1. Chapter 3 Data Representation
Dr. Bernard Chen Ph.D.
University of Central Arkansas
Spring 2009

2. Data Types
The data types stored in digital computers may be classified as being one of the following categories:
numbers used in arithmetic computations,
letters of the alphabet used in data processing, and
other discrete symbols used for specific purposes.
All types of data are represented in computers in binary-coded form.

3. 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.

4. Radix representation of numbers
Example: convert the following number to the radix 10 format
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
= 9x x x x x x x10-2

5. 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

6. Example
Use radix representation to convert the binary number (101.01) into decimal.
The position value is power of 2
    
/22 = (101.01)2  (5.25)10
= x x x x 2-2 6

7. Binary Addition Example Add (11110)2 to (10111)2 1 1 1 1 1 1 carries 1 1 1
carry
(111101)2 + (10111) 2 = ( )2

8. Binary Subtraction 1+1=2 1 10 borrows 0 10 10 0 0 10 1 0 0 1 1 0 1
We can also perform subtraction (with borrows).
Example: subtract (10111) from ( )
1+1=2
borrows
( )2 - (10111)2 = ( )2

9. The Growth of Binary Numbers
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 9

10. 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 + 2x x8-2 10

11. Counting in Octal 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 20 21 22 23 24 25 26 27 11

12. Conversion Between Number Bases
Octal(base 8)
Decimal(base 10)
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 12

13. Converting an Integer from Decimal to Another Base
For each digit position:
Divide the decimal number by the base (e.g. 2)
The remainder is the lowest-order digit
Repeat the first two steps until no divisor remains.
For binary the even number has no remainder ‘0’, while the odd has ‘1’ 13

14. Converting an Integer from Decimal to Another Base
Quotient
Remainder
Coefficient
Example for (13)10:
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
Answer (13)10 = (a3 a2 a1 a0)2 = (1101)2

15. Converting a Fraction from Decimal to Another Base
For each digit position:
Multiply decimal number by the base (e.g. 2)
The integer is the highest-order digit
Repeat the first two steps until fraction becomes zero.

16. Converting a Fraction from Decimal to Another Base
Example for (0.625)10:
Integer
Fraction
Coefficient
0.625 x 2 = a-1 = 1
0.250 x 2 = a-2 = 0
0.500 x 2 = a-3 = 1
Answer (0.625)10 = (0.a-1 a-2 a-3 )2 = (0.101)2

17. 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 and
Remove 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 part
and remove the integer. Arrange the sequence of integers left to right from the period. (Most significant fraction bit first)

18. . (Example) (41.6875)10 ® (?)2 Integer = 41, Fraction = 0.6875 Integer
Overflow
Fraction
X by 2
.6875 1
.3750
.750 .5
Integer
remainder /2 1 20 10 5 2
Closer to
the point .
The first procedure produces
41 =
= 1 x x x x x = (101001)

19. Converting an Integer from Decimal to Octal
For each digit position:
Divide decimal number by the base (8)
The remainder is the lowest-order digit
Repeat first two steps until no divisor remains.

20. Converting an Integer from Decimal to Octal
Example for (175)10:
Integer
Quotient
Remainder
Coefficient
175/8 = / a0 = 7
21/8 = / a1 = 5
2/8 = / a2 = 2
Answer (175)10 = (a2 a1 a0)2 = (257)8

21. Converting an Integer from Decimal to Octal
For each digit position:
Multiply decimal number by the base (e.g. 8)
The integer is the highest-order digit
Repeat first two steps until fraction becomes zero.

22. Converting an Integer from Decimal to Octal
Example for (0.3125)10:
Integer
Fraction
Coefficient
x 8 = a-1 = 2
x 8 = a-2 = 4
Answer (0.3125)10 = (0.24)8
Combine the two ( )10 = (257.24)8
Remainder of division
Overflow of multiplication

23. 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 = 3x x x x160 =
Hexadecimal with fractions:
(2D3.5)16 = 2x x x x16-1 =
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

24. Example
Convert the decimal number ( )10 into hexadecimal number.
( )10  (6B.01)16
Overflow
Fraction
X by 16
.0625 1
.0000
Integer
remainder
107
Divide/16 6
11=B
Closer to
the period .

25. 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

26. Converting between Base 16 and Base 2
3A9F16 = 3 A 9 F
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!

27. Converting between Base 16 and Base 8
3A9F16 = 3 A 9 F = 3 5 2 3 7
Convert from Base 16 to Base 2
Regroup bits into groups of three starting from right
Ignore leading zeros
Each group of three bits forms an octal digit (8 is represented by 3 binary bits).

28. Example Convert 101011110110011 to a. octal number
b.      hexadecimal number
a.       Each 3 bits are converted to octal :
(101) (011) (110) (110) (011)
    
= (53663)8
b.      Each 4 bits are converted to hexadecimal:
(0101) (0111) (1011) (0011)
    B
= (57B3)16
Conversion from binary to hexadecimal is similar except that the bits divided into groups of four.

29. Binary Coded Decimal
Binary coded decimal (BCD) represents each decimal digit with four bits
Ex =
This is NOT the same as
Why use binary coded decimal? Because people think in decimal. 3 2 9
Digit
BCD Code
0000 5
0101 1
0001 6
0110 2
0010 7
0111 3
0011 8
1000 4
0100 9
1001 29

30. 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 30

31. 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
0000 1
0001 2
0010
0011 3 4
0100
0110 5
0101
0111 6 7 8
1000
1100 9
1001
1101 10
1010
1111 11
1011
1110 12 13 14 15 31

32. 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 41 65
101 B 42 66
102 C 43 67
103 … Z a 1 ‘ 32