1. Chapter 3 Data Representation
Dr. Bernard Chen Ph.D.
University of Central Arkansas
Spring 2010

2. Outline
Data Representation
Compliments

3. 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.

4. 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.

5. 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

6. 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

7. 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 7

8. 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

9. 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

10. 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 10

11. 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 11

12. 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 12

13. 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 13

14. 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’ 14

15. 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

16. 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.

17. 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

18. 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)

19. . (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)

20. 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.

21. 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

22. 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.

23. 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

24. 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

25. 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 .

26. 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

27. 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!

28. 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).

29. 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.

30. 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 30

31. 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 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

33. Outline
Data Representation
Compliments

34. Subtraction using addition
Conventional addition (using carry) is easily
implemented in digital computers.
However; subtraction by borrowing is difficult and inefficient for digital computers.
Much more efficient to implement subtraction using ADDITION OF the COMPLEMENTS of numbers.

35. Complements of numbers
(r-1 )’s Complement
Given a number N in base r having n digits,
the (r- 1)’s complement of N is defined as
(rn - 1) - N
For decimal numbers the base or r = 10 and r- 1= 9,
so the 9’s complement of N is (10n-1)-N
99999……. - N 9 9 9 9 9 -
Digit n
Digit n-1
Next digit
Next digit
First digit

36. 9’s complement Examples- 5 4 6 7
2- Find the 9’s complement of and 12389
The 9’s complement of is =
and the 9’s complement of is = 4 5 3 2 9 9 9 9 9 9 9 - 1 2 3 8 9 8 7 6 1

37. l’s complement - For binary numbers, r = 2 and r — 1 = 1,
r-1’s complement is the l’s complement.
The l’s complement of N is (2^n- 1) - N.
Bit n-1
Bit n-2
…….
Bit 1
Bit 0 1 1 1 1 1 -
Digit n
Digit n-1
Next digit
Next digit
First digit

38. l’s complement
Find r-1 complement for binary number N with four binary digits.
r-1 complement for binary means 2-1 complement or 1’s complement.
n = 4, we have 24 = (10000)2 and = (1111)2.
The l’s complement of N is (24 - 1) - N. = (1111) - N

39. l’s complement - - The complement 1’s of 1011001 is 01001101 1 1
The complement 1’s of is 1 1 1 1 1 1 1 1 1 1 - 1 1 1 1
The 1’s complement of is 1 1 1

40. r’s Complement - Given a number N in base r having n digits,
the r’s complement of N is defined as
rn - N.
For decimal numbers the base or r = 10,
so the 10’s complement of N is 10n-N.
100000……. - N 1 -
Digit n
Digit n-1
Next digit
Next digit
First digit

41. 10’s complement Examples
Find the 10’s complement of and 12389
The 10’s complement of is =
and the 10’s complement of is =
Notice that it is the same as 9’s complement + 1. 1 - 5 4 6 7 4 5 3 3 1 - 1 2 3 8 9 8 7 6 1 1

42. 2’s complement - For binary numbers, r = 2,
r’s complement is the 2’s complement.
The 2’s complement of N is 2n - N. 1 -
Digit n
Digit n-1
Next digit
Next digit
First digit

43. 2’s complement Example - - The 2’s complement of 1011001 is 0100111-
The 2’s complement of is 1 1 1 1 1 1 1 1 1 -
The 2’s complement of is 1 1 1 1 1 1 1 1

44. Fast Methods for 2’s Complement
The 2’s complement of binary number is obtained by adding 1 to the l’s complement value.
Example:
1’s complement of is (invert the 0’s and 1’s)
2’s complement of is =

45. Fast Methods for 2’s Complement
The 2’s complement can be formed by leaving all least significant 0’s and the first 1 unchanged, and then replacing l’s by 0’s and 0’s by l’s in all other higher significant bits.
Example:
The 2’s complement of is
Leave the two low-order 0’s and the first 1 unchanged, and then replacing 1’s by 0’s and 0’s by 1’s in the four most significant bits.

46. Examples
Finding the 2’s complement of ( )2
Method 1 – Simply complement each bit and then add 1 to the result.
( )2
[N] = 2’s complement = 1’s complement ( )2 +1
=( )2
Method 2 – Starting with the least significant bit, copy all the bits up to and including the first 1 bit and then complement the remaining bits.
N =
[N] = 46