Early computer systems used electrical switches and when
electrical switches were replaced by less mechanical devices such as vacuum
tubes, than the transistor, the integrated circuit, the concept of switching on
and off remained with computers but a representation of the on/off behavior of
computers had to be made.
The term, digital, in computing and electronics applies to converting
real world information to binary numeric form. The binary (base 2)
number system represents two discreet values using two symbols or digits i.e.,
0 and 1. The binary number system, where a zero symbolizes no electrical
current (OFF) and one represents electrical current exists (ON).
All computer data (alpha-numeric, symbols and characters, audio,
graphics and video) are represented or encoded using sequences of binary digits
that are interpreted according to appropriate software. Computers are made up
of electronic devices and electronic device can exist or either in ON or OFF
state. For our convenience an ON State is represented by the code ‘1’ and OFF
state is represented by the code‘0. They are called ‘bits’, an abbreviation for
Binary Digit. The numbers represented by bits are known as binary numbers.
Basically number systems are classified in two
types :
1)
Non –
Positional Number System
2)
Positional
Number System
Non-Positional
Number System :
In this number system any number can be represented by arranging
symbols in various positions. Each symbol represents a definite value
irrespective of the position in which they appear. For example : Roman Number
System
Positional
Number System :
In a positional number system, a number is represented by a set of
symbols. Each symbol represents a particular value, depending on its position.
The actual number of symbols used in a position system depends on its base.
Base :
The number of digits or basic symbols used in a positional number
system is known as the
BASE or Radix of the system.
1.
Decimal
Number System (Base 10) :
It is a positional number system with base 10 and thus it uses 10
symbols i.e., 0 to 9. Any number can be represented by arranging symbols in
various positions. In the decimal system, each position represents a specific
power of 10.
For example : The decimal number 654.52,
written as (654.52)10 or 654.52(10) to specify base 10 is
represented as follows :
|
Increasing Powers of 10 |
Decreasing Powers of 10 |
||||
|
|
Hundreds |
Tens |
Units |
One Tenth |
One
Hundredth |
|
Weights |
102 |
101 |
100 |
10-1 |
10-2 |
|
Digits |
6 |
5 |
4 |
5 |
2 |
Thus, the expanded notation of
654.52(10)= 6 X 102
+ 5 X 101 + 4 X 100 + 5 X 10-1 + 2 X 10-2
= 6 X 100 + 5 X 10 + 4 X 1 + 5 X
0.1 + 2 X 0.01
= 600 + 50 + 4 + 0.5 + 0.02
= 654.52
Note : In a positional number system all bits or digits to the
left of the decimal or binary point have weights that are positive powers of
base and those to the right have weights that are negative powers of base. The
base is also called Radix and fractional
point is called as Radix point.
2.
Binary
Number System (Base 2) :
The binary system is a positional system to the base 2. It uses
two symbols 0 and 1. Each position represents specific power of 2.
For example
: The binary
number 1 1 0 1 . 1
1 written as 1
1 0 1 . 1 1 (2)
or (1 1 0 1 . 1 1)2 to specify base 2, is represented as follows :
|
Increasing
Powers of 2 |
Decreasing
Powers of 2 |
|||||
|
Weights |
23 |
22 |
21 |
20 |
2-1 |
2-2 |
|
Digits |
1 |
1 |
0 |
1 |
1 |
1 |
Thus, the expanded notation of 1 1 0 1 . 1 1 (2)
= 1 X 23 + 1 X 22 + 0 X 21 + 1 X
20 + 1 X 2-1 + 1 X 2-2
3.
Octal
Number System (Base 8) :
The octal number system is a positional number system with base 8.
It uses 8 symbols i.e., 0 to 7. In octal number system, each digit
position corresponding to a power of 8.
For example : The octal number 43.12 written as (43.12)8
or 43.12(8) to specify base 8, is
represented as follows :
|
Increasing
Powers of 8 |
Decreasing
Powers of 8 |
|||
|
Weights |
81 |
80 |
8-1 |
8-2 |
|
Digits |
4 |
3 |
1 |
2 |
Thus, the expanded notation of 43.12(8)
= 4 X 81 + 3 X 80 + 1 X 8-1 + 2 X 8-2
4.
Hexadecimal
Number System (Base 16) :
The hexadecimal number system is a positional number system to the
base 16. It uses 16 symbols to represent any number. The first 10 symbols are
represented by digits 0 to 9 and the remaining 6 symbols by the letters A to F,
representing the decimal values 10 to 15 respectively. Each position represents
a specific powers of 16.
For example, the hexadecimal number BA85.12
is written as (BA85.12)16 or BA85.12(16), is represented as follows :
|
Increasing
Powers of 16 |
Decreasing
Powers of 16 |
|||||
|
Weights |
163 |
162 |
161 |
160 |
16-1 |
16-2 |
|
Digits |
B |
A |
8 |
5 |
1 |
2 |
Thus, the expanded notation of BA85.12(16)
=B X 163 + A X 162 + 8 X 161 + 5
X 160 + 1 X 16-1 + 2 X
16-2
CONVERSION OF NUMBER FROM ONE SYSTEM TO ANOTHER
INTEGER PART
1.
