Number conversion is the fundamental operation of any digital systems.

There are different bases like base2, base8, base10 and base 16

Base – 10 | Base 2 (Binary) | Base 8 (Octal) | Base 16 (Hexadecimal) |

0 | 0000 | 00 | 0 |

1 | 0001 | 01 | 1 |

2 | 0010 | 02 | 2 |

3 | 0011 | 03 | 3 |

4 | 0100 | 04 | 4 |

5 | 0101 | 05 | 5 |

6 | 0110 | 06 | 6 |

7 | 0111 | 07 | 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 |

The above table shows the various base systems.

Converting between one base to another base is of importance. Usually all the numbering systems of day to day life is done using Base 10 or Decimal system. So it is necessary to convert

- Other Base to Decimal
- Decimal to other base

Any number represented as

**a**_{n}a_{n-1}a_{n-2}a_{n-3…..}a_{1}a_{0 }. a_{-1}a_{-2}……..a_{-n} |

We will see one by one

__Decimal to Binary (Base 10 to Base 2)__

**(19.456)**_{10} – (?.?)_{2}

In the above, the number 19.456 has to be converted to binary or base2 which will have only 0s and 1s.

First we will take 19

Division | Quotient | Remainder | Remarks |

19/2 | 9 | 1 | a0=1 |

9/2 | 4 | 1 | a1=1 |

4/2 | 2 | 0 | a2=0 |

2/2 | 1 | 0 | a3=0 |

1/2 | 0 | 1 | a4=1 |

so the conversion is 10011

Secondly we will take (0.456)_{10}

Multiplication | Whole number | decimal | Remarks |

0.456 * 2 | 0 | 0.912 | a_{-1}=0 |

0.912 * 2 | 1 | 0.824 | a_{-2}=1 |

0.824 * 2 | 1 | 0.648 | a_{-3}=1 |

0.648 * 2 | 1 | 0.296 | a_{-4}=1 |

Therefore (0.456)_{10 }is (0.0111)_{2}

Finally (19.456)_{10} = (10011.0111)_{2}

__Binary (Base 2) to Decimal (Base 10) Conversion__

**(11110.0111)**_{2} = (?.?)_{10}

Let us take the whole portion 11110

1 | 1 | 1 | 1 | 0 | **.** | 0 | 1 | 1 | 1 |

a_{4} | a_{3} | a_{2} | a_{1} | a_{0} | **.** | a_{-1} | a_{-2} | a_{-3} | a_{-4} |

1 * 2^{4} | 1 * 2^{3} | 1 * 2^{2} | 1 * 2^{1} | 0 * 2^{0} | **.** | 0 * 2^{-1} | 1 * 2^{-2} | 1 * 2^{-3} | 1 * 2^{-4} |

16 | 8 | 4 | 2 | 0 | **.** | 0.5 | 0.25 | 0.125 | 0.0625 |

**Total is 30.9375**

__Decimal to Octal (Base 10 to Base 8)__

**(19.456)**_{10} – (?.?)_{8}

In the above, the number 19.456 has to be converted to binary or base2 which will have only 0s and 1s.

First we will take 19

Division | Quotient | Remainder | Remarks |

19/8 | 2 | 3 | a0=3 |

2/8 | 0 | 2 | a1=2 |

so the conversion is 23

Secondly we will take (0.456)_{10}

Multiplication | Whole number | decimal | Remarks |

0.456 * 8 | 3 | 0.648 | a_{-1}=3 |

0.648 * 8 | 5 | 0.184 | a_{-2}=5 |

0.824 * 8 | 1 | 0.472 | a_{-3}=1 |

Therefore (0.456)_{10 }is (0.351)_{8}

Finally (19.456)_{10} = (23.351)_{8}

__Octal (Base 8) to Decimal (Base 10) Conversion__

**(337.64)**_{8} = (?.?)_{10}

Let us take the whole portion 11110

3 | 3 | 7 | **.** | 6 | 4 |

a_{2} | a_{1} | a_{0} | **.** | a_{-1} | a_{-2} |

3 * 8^{2} | 3 * 8^{1} | 7 * 8^{0} | **.** | 6 * 8^{-1} | 4 * 8^{-2} |

192 | 24 | 7 | **.** | 0.75 | 0.0625 |

**Total is 267.8125**

__Decimal to Hexadecimal (Base 10 to Base 16)__

**(19.456)**_{10} – (?.?)_{16}

In the above, the number 19.456 has to be converted to binary or base2 which will have only 0s and 1s.

First we will take 19

Division | Quotient | Remainder | Remarks |

19/16 | 1 | 3 | a0=3 |

1/16 | 0 | 1 | a1=1 |

so the conversion is (13)_{16}

Secondly we will take (0.456)_{10}

Multiplication | Whole number | decimal | Remarks |

0.456 * 16 | 7 | 0.296 | a_{-1}=7 |

0.296 * 16 | 4 | 0.736 | a_{-2}=4 |

0.736 * 16 | B | 0.776 | a_{-3}=B |

Therefore (0.456)_{10 }is (0.74B)_{16}

Finally (19.456)_{10} = (13.74B)_{16}

__Octal (Base 8) to Decimal (Base 10) Conversion__

**(1AB.62)**_{16} = (?.?)_{10}

Let us take the whole portion 11110

1 | A | B | **.** | 6 | 2 |

a_{2} | a_{1} | a_{0} | **.** | a_{-1} | a_{-2} |

1 * 16^{2} | 10 * 16^{1} | 11 * 16^{0} | **.** | 6 * 16^{-1} | 2 * 16^{-2} |

256 | 160 | 11 | **.** | 0.375 | 0.007 |

**Total is (427.382)**_{10}