CHAPTER – 03
DATA REPRESENTATION
Prepared By:
Abhilash Kumar Thakre
PGT (Computer Science)
Introduction:
Number system have been introduced to represent the
digits in electronic form. These number system are
acceptable to all type of machine. They are as follows-
‡ Decimal.
‡ Binary
‡ Octal
‡ Hexadecimal
Common Number Systems
Used by Used in
System Base Symbols humans? computers?
Decimal 10 0, 1, … 9 Yes No
Binary 2 0, 1 No Yes
Octal 8 0, 1, … 7 No No
Hexadecimal 16 0, 1, … 9,A, B, … F No No
Decimal Number System:
It is composed from 10 numerals or symbols. It is also known as
base 10 system because it has 10 digits. It use
0,1,2,3,4,5,6,7,8,9,10 symbols. Decimal Point
103 102 101 100 10-1 10-2 10-3
2 1 7 8 0 2 8
Binary Number System:
It is very difficult to handle 10 voltage level in decimal number
system and convenient implementation of digital system needed
minimum level of voltage. Base 2 or binary system has 2 (0,1)
voltage level and it express the weight as a power of 2.
Decimal Point
23 22 21 20 2-1 2-2 2-3
0 1 1 1 0 1 0
Octal Number System:
It is composed from 8 numerals or symbols. It is also known as base
8 system because it has 10 digits. It use 0,1,2,3,4,5,6,7 symbols.
A B C OCT
0 0 0 0
0 0 1 1
0 1 0 2
0 1 1 3
1 0 0 4
1 0 1 5
1 1 0 6
1 1 1 7
Hexadecimal Number System: A B C D HEX
It is composed from 16 numerals or symbols. 0 0 0 0 0
It is also known as base 16 system because it 0 0 0 1 1
has 10 digits. It use 0 0 1 0 2
0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 symbols. 0 0 1 1 3
0 1 0 0 4
0 1 0 1 5
0 1 1 0 6
0 1 1 1 7
1 0 0 0 8
1 0 0 1 9
1 0 1 0 A(10)
1 0 1 1 B(11)
1 1 0 0 C(12)
1 1 0 1 D(13)
1 1 1 0 E(14)
1 1 1 1 F(15)
Conversion Among Bases
• The possibilities:
Decimal Octal
Binary Hexadecimal
Quick Example
2510 = 110012 = 318 = 1916
Base
Decimal to Decimal (just for fun)
Decimal Octal
Binary Hexadecimal
Weight
12510 => 5 x 100 = 5
2 x 101 = 20
1 x 102 = 100
125
Base
Decimal to Binary
Decimal Octal
Binary Hexadecimal
Decimal to Binary
Technique
Divide by two, keep track of the remainder
First remainder is bit 0 (LSB, least-significant bit)
Second remainder is bit 1
Etc.
Example
12510 = ?2 2 125
2 62 1
2 31 0
2 15 1
2 7 1
2 3 1
2 1 1
0 1
12510 = 11111012
Decimal to Octal
Decimal Octal
Binary Hexadecimal
Decimal to Octal
Technique
Divide by 8
Keep track of the remainder
Example
123410 = ?8
8 1234
8 154 2
8 19 2
8 2 3
0 2
123410 = 23228
Decimal to Hexadecimal
Decimal Octal
Binary Hexadecimal
Decimal to Hexadecimal
Technique
Divide by 16
Keep track of the remainder
Example
123410 = ?16
16 1234
16 77 2
16 4 13 = D
0 4
123410 = 4D216
Binary to Octal
Decimal Octal
Binary Hexadecimal
Binary to Octal
Technique
Group bits in threes, starting on right
Convert to octal digits
Example
10110101112 = ?8
1 011 010 111
1 3 2 7
10110101112 = 13278
Binary to Decimal
Decimal Octal
Binary Hexadecimal
Binary to Decimal
Technique
Multiply each bit by 2n, where n is the “weight” of the bit
