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 10⁷, 10⁶, 10⁵, 10⁴, 10³, 10², 10¹, 10⁰
• Base 2 positions are worth 2⁷, 2⁶, 2⁵, 2⁴ 2³, 2², 2¹, 2⁰
• Base 16 positions are worth 16⁷, 16⁶, 16⁵, 16⁴, 16³, 16², 16¹, 16⁰

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: 34₁₀ means (3 x 10) + (4 x 1) whereas 34₁₆ 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.

3C₁₆ is the same as (3 x 16) + (12 x 1) = 60₁₀

Example 2: convert 25₁₆ into decimal.

25₁₆ is the same as (2 x 16) + (5 x 1) = 37₁₀

Example 3: convert 8₁₆ into decimal.

8₁₆ is the same as (8 x 1) = 8₁₀

Example 4: convert 3AF₁₆ into decimal.

3AF₁₆ is the same as (3 x 256) + (10 x 16) + (15 x 1) = 943₁₀

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 125₁₀ into a hex number.

You should always check the hex answer you got. 7D₁₆ = (7 x 16) + (13 x 1) = 125₁₀ 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. 4B₁₆ = (4 x 16) + (11x 1) = 75₁₀ so our answer is correct. This may seem a little long-winded 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) 36₁₆      b) 3₁₆     c) FA₁₆     d) 15 ₁₆
Q2. Convert these decimal numbers into hex: a) 103 ₁₀      b) 14₁₀      c) 58₁₀     d) 7 ₁₀
Q3. Why are nibbles important when using hex?