Number System

1. Decimal Number System

It is used by humans
It uses 10 different digits from 0 to 9 to represent numbers.
Decimal number system has base 10
Each place to the left is ten times greater than the place to
its right which means each place represent a specific
power of the base (10).

Following figure represents the Integer and fractional part of

decimal number 1253.76 alongwith calculation of the decimal
number using positional values

Positional Value Representation of Decimal Number

2. Binary Number System

It is used by Computer and other Digital devices.
It uses two different digits 0 and 1 called as bit to represent
any information
Binary number system has base 2
Each place in a binary code represents a specific power of
the base (2).
Binary Number System is also referred as Machine
Language.
Application of Binary Number System
Used as Machine Language for Computer Hardware and
other Electronic Devices.
Used in ASCII Code and Unicode
Used in Internet Protocols.
Used in Image processing, High-end audio recordings, HD
Videos recordings and data processing and storage.
Used in Boolean algebra that is a branch of Mathematics.
 Number System

3. Octal Number System

It is used to represent large and complex binary codes
into concise manner.
It uses 8 different digits from 0 to 7 to represent
information.
Octal Number System has base 8.
Each place in Octal code represents a specific power of
the base (8).
Applications of Octal Number System
It is used in register and flip flop of Computer
Motherboards.
It is used in Mainframe such as UNIVAC, PDP etc.

4. Hexadecimal Number System

It is used to represent large and complex binary codes
into concise manner.
It uses 16 different literals which are
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F to represent information.
Hexadecimal Number System has base 16.
Each place in a Hexadecimal code represents a specific
power of the base (16).
Applications of Hexadecimal Number System
It is used to represent Memory Address for storage
devices used in computers.
It is used to describe colors in the Web Pages.
It is used to represent Media Access Control (MAC)
addresses.
It is used to display error messages.
 Conversion Between Number System
1. Decimal to Binary Conversion
Following are the steps to convert Decimal Number System to
Binary Number System:
Divide the number by the base value of the Binary Number
System i.e. 2 you want to convert it from
Note the remainder
Keep dividing the quotient by the base value 2 untill it
becomes zero and note the remainder of each division
Write all noted remainders in reverse order (bottom to
top)
Example 1: Convert (45) to Binary Number.
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Example 2: Convert (234) to Binary Number

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 Conversion Between Number System
2. Decimal to Octal Conversion
Following are the steps to convert Decimal Number System to
Octal Number System:
Divide the number by the base value of the Octal Number
System i.e. 8 you want to convert it from
Note the remainder
Keep dividing the quotient by the base value 8 until it
becomes zero and note the remainder of each division
Write all noted remainders in reverse order (bottom to
top)
Example 1: Convert (186) to Octal Number.
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Example 2: Convert (1683) to Octal Number.

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 Conversion Between Number System
3. Decimal to Hexadecimal Conversion
Following are the steps to convert Decimal Number System to
Hexadecimal Number System:
Divide the number by the base value of the Hexadecimal
Number System i.e. 16 you want to convert it from
Note the remainder
Keep dividing the quotient by the base value 16 untill it
becomes zero and note the remainder of each division
Write all noted remainders in reverse order (bottom to
top)
Example 1: Convert (763) to Hexadecimal Number.
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Example 2: Convert (2940) to Hexadecimal Number.

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 Conversion Between Number System
4. Binary to decimal Conversion
Following are the steps for converting Binary to Decimal
number:
1. Write down binary number
2. Starting from right to left, Write position number of each
bit of given binary code
3. Find positional value of each bit by raising its position
number as power to the base 2
4. Multiply each bit with its positional value and find its
corresponding decimal number
5. Add all these decimal numbers to find equivalent Decimal
number of given Binary Number.

Example 1: Convert (10011) to Decimal Number

2

5. Octal to decimal Conversion

Following are the steps for converting octal to decimal
number:
1. Write down octal number
2. Starting from right to left, Write position number of each
digit of given octal code
3. Find positional value of each bit by raising its position
number as power to the base 8
4. Multiply each bit with its positional value and find its
corresponding decimal number
5. Add all these decimal numbers to find equivalent Decimal
number of given Octal Number.
 Conversion Between Number System
Example 1: Convert (336) to Decimal Number
8

6. Hexadecimal to decimal Conversion

Following are the steps for converting Hexadecimal to
Decimal number:
1. Write down Hexadecimal number
2. Starting from right to left, Write position number of each
symbol (digit or character) of given hexadecimal code
3. Find positional value of each symbol (digit or character)
by raising its position number as power to the base 16
4. Multiply each bit with its positional value and find its
corresponding decimal number
5. Add all these decimal numbers to find equivalent Decimal
number of given Hexadecimal Number.

