## Decimal Numbering System

Decimal is a base 10 numbering system which is made up of 10 numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. It is the most commonly used numbering system. The reason behind that is convenience. We have 10 fingers that we use for counting, so it is easier to count with a base 10 numbering system, thus decimal is widely used.

## Hexadecimal Numbering System

Hexadecimal is a base 16 numbering system which is made up of 16 digits: 0 – 9 and six more, which is A through F.

## Conversion from Decimal to Hexadecimal

Conversion can be done by dividing the decimal number by 16 repeatedly until the final result is 0.

For example, the decimal number 357 is converted to hexadecimal number as follows:

Division | Result | Remainder |
---|---|---|

357 / 16 | 22 | 5 |

22 / 16 | 1 | 6 |

1 / 16 | 0 | 1 |

Hexadecimal number is taken from the remainder starting from the last to the first, or in the illustration above, from bottom to top, which is 165.

*Note: For remainder between 0 to 9, hexadecimal equivalent is the same, see table below.*

Hexadecimal | Decimal |
---|---|

0 | 0 |

1 | 1 |

2 | 2 |

3 | 3 |

4 | 4 |

5 | 5 |

6 | 6 |

7 | 7 |

8 | 8 |

9 | 9 |

So decimal number 357 is 165 in hexadecimal number.

*357 _{10 }= 165_{16}*

Another example, with remainder greater than 9, in order to have an alphabet result.

Decimal number 2856 is converted to hexadecimal number as follows:

Division | Result | Remainder |
---|---|---|

2856 / 16 | 178 | 8 |

178 / 16 | 11 | 2 |

11 / 16 | 0 | 11 |

Hexadecimal number is taken from the remainder starting from the last to the first, or in the illustration above, from bottom to top, which is 1128. Looking at the table below, 11 is B in hexadecimal.

Hexadecimal | Decimal |
---|---|

A | 10 |

B | 11 |

C | 12 |

D | 13 |

E | 14 |

F | 15 |

So decimal number 2856 is B28 in hexadecimal number.

*2856 _{10 }= B28_{16}*

## Conversion from Hexadecimal to Decimal

Conversion can be done by multiplying the hexadecimal number by 16. Each digit will be multiplied by the corresponding power of 16 value. It would be easier to convert by putting the hexadecimal on the power of 16 columns as follows:

Exponent | 16^{2} |
16^{1} |
16^{0} |
---|---|---|---|

Value | 256 | 16 | 1 |

Hexadecimal | 1 | 6 | 5 |

Multiply each value by the hexadecimal digit as follows:

(1 x 256) = 256

(6 x 16) = 96

(5 x 1) = 5

Total value = (256 + 96 + 5) = 357

So, the hexadecimal number 165 is 357 in decimal number.

*165 _{16 }= 357_{10}*

Another example, with alphabet hexadecimal digit.

Conversion can be done by multiplying the hexadecimal number by 16. Each digit will be multiplied by the corresponding power of 16 value. It would be easier to convert by putting the hexadecimal on the power of 16 columns as follows:

Exponent | 16^{2} |
16^{1} |
16^{0} |
---|---|---|---|

Value | 256 | 16 | 1 |

Hexadecimal | B | 2 | 8 |

Multiply each value by the hexadecimal digit as follows:

(B x 256) = (11 x 256) = 2,816

(2 x 16) = 32

(8 x 1) = 8

Total value = (2,816 + 32 + 8) = 2,856

So, the hexadecimal number B28 is 2,856 in decimal number.

*B28 _{16 }= 2,856_{10}*

Conversion may seem difficult at first, but with just a little practice and understanding the principles behind the conversion, it would be easier over time. Of course, there are online conversion tools and calculators, but it would be handy to know how it is done manually and understand the concept behind the different numbering systems.