We know that a digit's worth depends on what position it is in relative to the other digits in the number.

- Base 10 positions are worth 10
^{7}, 10^{6}, 10^{5}, 10^{4}, 10^{3}, 10^{2}, 10^{1}, 10^{0} - Base 2 positions are worth 2
^{7}, 2^{6}, 2^{5}, 2^{4}2^{3}, 2^{2}, 2^{1}s, 2^{0} - Base 16 positions are worth 16
^{7}, 16^{6}, 16^{5}, 16^{4}, 16^{3}, 16^{2}, 16^{1}, 16^{0}

**How does the hexadecimal system work?**

The first thing to note is that there are 16 'numbers' in this system: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F. It may well seem a little odd using letters to represent numbers: 10=A, 11=B, 12=C, 13=D, 14=E, 15=F. With a little practice, you will see what an excellent system this is.

Just to remind you, to show what system is being used when you write down a number, it is common to use a subscript. So for example: 34_{10} means (3 x 10) + (4 x 1) whereas 34_{16} means (3 x 16) + (4 x 1)

As you know, when we write down numbers in our daily life, we omit the subscript because we assume that every one is using base 10. Sometimes, especially in computer circles, it is a dangerous assumption to make! If there is any doubt, then add a subscript!

__When doing exam questions, always use a subscript, just to show how clever you are__!

Let's convert a few hex numbers into denary. For the first few you do, you should write down the worth of each position. Then write the number you are converting underneath it. Finally, do the conversion.

**Example 1: convert 3C _{16} into decimal.**

Worth of each position | 256 (16^{2}) |
16 (16^{1}) |
1 (16^{0}) |

Number to convert | 3 | C |

3C_{16} is the same as (3 x 16) + (12 x 1) = 60_{10}

**Example 2: convert 25 _{16 }into decimal.**

Worth of each position | 256 (16^{2}) |
16 (16^{1}) |
1 (16^{0}) |

Number to convert | 2 | 5 |

25_{16} is the same as (2 x 16) + (5 x 1) = 37_{10}

_{
}

**Example 3: convert 8 _{16} into decimal.**

Worth of each position | 256 (16^{2}) |
16 (16^{1}) |
1 (16^{0}) |

Number to convert | 8 |

8_{16} is the same as (8 x 1) = 8_{10}

_{
}

**Example 4: convert 3AF _{16} into decimal.**

Worth of each position | 256 (16^{2}) |
16 (16^{1}) |
1 (16^{0}) |

Number to convert | 3 | A | F |

3AF_{16} is the same as (3 x 256) + (10 x 16) + (15 x 1) = 943_{10}

Q1. Convert these numbers into their denary form: a) 36_{16} b) 3_{16 } c) FA_{16} d) 15_{16}

Q2. Convert these decimal numbers into hex: a) 103_{10} b) 14_{10} c) 58_{10} d) 7_{10}

Q3. Why are nibbles important when using hex?