Example 1: Convert (7AC) to Decimal Number

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Number Table
 Conversion Between Number System
Binary to Octal Conversion
1. Write Binary Number
2. From right to left, group all the bits of binary in the set of
three
3. Add 0 to the left of the last bit of given binary number
incase set of 3 bit is not formed
4. Write octal no for each corresponding set of 3 bits
5. Group these octal digits together to form equivalent octal
number
Example 1: Convert (100110101) in Octal Number
2

Octal to Binary Conversion

1. Write octal number
2. Replace each digit of octal number with its equivalent set
of 3-bit
3. From left to write Arrange and write all 3-bit sets together
and form binary equivalent of given given octal number.

Example 1: convert (745) into binary number

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 Conversion Between Number System
Binary to Hexadecimal Conversion
1. Write Binary Number
2. From right to left, group all the bits of binary in the set of
four
3. Add 0 to the left of the last bit of given binary number
incase set of 4 bit is not formed
4. Write hexdecimal no for each corresponding set of 4 bits
5. Group these hexadecimal digits together to form
equivalent octal number
Example 1: Convert (1100110101) in hexadecimal Number
2

Hexadecimal to binary conversion

1. Write hexadecimal number
2. Replace each digit of hexadecimal number with its
equivalent set of 4-bit
3. From left to write Arrange and write all 4-bit sets together
and form binary equivalent of given hexadecimal
number.
Example 1: convert (CA12) into binary number
16
 Conversion Between Number System

Conversion of Decimal Number with fractional part to Binary

number
Following are the steps to convert the fractional part of a
decimal number to binary number system:
multiply the fractional part by the base value 2 repeatedly
till the fractional part becomes 0.
from top to bottom, Write integer part of the number to get
equivlent binary number.
If the fractional part does not become 0 in successive
multiplication, then stop after 10 multiplications. In some
cases, fractional part may start repeating, then stop
further calculation.
Example 1: convert (0.625) to binary number.
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Example 2: convert (.36) to binary number

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 Conversion Between Number System

Conversion of Decimal Number with fractional part to Octal

number
Following are the steps to convert the fractional part of a
decimal number to Octal number system:
multiply the fractional part by the base value 8 repeatedly
till the fractional part becomes 0.
from top to bottom, Write integer part of the number to get
equivlent Octal number.
If the fractional part does not become 0 in successive
multiplication, then stop after 10 multiplications. In some
cases, fractional part may start repeating, then stop
further calculation.
Example 1: convert (0.175) to Octal number.
10

Example 2: convert (0.345) to octal number

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 Conversion Between Number System

Conversion of Decimal Number with fractional part to

Hexadecimal number
Following are the steps to convert the fractional part of a
decimal number to Hexadecimal number system:
multiply the fractional part by the base value 16
repeatedly till the fractional part becomes 0.
from top to bottom, Write integer part of the number to get
equivlent Hexadecimal number.
If the fractional part does not become 0 in successive
multiplication, then stop after 10 multiplications. In some
cases, fractional part may start repeating, then stop
further calculation.
Example 1: convert (0.175) to Hexadecimal number.
10

Example 2: convert (0.220) to Hexadecimal number

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 Conversion Between Number System
Binary with fractional part to decimal Conversion
Following are the steps for converting Binary with Fractional
part to Decimal number:
1. Write down binary number
2. Find positional value of each bit by raising its position
number as power to the base 2
3. Multiply each bit with its positional value and find its
corresponding decimal number
4. Add all these decimal numbers to find equivalent Decimal
number of given Binary Number.
Example 1: Convert (10011.11) to Decimal Number
2

Binary with fractional part to Octal Conversion

Write Binary Number
Group all the bits of binary in the set of three
Add 0 to the left of the last bit of integer part and to the
right of the last bit if the fractional part of given primary
number, incase set of 3 bit is not formed
Write octal no for each corresponding set of 3 bits
Group these octal digits together to form equivalent octal
number
Example 1: Convert (10101.01101) in Octal Number
2
 Conversion Between Number System

Binary with fractional part to Hexadecimal Conversion

1. Write Binary Number
2. Group all the bits of binary in the set of four
3. Add 0 to the left of the last bit of integer part and to the
right of the last bit if the fractional part of given binary
number, incase set of 4 bit is not formed
4. Write hexadecimal no for each corresponding set of 4 bits
5. Group these hexadecimal digits together to form
equivalent hexadecimal number

Example 1: Convert (110101.011011) in hexadecimal

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