The weight is the position of the bit, starting from 0 on the right
Add the results
Example
Bit “0”
1010112 => 1 x 20 = 1
1 x 21 = 2
0 x 22 = 0
1 x 23 = 8
0 x 24 = 0
1 x 25 = 32
4310
Binary to Hexadecimal
Decimal Octal
Binary Hexadecimal
Binary to Hexadecimal
Technique
Group bits in fours, starting on right
Convert to hexadecimal digits
Example
10101110112 = ?16
10 1011 1011
2 B B
10101110112 = 2BB16
Octal to Decimal
Decimal Octal
Binary Hexadecimal
Octal to Decimal
Technique
Multiply each bit by 8n, where n is the “weight” of the bit
The weight is the position of the bit, starting from 0 on the right
Add the results
Example
7248 => 4 x 80 = 4
2 x 81 = 16
7 x 82 = 448
46810
Hexadecimal to Decimal
Decimal Octal
Binary Hexadecimal
Hexadecimal to Decimal
Technique
Multiply each bit by 16n, where n is the “weight” of the bit
The weight is the position of the bit, starting from 0 on the right
Add the results
Example
ABC16 => C x 160 = 12 x 1 = 12
B x 161 = 11 x 16 = 176
A x 162 = 10 x 256 = 2560
274810
Octal to Binary
Decimal Octal
Binary Hexadecimal
Octal to Binary
Technique
Convert each octal digit to a 3-bit equivalent binary
representation
Example
7058 = ?2
7 0 5
111 000 101
7058 = 1110001012
Hexadecimal to Binary
Decimal Octal
Binary Hexadecimal
Hexadecimal to Binary
Technique
Convert each hexadecimal digit to a 4-bit equivalent binary
representation
Example
10AF16 = ?2
1 0 A F
0001 0000 1010 1111
10AF16 = 00010000101011112
Octal to Hexadecimal
Decimal Octal
Binary Hexadecimal
Octal to Hexadecimal
Technique
Use binary as an intermediary
Example
10768 = ?16
1 0 7 6
001 000 111 110
2 3 E
10768 = 23E16
Hexadecimal to Octal
Decimal Octal
Binary Hexadecimal
Hexadecimal to Octal
Technique
Use binary as an intermediary
Example
1F0C16 = ?8
1 F 0 C
0001 1111 0000 1100
1 7 4 1 4
1F0C16 = 174148
Exercise – Convert ...
Hexa-
Decimal Binary Octal decimal
33
1110101
703
1AF
Don’t use a calculator!
Skip answer Answer
Exercise – Convert …
Answer
Hexa-
Decimal Binary Octal decimal
33 100001 41 21
117 1110101 165 75
451 111000011 703 1C3
431 110101111 657 1AF
Binary Addition (1 of 2)
Two 1-bit values
A B A+B
0 0 0
0 1 1
1 0 1
1 1 10
“two”
pp. 36-38
Binary Addition (2 of 2)
Two n-bit values
Add individual bits
Propagate carries
E.g.,
1 1
10101 21
+ 11001 + 25
101110 46
Fractions
Decimal to decimal (just for fun)
3.14 => 4 x 10-2 = 0.04
1 x 10-1 = 0.1
3 x 100 = 3
3.14
pp. 46-50
Fractions
Binary to decimal
10.1011 => 1 x 2-4 = 0.0625
1 x 2-3 = 0.125
0 x 2-2 = 0.0
1 x 2-1 = 0.5
0 x 20 = 0.0
1 x 21 = 2.0
2.6875
pp. 46-50
Fractions
Decimal to binary
.14579
x 2
3.14579 0.29158
x 2
0.58316
x 2
1.16632
x 2
0.33264
x 2
0.66528
x 2
1.33056
11.001001... etc.
p. 50
Exercise – Convert ...
Hexa-
Decimal Binary Octal decimal
29.8
101.1101
3.07
C.82
Don’t use a calculator!
Skip answer Answer
Exercise – Convert …
Answer
Hexa-
Decimal Binary Octal decimal
29.8 11101.110011… 35.63… 1D.CC…
5.8125 101.1101 5.64 5.D
3.109375 11.000111 3.07 3.1C
12.5078125 1100.10000010 14.404 C.82