Divide the
integer part of the decimal number by the base ‘b’ of the new system. The
remainder will give the right most digit of the integer part of the new number.
2.
Divide the
quotient again by the base ‘b’. The remainder is the next digit from right.
3.
Repeat step
2 until a zero quotient is obtained. Last remainder is the left most digit of
the new number.
FRACTIONAL PART
1.
Multiply the
fractional part of the decimal number by the
base ‘b’ of the new system. The integer part of the product gives the left most
digit of the fractional part of the new number.
2.
Multiply the
fractional part of the product by the
base ‘b’. The integer part of the resultant product is the next digit from left.
3.
Repeat step
2 until a zero fractional part or a repeated fractional part or a
non-terminating fractional part occurs.
1.
Decimal
to Binary Conversion :
1) 25. 625 (10) = ? (2)
|
|
2 5 |
|
|
Product |
Integer |
Fractional-Part |
|
2 |
1 2 |
|
0.625 X 2 |
1.250 |
|
0.250 |
|
2 |
6 |
0 |
0.250 X 2 |
0.500 |
0 |
0.500 |
|
2 |
3 |
0 |
0.500 X 2 |
1.000 |
1 |
0.000 |
|
2 |
1 |
1 |
|
|
|
|
|
2 |
0 |
1 |
Therefore,
25(10) = 1 1 0 0 1 Thus, the
result is 25.625(10) |
& = |
0.625(10)
= 1 0 1 1 1 0 0 1
. 1 0 1(2) |
|
2.
Binary
to Decimal Conversion :
1) 1 1 . 1 0 1 1 (2) = ? (10)
1 1 . 1 0 1 1(2) = 1 X 21 + 1 X
20 + 1 X 2-1 + 0 X 2-2 + 1 X 2-3 +
1 X 2-4
= 1 X 2 + 1 X 1 + 1 X 0.5 + 0 X
0.25 + 1 X 0.125 + 1 X 0.0625
= 2 + 1 + 0.5 + 0 + 0.125 + 0.0625
= 3.6575(10)
3.
Binary
to Octal Conversion :
1) 0 1 1 1 0 1 (2) = ?(8)
|
Octal |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
|
Binary |
000 |
001 |
010 |
011 |
100 |
101 |
110 |
111 |
Note : Place the 3 binary bits into one group from right side and
write equivalent octal digit.
0 1 1 1 0 1
3 5
Thus, the result is 0 1 1 1 0 1 (2) = 3 5 (8)
4.
Binary
to Hexadecimal Conversion :
1) 01101101(2) = ?(16)
|
Hexa Decimal |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
A |
B |
C |
D |
E |
F |
|
Binary |
0000 |
0001 |
0010 |
0011 |
0100 |
0101 |
0110 |
0111 |
1000 |
1001 |
1010 |
1011 |
1100 |
1101 |
1110 |
1111 |
Note : Place the 4 binary bits into one group from right side and
write equivalent Hexadecimal digit.
0 1 1 0 1 1 0 1
6 D
Thus, the result
is 0 1 1 0 1 1 0 1 (2) =
6D (16)
5.
Octal
to Decimal Conversion :
1) 43.12(8) = ? (10)
43.12(8) = 4
X 81 + 3 X 80 + 1 X 8-1 + 2 X 8-2
= 4 X 8 + 3 X 1 + 1 X 0.125 + 2 X 0.015625
= 32 + 3 + 0.125 + 0.03125
= 35.15625 (10)
6.
Octal
to Binary Conversion :
|
Octal |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
|
Binary |
000 |
001 |
010 |
011 |
100 |
101 |
110 |
111 |
Note : Place the 3 bit binary equivalent of each digit below the
number 1) 67.35(8) = ?(2)
|
6 |
7 . |
3 5 |
|
110 |
111 . |
011 101 |
Thus the result is 67.35(8) = 110111.011101(2)
7.
Octal
to Hexadecimal Conversion :
(a)
First Covert Octal
to Decimal 43.12(8) = ? (10)
= 4 X 81 + 3 X 80 + 1 X 8-1 + 2 X
8-2
= 4 X 8 + 3 X 1 + 1 X 0.125 + 2 X 0.015625
= 32 + 3 + 0.125 + 0.031250
= 35.15625 (10)
(b)
Then
Convert Decimal to Hexadecimal
35.15625 (10) = ? (16)
|
16 |
3 5 |
|
|
Product |
Integer |
Fractional-Part |
|
16 |
2 |
3 |
0.15625 X
16 |
2.500 |
|
0.500 |
|
|
0 |
2 |
0.50000 X
16 |
8.000 |
0 |
0.000 |
Therefore, 35(10) = 2 3 & 0.15625(10) = 2 0
Thus, the result is 35.15625(10) = 2 3 . 2 0
(16)
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