Lecture 04 number and information representation

Lecture No. 04
Number and Information Representation

Md. Manowarul Islam, Dept. of CSE, JnU 2
Information Representation
 Numbers are important to computers
 represent information precisely
 can be processed
 For example:
 to represent yes or no: use 0 for no and 1 for yes
 to represent 4 seasons: 0 (autumn), 1 (winter), 2(spring) and 3
(summer)
 NRIC number: a letter, 7 digits, and a check code
 matriculation number (8 alphanumeric) to represent individual
students
Md. Obaidur Rahman, Dept. of CSE, EUB

 Elementary storage units inside computer are
electronic switches. Each switch holds one of two
states: on (1) or off (0).
 We use a bit (binary digit), 0 or 1, to represent the
state.
ON OFF

 Storage units can be grouped together to cater for larger
range of numbers. Example: 2 switches to represent 4
values.
0 (00)
1 (01)
2 (10)
3 (11)

Positional Notations
 Position-independent notation
 each symbol denotes a value independent of its position:
Egyptian number system
 Relative-position notation
 Roman numerals symbols with different values: I (1), V (5), X
(10), C (50), M (100)
 Examples: I, II, III, IV, VI, VI, VII, VIII, IX
 Relative position important: IV = 4 but VI = 6
 Computations are difficult with the above two notations

 Weighted-positional notation
 Decimal number system, symbols = { 0, 1, 2, 3, …, 9 }
 Position is important
 Example:(7594)10 = (7x103) + (5x102) + (9x101) + (4x100)
 The value of each symbol is dependent on its type and its
position in the number
 In general,
(anan-1… a0)10 = (an x 10n) + (an-1 x 10n-1) + … + (a0 x 100)

 Fractions are written in decimal numbers after the
decimal point.
 (2.75)10 = (2 x 100) + (7 x 10-1) + (5 x 10-2)
 In general,
(anan-1… a0 . f1f2 … fm)10 =
(an x 10n) + (an-1x10n-1) + … + (a0 x 100) +
(f1 x 10-1) + (f2 x 10-2) + … + (fm x 10-m)
 The radix (or base) of the number system is the total
number of digits allowed in the system.

Md. Manowarul Islam, Dept. of CSE, JnU
Common Number Systems
System Base Symbols
Decimal 10 0, 1, … 9
Binary 2 0, 1
Octal 8 0, 1, … 7
Hexa-decimal 16 0, 1, … 9, A, B, … F

Md. Manowarul Islam, Dept. of CSE, JnUCS1104-2 Information 9
 The digits are consecutive.
 The number of digits is equal to the size of the base.
 Zero is always the first digit.
 The base number is never a digit.
 When 1 is added to the largest digit, a sum of zero and a
carry of one results.

Quantities/Counting
Decimal Binary Octal
Hexa-
decimal
0 0 0 0
1 1 1 1
2 10 2 2
3 11 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7

Quantities/Counting
Decimal Binary Octal
Hexa-
decimal
8 1000 10 8
9 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

 Positional Number System
 The word decimal is a derivative of decem, which is the
Latin word for ten.
 Number’s value = a weighted sum of the digits
 In general,
 The powers of 10 increment from 0, 1, 2, etc. as you move
right to left
Decimal Number Systems

 Positional Number System
 The word decimal is a derivative of decem, which is the
Latin word for ten.
 Number’s value = a weighted sum of the digits
 In general,
 The powers of 10 increment from 0, 1, 2, etc. as you
move right to left
Decimal Number System

 Example

 With only 2 values, can be widely separated, therefore
clearly differentiated
 Binary numbers are made of binary digits
 Binary DigiTs (BITs) can be represented
 electronically:
Binary Number System

 Base is 2 or ‘b’ or ‘B’ or ‘Bin’
 Two symbols: 0 and 1
 Each place is weighted by the power of 2
 All the information in the digital computer is represented
as bit patterns
 What is a bit pattern?
 01010101

 Look at this bit pattern
0101 0101
 How many bits are present ?
 Count the number of ones and the zeros in the above
pattern
 Answer = Total 8 bits
Bit7 Bit 6 Bit 5 Bit 4 Bit 3 Bit 2 Bit 1 Bit 0
0 1 0 1 0 1 0 1

 A single bit can represent two states:0 1
 Therefore, if you take two bits, you can use them
to represent four unique states:
00, 01, 10, & 11
 And, if you have three bits, then you can use
them to represent eight unique states:
000, 001, 010, 011, 100, 101, 110, & 111

 With every bit you add, you double the number of states
you can represent.
 Therefore, the expression for the number of states with n
bits is 2n.
 Most computers operate on information in groups of 8
bits,

0 1 0 1 0 1 0 1
 There are 8 bits in the above table
 Group of 4 bits = 1 Nibble
 Group of 8 bits = 1 Byte
 Group of 16 bits = 1 Word 2 Bytes = 1 Word

27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1
Bit positions and their values

 Base = 8 or ‘o’ or ‘Oct’
 8 symbols: { 0, 1, 2, 3, 4, 5, 6, 7}
 Example 123, 567, 7654 etc
 987 This is incorrect why?
 How to represent a Decimal Number using a Octal
Number System ?
Octal Number System

 Base = 16 or ‘H’ or ‘Hex’
 16 symbols: { 0, 1, 2, 3, 4, 5, 6, 7,8,9 }
 { 10=A, 11=B, 12=C, 13=D, 14=E, 15= F}
 Example AB12, 876F, FFFF etc
Hexadecimal Number System

Lecture 04 number and information representation

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