Lesson Objectives

• Understand the hexadecimal numbering system

• Convert numbers between hexadecimal and denary, and vice versa

• ALL students be able to count in hex from 1 to 16

• MOST students will convert hex numbers into denary

• SOME students will convert numbers between hex, denary and binary

• You already know about base 10 (Decimal/Denary)

• And you’ve just learnt about base 2 (Binary)

1 2 3 4101000 1100

x10x10 x10

1 0 1 128 14

x2x2 x2

Why do we need binary numbers?

• Because computers work on the principle of 2 states, that something is either ON/TRUE or OFF/FALSE.

• This can only be done with base 2 (binary)

• If it was done with decimal/base 10 there would be 10 different states!

The problem with binary…

• There is one big problem with binary…numbers can become VERY long!

• In order to make it easier for a human programmers to work with binary numbers, they use the hexadecimal system (like a binary shortcut)

Hexadecimal

0 1 2 3164096 1256

x16x16 x16 As we move left, the column headings increase by a factor of sixteen

This number is:

1 x 256 + 2 x 16 + 3 x 1 = 291

It’s still two hundred and ninety-one, it’s just written down differently

http://www.advanced-ict.info/interactive/hexadecimal.html

How can there be sixteen possible digits in each column, when there are only ten digits?

Hexadecimal

• Hexadecimal uses the digits 0-9 and the letters A-F to represent the denary numbers 0-15

Den Hex Den Hex0 0 8 8

1 1 9 9

2 2 10 A

3 3 11 B

4 4 12 C

5 5 13 D

6 6 14 E

7 7 15 F

Notice how 0 is classed as a

digit, so there are 16 numbers in total from 0 to

15

Making bigger numbers

• You do it in exactly the same way

16 1 Den

1 0 1x16 16

1 1 1x16 + 1x1 17

1 A 1x16 + 1xA(10) 26

A 0 A(10)x16 160

2 B 2x16 + 1xB(11) 43

Where is it used?

• When have you seen numbers being represented as letters?

• Hex is often used for 32-bit colour values, especially on web pages

• FF00EE99 instead of 11111111000000001110111010011001.

• http://www.advanced-ict.info/interactive/colours.html

255

• Denary – 255

• Binary– 11111111

• Hexadecimal– FF

• Large binary numbers are hard to remember

• Programmers use hexadecimal values because:– each digit represents exactly 4 binary digits;

– hexadecimal is a useful shorthand for binary numbers;

– hexadecimal still uses a multiple of 2, making conversion easier whilst being easy to understand;

– converting between denary and binary is relatively complex;

– hexadecimal is much easier to remember and recognise than binary;

– this saves effort and reduces the chances of making a mistake.

You convert denary to hex in the same was as binary

16 1D

Convert the denary number 45 into a hex number

Step 1: How many times does 16 go into 45? 45 / 16 = 2 (with 13 remaining)

Step 2: How many times does 13 go into 1?13! 13 in hex is D

2

Let’s do another one

16 17

Convert the denary number 199 into a hex number

Step 1: How many times does 16 go into 199? 199 / 16 = 12 (with 7 remaining)

Step 2: 12 in hex is C

Step 2: How many times does 7 go into 1?7! 7 in hex is 7!

C

Lesson task:

• Complete the denary to hex conversions in your workbook.

• Extension: If you complete, have a go at the cross word task in your booklet.

Hex to binary• To convert from hexadecimal to binary treat

each digit separately. It may be easier to go via denary to get a binary number.

• So DB in hexadecimal is 11011011 in binary.

Hex D B

Denary 13 11

Binary 1 1 0 1 1 0 1 1