Data Representation
Notes
Number systems are the technique to represent numbers in the computer system architecture, every value that you are saving or getting into/from computer memory has a defined number system.
Computer architecture supports following number systems.
Binary number system Octal number system Decimal number system Hexadecimal (hex) number system
1) Binary Number System
A Binary number system has only two digits that are 0 and 1. Every number (value) represents with 0 and 1 in this number system. The base of binary number system is 2, because it has only two digits.
2) Octal number system
Octal number system has only eight (8) digits from 0 to 7. Every number (value) represents with 0,1,2,3,4,5,6 and 7 in this number system. The base of octal number system is 8, because it has only 8 digits.
3) Decimal number system Decimal number system has only ten (10) digits from 0 to 9. Every number (value) represents with 0,1,2,3,4,5,6, 7,8 and 9 in this number system. The base of decimal number system is 10, because it has only 10 digits.
4) Hexadecimal number system
A Hexadecimal number system has sixteen (16) alphanumeric values from 0 to 9 and A to F. Every number (value) represents with 0,1,2,3,4,5,6, 7,8,9,A,B,C,D,E and F in this number system. The base of hexadecimal number system is 16, because it has 16 alphanumeric values. Here A is 10, B is 11, C is 12, D is 13, E is 14 and F is 15.
Table of the Numbers Systems with Base, Used Digits, Representation,
Number system Base Used digits Example
Binary 2 0,1 (11110000)2
Octal 8 0,1,2,3,4,5,6,7 (360)8
Decimal 10 0,1,2,3,4,5,6,7,8,9 (240)10
Hexadecimal 16 0,1,2,3,4,5,6,7,8,9, (F0)16 A, B, C, D, E, F
NUMBER CONVERSIONS :
To convert Number system from Decimal Number System to Any Other Base is quite easy; you have to follow just two steps:
A) Divide the Number (Decimal Number) by the base of target base system (in which you want to convert the number: Binary (2), octal (8) and Hexadecimal (16)). B) Write the remainder from step 1 as a Least Signification Bit (LSB) to Step last as a Most Significant Bit (MSB).
Decimal Number is Binary Number is : (12345)10 (11000000111001)2
Decimal to Octal Conversion Result
Decimal Number is : (12345)10 Octal Number is (30071)8
Decimal to Hexadecimal Conversion Result Example 1 Hexadecimal Decimal Number is : (12345)10 Number is (3039)16
For example:
a. (149)10 2 149 1 2 74 0 2 37 1 2 18 0 2 9 1 2 4 0 2 2 0 2 1 1
0
Therefore, (149)10 = (10010101)2 b. Here, (804)10 8 804 4 8 100 4 8 12 4 8 1 1
0
Therefore, (804)10 = (1444)8 c. Here, (1600)10 16 1600 0 16 100 4 16 6 6
0
Therefore, (1600)10 = (640)16 d. 5 4 3 2 1 0 Here, (100100)2 = 2 x 1 + 2 x 0 + 2 x 0 + 2 x 1 + 2 x 0 + 2 x 0 = 32 + 0 + 0 + 4 + 0 + 0 = (36)10 e. 3 2 1 0 Here, (2040)8 = 8 x 2 + 8 x 0 + 8 x 4 + 8 x 0 = 1024 + 32 = (1056)10 f. 3 2 1 0 Here, (1E0D)16 = 16 x 1 + 16 x E + 16 x 0 + 16 x D = 4096 + 256 x 14 + 0 + 1 x 13 = 4096 + 3584 + 13 = (7693)10 g. Here, (110111101)2 110 111 101 6 7 5
Therefore, (110111101)2 = (675)8 h. Here, (1001110111)2 0010 0111 0111 2 7 7
Therefore, (1001110111)2 = (277)16 i. Here, (375)8 3 7 5 011 111 101
Therefore, (375)8 = (11111101)2
j. Here, (ABC)16 A B C 1010 1011 1100
Therefore, (ABC)16 = (101010111100)2 k. Here, (555)8 5 5 5 101 101 101
= (101101101)2 0001 0110 1101 1 6 D
