Chapter 1: Introduction to Networking

The Hexadecimal Number System

These notes guide you through understanding the hexadecimal number system.

Introduction
Overview of Hexadecimal
Hexadecimal Number System

~~Hex and Decimal~~
          Hex to Decimal Conversions
          Decimal To Hex Conversions

~~Hex and Binary~~
          Hex to Binary Conversions
          Binary To Hex Conversions

Summary

Introduction

On completion of these notes you should be

• Understand the hexadecimal number system
• Be able to convert from hex to decimal and vice versa
• Be able to convert from hex to binary and vice versa

Overview

Quite often it is convenient for programmers to write a computer's binary code in octal or hexadecimal.  Converting octal or hexadecimal numbers to binary is easier than converting decimal to binary.  Hexadecimal is used in preference to octal because computers organize memory in groups of 8 bits (bytes) and these can be conveniently divided into groups of four bits (nibbles).  

Four bits is easily coded in hexadecimal form.

Hexadecimal Number System

The hexadecimal number system has sixteen symbols...

0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

This means the base of the hexadecimal system must be 16.  So, each place value is weighted by a power of sixteen.  A place value table for the hexadecimal system would look like...
   

So, the hexadecimal number 1016 means 1 sixteen +  0 units.  

Again, this number 1016 is not the same as the number 1010 in the denary system.  We will see how to convert the number to decimal later on.

Hex to Decimal Conversions

Suppose we have the hexadecimal number 3A16.

Well 3A16 is equivalent to 3×16 + A×1.  Which is 48 + (10×1) = 5810.  

So to convert a hexadecimal number to denary, all we do is multiply each hexadecimal digit by it's place value and add them all together.

Here are a few more examples.

B316 = B×16 + 3×1 = (11×16) + 3 = 17910

5F16 = 5×16 + F×1 = 80 + 15 = 9510

CD16 = C×16 + D×1 = 12×16 + 13×1 = 8510

~Now try the activity~

Decimal to Hex Conversions

How do we convert a denary number to hexadecimal?  Using a place value table, this is similar to converting a denary number to binary.  

As an example, suppose we want to convert 3510 to hexadecimal?  Lets look at a place value table..

Looking at the place value table, the nearest place value less than or equal to 35 is 16.  35 divided by 16 is 2 leaving a remainder of 3.  So we can write a 2 below the place value of 16.  

Now, considering the remainder of 3.  The nearest place value less than or equal to 3 is 1.  Since we need 3 amounts of 1 we can write a 3 below the place value of 1.   There is no remainder.  

So, 3510 is equivalent to 2316 

~Now try the activity~

Binary to Hex Conversions

How do we convert a binary number to hexadecimal?  

If a binary number is a byte long (8 bits) then first divide the byte into two nibbles.  Thus:-

111001102   =   1110 01102    =   E 616

I.e. separately 

11102 = E16 

01102    =   616

Combined, this is E616

Here are some more examples:-

    
~Now try the activity~

Hex to Binary Conversions

How do we convert a hexadecimal number to binary?  

The trick is to assign a 4-bit binary number to each hex digit.  Thus:-

F816  =  1111 10002

i.e. The first hex number is F. The number F in decimal is 15. When you convert the number 15 to binary you get 1111

The second hex number is 8. In decimal this is also 8. When you convert the number 8 to binary you get 1000

Combining the numbers together gives 1111 10002

The trick is to convert each separate hex number to binary, then join the binary numbers together in the correct order. Also, each separate binary number must be four bits long. If a binary number ends up less than four bits long, just add leading zeros. As an example of this...

F416  =  1111 01002

We worked out in the previous example that the number F in binary is 1111. What about the hex number 4? Well, in decimal this is also 4. When you convert the number 4 to binary you get 100. Since this is not four bits long, I just add a leading zero - 0100.

One final example. Lets convert 7D416 to binary.

The first hex number is 7. The number 7 in decimal is 7. When you convert the number 7 to binary you get 0111 (notice the leading zero).

The second hex number is D. In decimal this is 13. When you convert the number 13 to binary you get 1101

The third hex number is 4. In decimal this is 4. When you convert the number 4 to binary you get 0100

7D416  =  0111 1101 01002
  

~Now try the activity~

Summary

On completing these notes you should have:-

• Learnt that the hexadecimal number system uses base 16
• Practiced converting from hexadecimal to decimal and vice-versa
• Practiced converting from hexadecimal to binary and vice-versa