Decimal system
*In the decimal system there are actually have 10 numbers
used write a number i.e 0,1,2,……,9.
*This system is used in daily life like calculators.
*This is also called base ten system.
Binary system
*This system contain only two numbers 0,1 these two numbers
are used to wirte any number.
*Basic elements……..
Binary
system
|
0
|
1
|
10
|
11
|
100
|
101
|
101
|
111
|
1000
|
1001
|
Decimal
system
|
0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
* These are remainders when we divide number by 2.so,there
are only one’s and two’s.
*This system is used in computer and electronic machines
where number comes into play.
*The binary system is useful in computer science and
electrical engineering. Transistors operate from the binary system, and
transistors are found in practically all electronic devices. A 0 means no
current, and a 1 means to allow current. With various transistors turning on
and off, signals and electricity is sent to do various things such as making a
call or putting these letters on the screen.
*Computers and electronics work with bytes or eight digit
binary numbers. Each byte has encoded information that a computer is able to
understand. Many bytes are stringed together to form digital data that can be
stored for use later.
Octal number system
This is another number system where only 8 numbers are used
that is 0,1,2,….,8.
This is called base eight system.
*Basic elements……..
Octal
system
|
0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
10
|
11
|
Decimal
system
|
0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
These numbers divide by 8 and collecting the
remainders.
Hexadecimal system
This is to base 16.
Therefore the numbers invoved over there are 0,1,2,3,4,5,6,7,8,9,A,B,C,D,F,E.
A corresponds to remainder 10 as when divided by 16,B-11,C-12,D-13,E-14,F-15.
DECIMAL
SYSTEM
|
BINARY SYSTEM
|
OCTAL SYSTEM
|
HEXADECIMAL
SYSTEM
|
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
|
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
|
16
|
10000
|
20
|
10
|
17
|
10001
|
21
|
11
|
18
|
10010
|
22
|
12
|
19
|
10011
|
23
|
13
|
20
|
10100
|
24
|
14
|
Conversion from one form to other form
Here we will take a number and I show how to convert and
reconvert
We will take 588 a three digit number.
588 in decimal system
Decimal to binary
divisor
|
quotient
|
remainders
|
2
|
588
|
|
2
|
294
|
0
|
2
|
147
|
0
|
2
|
73
|
1
|
2
|
36
|
1
|
2
|
18
|
0
|
2
|
9
|
0
|
2
|
4
|
1
|
2
|
2
|
0
|
1
|
0
|
|
Here we have taken the number 588 and first it is divided by
2 continuously until remainder is 1.
Here for binary number we have to take remainders from bottom
to top. Therefore the number is (1001001100)2.
Re conversion into decimal system
Now we are given with (1001001100)2 and we have
reconvert.
29
|
28
|
27
|
26
|
25
|
24
|
23
|
22
|
21
|
20
|
1
|
0
|
0
|
1
|
0
|
0
|
1
|
1
|
0
|
0
|
512
|
0
|
0
|
64
|
0
|
0
|
8
|
4
|
0
|
0
|
Multiply the 1st row and 2nd row
numbers correspondingly we get the 3rd row numbers.
Adding the 3rd row numbers give 588.
Conversion from decimal to octal decimal system
8
|
588
|
|
8
|
73
|
4
|
8
|
9
|
1
|
8
|
1
|
1
|
Therefore the number is (1114)8.
Re conversion into decimal system from octal decimal
system
This of octal number system to decimal system.
83
|
82
|
81
|
80
|
1
|
1
|
1
|
4
|
512
|
64
|
8
|
4
|
*Adding the number in the 3rd row we get 588
Conversion from decimal to hexadecimal system
16
|
588
|
|
16
|
36
|
C(12)
|
2
|
4
|
Therefore
the number in hexadecimal system is (24C)16.
hexadecimal number
system to decimal system
162
|
161
|
160
|
2
|
4
|
C(12)
|
512
|
64
|
12
|
*Adding the numbers in 3rd row we get 588
