Introduction
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}, 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 
1 
Number to convert 
3 
A 
F 
3AF_{16} is the same as (3 x 256) + (10 x 16) + (15 x 1) = 943_{10}
converting from decimal to hex
Going from hex to denary is relatively easy after you've done a few of them. You have to think a little bit harder going the other way, from denary to hex. But there is a great trick you can use  if you can use binary.
Binary and hex are actually very closely related, much more so than first appears. Each hex digit is just a group of four bits!! As long as we can do binary to denary conversion off the top of our heads, there is a method for converting denary to hex (and also back again) very quickly. See if you can follow this example. We are going to convert 125_{10} into a hex number.
You should always check the hex answer you got. 7D_{16} = (7 x 16) + (13 x 1) = 125_{10} so our answer is correct.
(Of course, you could always check your answer using a calculator! In Windows, Go to WINDOWS  ACCESSORIES  CALCULATOR  VIEW  SCIENTIFIC).
You should always check the hex answer you got. 4B_{16} = (4 x 16) + (11x 1) = 75_{10} so our answer is correct. This may seem a little longwinded to start with, but this method is very mechanical and always works. Once you've done a few, you'll be an expert. Besides, it's good practice for binary conversion!
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?
