Number System Conversion | Conversion between number system

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)₁₀ to Binary Number.

Example 2: Convert (234)₁₀ to Binary Number

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)₁₀ to Octal Number.

Example 2: Convert (1683)₁₀ to Octal Number.

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)₁₀ to Hexadecimal Number.

Example 2: Convert (2940)₁₀ to Hexadecimal Number.

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)₂ to Decimal Number

Example 2: Convert (111100)₂ to Decimal Number

1 x 2⁵  +   1 x 2⁴   +   1 x 2³   +   1 x 2²   +   0 x 2¹   +   0 x 2⁰ 
=  32 +     16       +       8       +       4       +       0        +      0
= (61)₁₀

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.

Example 1: Convert (336)₈ to Decimal Number

Example 2: Convert (1325)₈ to Decimal Number

1 x 8³  +   3 x 8²   +   2 x 8¹   +   5 x 8⁰  
=  512 +      192    +       16     +       5   
= (725)₁₀

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)₁₆ to Decimal Number

Example 2: Convert (1BF6)₁₆ to Decimal Number

    1 x 16³  +   11 x 16²   +   15 x 16¹   +   6 x 8⁰ 
=  4096      +   2816        +    240           +   6      
= (7158)₁₀

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)₂ in Octal Number

Example 2: Convert (1011)₂ in Octal Number

1          0     1    1
001      011
1 3
=(13)₈

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)₈ into binary number

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)₂ in hexadecimal Number

Example 2: Convert (11101001101)2 in Hexadecimal no

1      1     1     0     1     0     0     1     1     0     1

0111         0100         1101

7                  4                D

= (74D)₁₆

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)₁₆ into binary number

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)₁₀ to binary number.

Example 2: convert (.36)₁₀ to binary number

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)₁₀ to Octal number.

Example 2: convert (0.345)₁₀ to octal number

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)₁₀ to Hexadecimal number.

Example 2: convert (0.220)₁₀ to Hexadecimal number

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)₂ to Decimal Number

Example 2: Convert (101.01)₂ to Decimal Number

1×2²   +   0x2¹   +   1×2⁰   .   0x2^-1   +   1×2^-2 
= 4  +  0  +  1 . 0 + 0.25
= (5.25)₁₀

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)₂ in Octal Number

Example 2: Convert (1011.10)₂ in Octal Number

1   0    1    1    .    1    0
011   011   .   100
  3       3     .      4
=(33.4)₈

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)₂ in Octal Number

Example 2: Convert (1011111011.1011010)₂ in Octal Number

1   0   1   1   1   1   1   0   1   1   .   1   0   1   1   0   1   0
0010   1111   1011   .   1011   0100
2          F          B     .       B         4        
=(2FB.B4)₁₆