Mega
Giga
Tera
Octal Numbers
Octal numbers (Radix or base=8) are made of octal digits: (0,1,2,3,4,5,6,7)
How many items does an octal number represent?
Convert the following octal number to decimal(465.27)8 = 4x82 + 6x81 + 5x80 + 2x8-1 + 7x8-2
Counting in Octal
0 1 2 3 4 5 6 7
10 11 12 13 14 15 16 17
20 21 22 23 24 25 26 27
Conversion Between Number Bases
Decimal(base 10)
Octal(base 8)
Binary(base 2)
Hexadecimal
(base16)° We normally convert to base 10
because we are naturally used to the decimal number system.
° We can also convert to other number systems
Converting an Integer from Decimal to Another Base
1. Divide the decimal number by the base (e.g. 2)
2. The remainder is the lowest-order digit
3. Repeat the first two steps until no divisor remains.
4. For binary the even number has no remainder ‘0’, while the odd has ‘1’
For each digit position:
Converting an Integer from Decimal to Another Base
Example for (13)10:
IntegerQuotient
13/2 = (12+1)½ a0 = 1
6/2 = ( 6+0 )½ a1 = 0
3/2 = (2+1 )½ a2 = 1 1/2 = (0+1) ½ a3 = 1
Remainder Coefficient
Answer (13)10 = (a3 a2 a1 a0)2 = (1101)2
Converting a Fraction from Decimal to Another Base
1. Multiply decimal number by the base (e.g. 2)
2. The integer is the highest-order digit
3. Repeat the first two steps until fraction becomes zero.
For each digit position:
Converting a Fraction from Decimal to Another Base
Example for (0.625)10:
Integer
0.625 x 2 = 1 + 0.25 a-1 = 10.250 x 2 = 0 + 0.50 a-2 = 00.500 x 2 = 1 + 0 a-3 = 1
Fraction Coefficient
Answer (0.625)10 = (0.a-1 a-2 a-3 )2 = (0.101)2
DECIMAL TO BINARY CONVERSION(INTEGER+FRACTION)
(1) Separate the decimal number into integer and fraction parts.
(2) Repeatedly divide the integer part by 2 to give a quotient and a remainder andRemove the remainder. Arrange the sequence of remainders right to left from the period. (Least significant bit first)
(3) Repeatedly multiply the fraction part by 2 to give an integer and a fraction partand remove the integer. Arrange the sequence of integers left to right from the period. (Most significant fraction bit first)
(Example) (41.6875)10 (?)2
Integer = 41, Fraction = 0.6875
The first procedure produces
41 = 32+8+1
= 1 x 25 + 0 x 24 + 1 x 23 + 0 x 22 + 0 x 2 + 1 = (101001)
Integer remainder
41 /2 1
20 0
10 0
5 1
2 0
1 1
Overflow Fraction
X by 2 .6875
1 .3750
0 .750
1 .5
1 0
.Closer to
the point
Converting an Integer from Decimal to Octal
1. Divide decimal number by the base (8)
2. The remainder is the lowest-order digit
3. Repeat first two steps until no divisor remains.
For each digit position:
Converting an Integer from Decimal to Octal
Example for (175)10:
IntegerQuotient
175/8 = 21 + 7/8 a0 = 7 21/8 = 2 + 5/8 a1 = 5 2/8 = 0 + 2/8 a2 = 2
Remainder Coefficient
Answer (175)10 = (a2 a1 a0)2 = (257)8
Converting an Integer from Decimal to Octal
1. Multiply decimal number by the base (e.g. 8)
2. The integer is the highest-order digit
3. Repeat first two steps until fraction becomes zero.
For each digit position:
Converting an Integer from Decimal to Octal
Example for (0.3125)10:
Integer
0.3125 x 8 = 2 + 0.5 a-1 = 20.5000 x 8 = 4 + 0 a-2 = 4
Fraction Coefficient
Answer (0.3125)10 = (0.24)8
Combine the two (175.3125)10 = (257.24)8
Remainder of division
Overflow of multiplication
Hexadecimal Numbers Hexadecimal numbers are made of 16 symbols:
(0,1,2,3,4,5,6,7,8,9,A, B, C, D, E, F) Convert a hexadecimal number to decimal
(3A9F)16 = 3x163 + 10x162 + 9x161 + 15x160 = 1499910 Hexadecimal with fractions:
(2D3.5)16 = 2x162 + 13x161 + 3x160 + 5x16-1 = 723.312510
Note that each hexadecimal digit can be represented with four bits.
(1110) 2 = (E)16
Groups of four bits are called a nibble. (1110) 2
Example Convert the decimal number
(107.00390625)10 into hexadecimal number.
(107.00390625)10 (6B.01)16
Integer remainder
107 Divide/16
6 11=B
0 6
Overflow
Fraction
X by 16 . 00390625
0 .0625
1 .0000
.Closer to
the period
One to one comparison Binary, octal, and
hexadecimal similar Easy to build circuits
to operate on these representations
Possible to convert between the three formats
Converting between Base 16 and Base 2
° Conversion is easy!
Determine 4-bit value for each hex digit
° Note that there are 24 = 16 different values of four bits which means each 16 value is converted to four binary bits.
° Easier to read and write in hexadecimal.
° Representations are equivalent!
3A9F16 = 0011 1010 1001 11112
3 A 9 F
Converting between Base 16 and Base 8
1. Convert from Base 16 to Base 2
2. Regroup bits into groups of three starting from right
3. Ignore leading zeros
4. Each group of three bits forms an octal digit (8 is represented by 3 binary bits).
352378 = 011 101 010 011 1112
5 2 3 73
3A9F16 = 0011 1010 1001 11112
3 A 9 F
Example Convert 101011110110011 to a. octal numberb. hexadecimal number a. Each 3 bits are converted to octal :
(101) (011) (110) (110) (011)
5 3 6 6 3 101011110110011 = (53663)8
b. Each 4 bits are converted to hexadecimal:(0101) (0111) (1011) (0011)
5 7 B 3
101011110110011 = (57B3)16
Conversion from binary to hexadecimal is similar except that the bits divided into groups of four.
Binary Coded Decimal• Binary coded decimal (BCD) represents each decimal digit
with four bits
– Ex. 0011 0010 1001 = 32910
• This is NOT the same as 0011001010012
• Why use binary coded decimal? Because people think in decimal.Digit BCD Code Digit BCD Code
0 0000 5 0101
1 0001 6 0110
2 0010 7 0111
3 0011 8 1000
4 0100 9 1001
3 2 9
BCD versus other codes
° BCD not very efficient
° Used in early computers (40s, 50s)
° Used to encode numbers for seven-segment displays.
° Easier to read?
(Example)
The decimal 99 is represented by 1001 1001.
Gray Code • Gray code is not a
number system.– It is an alternative
way to represent four bit data
• Only one bit changes from one decimal digit to the next
• Useful for reducing errors in communication.
• Can be scaled to larger numbers.
Digit Binary 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
ASCII Code• American Standard Code for Information
Interchange• ASCII is a 7-bit code, frequently used with an 8th
bit for error detection (more about that in a bit).Character ASCII (bin) ASCII
(hex)Decimal Octal
A 1000001 41 65 101
B 1000010 42 66 102
C 1000011 43 67 103
…
Z
a
…
1
‘