Therefore, (555)8 = (16D)16 l. Here, (BCA)16 B C A 1011 1100 1010
= (101111001010)16 101 111 001 010 5 7 1 2
Therefore, (BCA)16 = (5712)8
TO CONVERT DECIMAL PART: a. Here, (0.55)10 0.55 x 2 = 1.1 1 0.1 x 2 = 0.2 0 0.2 x 2 = 0.4 0 0.4 x 2 = 0.8 0 0.8 x 2 = 1.6 1 0.6 x 2 = 1.2 1
Therefore, (0.55)10 = (0.100011)2 b. Here, (234.997)10 8 234 2 8 29 5 8 3 3
0
Also 0.997 x 8 = 7.976 7 0.976 x 8 = 7.808 7 0.808 x 8 = 6.464 6 0.464 x 8 = 3.712 3 0.712 x 8 = 5.696 5 0.696 x 8 = 5.568 5
Therefore, (234.997)10 = (352.776355)2 c. Here, (689.336)10 16 689 1 16 43 11 = B 16 2 2
0 Also 0.336 x 16 = 5.376 5 0.376 x 16 = 6.016 6 0.016 x 16 = 0.256 0 0.256 x 16 = 4.096 4 0.096 x 16 = 1.536 1 0.536 x 16 = 8.576 8
Therefore, (689.336)10 = (2B1.560418)16 d. 2 1 0 -1 -2 -3 -4 Here, (101.1101)2 = 2 x 1 + 2 x 0 + 2 x 1 + 2 x 1 + 2 x 1 + 2 x 0 + 2 x 1 = 4 + 0 + 1 + 0.5 + 0.25 + 0 + 0.0625 = (5.8125)10 e. -1 -2 -3 -4 Here, (0.1042)8 = 8 x 1 + 8 x 0 + 8 x 4 + 8 x 2 = 0.125 + 0 + 0.0078125 + 0.00048828125 = (0.1333)10 f. 1 0 -1 -2 -3 Here, (FA.AEF)16 = 16 x 15 + 16 x 10 + 16 x 10 + 16 x 14 + 16 x 15 = 240 + 10 + 0.625 + 0.0546875 + 0.00366211 = 250.68335 g. Here, (101010.110111)2 101 010 110 111 5 2 6 7
Therefore, (101010.110111)2 = (52.67)8 h. Here, (10101.11011)2 0001 0101 1101 1000 1 5 D 8
Therefore, (10101.11011)2 = (15.D8)16 i. Here, (77.226)8 7 7 2 2 6 111 111 010 010 110
Therefore, (77.226)8 = (111111.010010110)2 j. Here, (0.376)8 3 7 6 011 111 110
= (011111110)2 0111 1111 7 F
Therefore, (0.376)8 = (0.7F)16 k. Here, (0.5AB)16 5 A B 0101 1010 1011
Therefore, (0.5AB)16 = (0.010110101011)2 l. Here, (0.226)16 2 2 6 0010 0010 0110
= (0.001000100110)2 001 000 100 110 1 0 4 6
Therefore, (0.226)16 = (0.1046)8
Binary Arithmetic Operations Like we perform the arithmetic operations in numerals, in the same way, we can perform addition, subtraction, multiplication and division operations on Binary numbers. Let us learn them one by one.
Binary Addition Adding two binary numbers will give us a binary number itself. It is the simplest method. Addition of two single-digit binary number is given in the table below.
Binary Numbers Addition
0 0 0
0 1 1
1 0 1
1 1 0; Carry →1 Let us take an example of two binary numbers and add them.
For example: Add 11012 and 10012.
Solution:
EXAMPLE : a. Here, 11111 + 10001 11111 + 10001
110000
b. Here, 1111 + 1111 1111 + 1111
11110
CHARACTER/STRING REPRESENTATION: ASCII: Stands for "American Standard Code for Information Interchange." ASCII is a character encoding that uses numeric codes to represent characters. These include upper and lowercase English letters, numbers, and punctuation symbols. ISCII: Indian Script Code for Information Interchange (ISCII) is a coding scheme for representing various writing systems of India. It encodes the main Indic scripts and a Roman transliteration. The supported scripts are: Assamese, Bengali (Bangla), Devanagari, Gujarati, Gurmukhi, Kannada, Malayalam, Oriya, Tamil, and Telugu. UNICODE: Unicode is a character encoding standard that has widespread acceptance. ... They store letters and other characters by assigning a number for each one. Before Unicode was invented, there were hundreds of different encoding systems for assigning these numbers. No single encoding could contain enough characters.