Number Systems:
A number system is a format for representing
different numerical values
Any number system calculates the value of a
number being represented by
1)
The digit used to represent
and
2)
The place where the digits
are present in the actual number.
How the value of the number is calculated
also depends on the base of the number system.
Definition of Base of a number system:
Base of a number system is the number of digits used to represent the number in
that number system. The base is also called radix
We use 10 digits to represent the numbers in
the decimal number system, so the base of the decimal number system is 10.
In case a number XYZ *where X ,Y, and Z are digits are represented
in some number system, which has a base “B”, its value would be calculated by
the expression: X x B^{2} + Y x B + Z
Decimal Number System:
The system generally used by the convention
is the decimal number system. The base of the decimal number system is 10(ten),
and the digits used are from 0(zero) to 9(nine)
In the decimal number system, a number can be
represented as
47218= 4 x 10^{4} + 7 x 10^{3}
+ 2 x 10^{2} + 1 x 10^{1} + 8 x 10^{0}
= 40000 + 7000 + 200 +10 +8
Binary Number System:
The digits used for the binary system would
be 0 and 1. We use only 2 digits in the binary number system, so the base of
the binary number system is 2.
The binary digits o and 1 are called bits.
In the binary number system, zero is
represented by 0, and 1 is represented by 1. After 1, there is no any digit to
represent two. Therefore, two is written is as 10, three written as 11. Again
the number four is represented as 100.
For a number represented as 10101 in the
binary system would have a value of
(1 x 2^{0}) + (0 x 2^{1}) + (
1 x 2^{2}) + (0 x 2^{3 }) + ( 1 x 2^{4)} which amounts to 1 +0 + 4 + +0 + 16 =21
Octal Number System:
The digits use for the octal system would be
0 through 7. The number of digits used in this system is 8, so base of the
octal number system is 8.
The number 8 of the decimal number system is
represented by 10, 9 by 11, and so on.
For example , a number represented as 123 in
the octal number system would have a value of 1 x 8^{2} + 2 x 8^{1}
+ 3 x 8^{0} which amounts to 64
+ 16 + 3=83 as represented in the decimal system.
Hexadecimal Number
System:
The base of the hexadecimal number system is
16. The ten digits from 0 to 9 and letters from A to F are used to represent
numbers in this system.
The digits 0 to 9 are same as those for the
decimal number system, and digits from 10 to 15 are represented by alphabets A
to F respectively.
For example, a number represented as 6A9 in the hexadecimal number system would
have a value of 6 x 16^{2} + A^{ }x^{ }16^{1 }+
9 x 16^{0 }which amounts to 6 x 256 + 10 x 16 + 9 = 1705 as represented
in the decimal number system.
Conversion of Base 10 to
Base 2 :
Conversion of Decimal Number to Binary Number:
In case a number ABC (with A, B, and C as the
digits) represented in the base 10 system, is to be converted to a binary
system, then it has to be repeatedly divided by 2. The remainders obtained on
each division provide the digits that would appear in the binary system.
Thus a number 25 can be repeatedly divided by
2 as follows.
Number

/

Base

Quotient

Remainder

25

/

2

12

1

12

/

2

6

0

6

/

2

3

0

3

/

2

1

1

1

/

2

0

1

Repeated division until we get the remainder
0, gives the remainders .1,0,0,1 and 1 in that order. On writing these in the
reverse order, we get the binary number 11001.
Example 2: Convert 125_{(10)} to equivalent binary
number.
2  125
2  62  1
2  31 0
2  15  1
2 
7  1
2 
3  1
2 
1  1
0  1
The
binary equivalent of 125_{(10)} is 1111101_{(2)}
Conversion
of a Decimal Fraction to Binary Fraction:
The
conversion of a decimal fraction to a binary fraction can be done using the
successive multiplication by 2 techniques.
Following
example will explain the procedure to convert a decimal fraction to a binary
fraction:
Fraction

Fraction
x2

Remainder
Fraction

Integer

0.25

0.5

0.5

0

0.5

1

0

1

0.25_{(10)}
= 0.01
Conversion of A Binary Number System To Decimal Number System
To
convert binary number to decimal number, follow the procedure:
Example :
Convert 11001_{(2)} to equivalent decimal number
Solution
: 11001 = 1 x 2^{4} + 1 x 2^{3} + 0 x 2^{2} + 0 x2^{1} + 1
=16 + 8 + 0 + 0 + 1
=25
Example 2:
Convert 10.101 to equivalent decimal number
Solution
: 10.101_{(2)} = 1 x 2 + 0 x 2 + 1 x 2^{1} + 0 x 2^{2}
+ 1^{ }x 2^{3}
^{ } = 1 x 2 + 0 x 2+ 1/2 + 0/4 + 1/8
= 2 + 0 +0.5 +0
+0.125
= 3.625
Conversion
of a decimal number to octal number :
How to convert a number from base 10 to base 8 :
In
case a number ABC (with A, B, and C as the digits) represented in the base 10
system, is to be converted to an octal system, then it has to be divided by 8
repeatedly. The remainders obtained on each division provide the digits that
would appear in the octal number.
Number

/

Base

Quotient

Remainder

205

/

8

25

5

25

/

8

3

1

3

/

2

1

1

We have to repeat the division until we get
the remainder zero. The remainders obtained are 5,1, and 1. On writing these in
the reverse order, we get the octal number 115.
Example 2:
Convert 463_{(10)} to equivalent octal number.
8  463
8  57 7
8 
7 1
0  7
The octal
equivalent of 463_{(10)} is 717_{(8)}
Conversion
of octal number to decimal number :
Example 1:Convert 716_{(8)} to
equivalent decimal number
Solution: 716_{(8)} = 7 x 8^{2} + 1 x 8^{1}
+ 6 x 8^{0}
= 7 x 64 + 1 x 8 + 6
= 448 + 8 + 6 = 462
Example 2 : Convert 65.24 to equivalent
decimal number
Solution
: 65.24_{(8)} = 6 x 8 + 5 +
2/8 + 4/8^{2}
=48 + 5 +0.25+0.0625
= 53.3125
Conversion of a decimal
number to Hexadecimal Number:
In
case a number ABC (with A,B, and C as the digits) represented in the base 10
system, is to be converted to an octal system, then it has to be divided by 16
repeatedly. The remainders obtained on each division provide the digits that
would appear in the octal number.
Number

/

Base

Quotient

Remainder

410

/

16

25

10

25

/

16

1

9

9

/

16

0

9

We have to repeat the division until we get
the remainder zero. The remainders obtained are 10, 9, and 9.
In the hexadecimal number system, the
equivalent number for 10 is A.
On writing these in the reverse order, we get
the octal number 99A.
Example 2: Convert 8655_{(10)}
to equivalent decimal number.
16  8655
16  540 15
16 
33 12
16 
2  1
0  2
The
hexadecimal equivalents to 12 and 15 are C, F
The
octal equivalent of 8655_{(10)} is 21CF_{(8)}
Conversion
of the Hexadecimal number to Decimal Number:
To
convert a hexadecimal number to a decimal number, multiply each digit of the
given hexadecimal number by powers of 16 as follows :
Example: Convert 90BA(16) to equivalent
decimal number
Solution
: 90BA(16) = 9 x 16^{3} + 0 x 16^{2}
+ B x 16^{1} + A x 16^{0}
= 9 x 4096 + 0 x
256 + 11 x 16 +10 x 1
=36864 + 0 + 176
+10 =37050