Session Objectives#7
Explain the use of hexadecimal numbers to represent binary numbers.
Define the term character set (ASCII) and describe the use of binary numbers to
represent characters
Convert numbers in binary, denary and hexadecimal
Explain the relationship between the number of bits per character in a
character set and the number of characters which can be represented.
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Number Systems
Starter Number Puzzle:
Find a 10-digit number where the first digit is how many zeros
in the number, the second digit is how many 1s in the number
etc. until the tenth digit, which is how many 9s in the number.
6210001000
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Number Systems
We already know that computers like to use 0 and 1 to count due
to the fact an electrical charge = 1 and no charge = 0.
This forms the base2 or binary system. Instead of base 10, the
decimal system (denary) where you have Units, Tens, Hundreds,
Thousands etc, we have use multiples of 2:
128
0
64
1
32
0
16
0
8
1
4
0
2
1
1
1
Using a column diagram like this it is easy to turn denary into
binary. In this example, 75 is 1 lot 64, 1 lot 8, 1 lot of 2 and
1 lot 1.
So the denary number 75 expressed in 8 bit binary is
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01001011
Data Representation/Number Systems
TASK
Now express the following denary numbers in 8 bit binary form
(commonly known as a byte) and show your working.
a) 13
b) 56
c) 143
a) 00001101
b) 00111000
c) 10001111
EXTENSION:
Create your own 8 bit binary number and convert to base10.
What is the largest denary number you can create in 8 bit
binary? How many decimal values can that represent?...
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ASCII Character Set
The common QWERTY keyboard is often used for typing text and
numbers etc and to enable computers to understand each other a
common standard was agreed in the 1960s and the ASCII character
set was formed (American Standard Code for Information
Interchange).
Key Term: Character set – The characters available to a computer
(i.e letters, LETTERS, numbers and punctuation,!~#etc)
127 characters in total are represented by ASCII codes, within
the capability of 8 bits (one byte, the amount of storage
required for any single character).
INVESTIAGTION: SO why use 8 bits per character, why not 7?
ASCII is not appropriate for all circumstances.
HWK – Research the meaning and need for the UNICODE and EBCDIC
character sets.
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Position Valued Representation
Position valued representation (PVR) is a system to represent
any value with a limited number of symbols. Take the decimal
system for example, we can represent any value that we need with
only 10 symbols (0....9).
Examples:100=1......Any number to the 0th power is 1
101=10............Any number to the 1th power is that number
102=100..........10*10
103=1000........10*10*10
104=10000......10*10*10*10
With the PVR system there are a number of rules
• The number of digits needed for the system is equal to the
base. (i.e. decimal (base 10) has 10 digits)
• The value of the largest digit is one less than the base.
• The first position to the left of the "base point" is worth 1.
(i.e. the decimal point is the base point of the decimal
system.)
• Every other position is worth base times the position value to
its immediate right.
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OCTAL NUMBER SYSTEM
If binary is a base 2 number system, what would the base be of
an octal system?
8
In the octal system the column headings are now changed to:
and denary 75 is:
512
0
64
1
8 1
1 3
These larger base systems are needed for computing larger
numbers.
TASK: Convert the following denary numbers into octal and
remember the PVR rule: The value of the largest digit is one less than the base
a) 263
b) 482
c) 4095
0407
0742
7777
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NB OCTAL IS NOT ON THE GCSE SPECIFICATION
Hexadecimal
Some information is stored in computers as numbers in base 16.
This is called hexadecimal (or hex for short). The principles
are the same except that the digits above 9 are represented by
capital letters A – F.
Binary
Hex
Decimal
Binary
Hex
Decimal
0000
0
0
1000
8
8
0001
1
1
1001
9
9
0010
2
2
1010
A
10
0011
3
3
1011
B
11
0100
4
4
1100
C
12
0101
5
5
1101
D
13
0110
6
6
1110
E
14
0111
7
7
1111
F
15
TASK: As you can see decimal 10 is A in Hex, so what would the
decimal value of hex 10 be?? Discuss...
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Hexadecimal
The column headings in hex (base 16) would now be:
256
16
1
And 75 denary will be: 4 lots of 16 and 11 ones or
4B.
TASK: What will the hex value of 126 be?
16 goes into 126 7 times with a remainder of 14 (E)
Therefore the hexcode is 7E
EXT: What is the base 16 of a) 34 b) 93
ANSWERS: a) 22
b) 5D
TASK: Convert the following hex to base 10 a) 23, b) 3C c) 5F
ANSWERS: a) 35
b) 60
EXT: TRY SOME MORE
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c) 95
Session Objectives#7
Explain the use of hexadecimal numbers to represent binary numbers.
Define the term character set (ASCII) and describe the use of binary numbers to
represent characters
Convert numbers in binary, denary and hexadecimal
Explain the relationship between the number of bits per character in a
character set and the number of characters which can be represented.
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Converting Binary & Hexadecimal
Why use different number systems??
This is an obvious question since we know computers like the use
of binary. Well, the reason is down to the application or use of
the computer.
Large binary numbers are difficult to remember and programmers
need something easier to recognise. Converting from base 10 to 2
and vice versa can be complex so it is easier to go from base 2
to 16 and vice versa. Since a byte usually has 8 bits, it is
easy to split into two 4 bit sections, called nibbles
Key Terms : Byte – 8 bits;
Nibble – 4 bits or half a byte
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Converting Binary to Hexadecimal
Denary 45 in binary is 32 + 8 + 4 + 1 or
128
0
64
0
32
1
16
0
8
1
4
1
2
0
1
1
Split into 2 nibbles and treated as 4 bits each:
8
0
4 2 1
0 1 0
= 2
8
1
=
4 2
1 0
= 13
1
1
2D
TASK: Using the same method convert a) 11101011 b) 10100011
a) EB
Now try the reverse, convert to binary a) A5
b) A3
b) 3B
a) 10100101 b)00111011
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The Application of HEX
Why does the 256 (in hex, base 16 to power 2) sound familiar?
Each pixel you see in your display is either Red, Green or
Blue (RGB) The light intensity of each is changed from 0 to
255 to determine the colours you see.
That’s a lot of colour choices!
Colour codes are represented in Hex in a set of 6 digits, 2
for Red, 2 for Green and 2 for Blue. What would the codes
be like in binary? Why use hexadecimal??
Follow the link to see codes for the various colours and
then attempt the quiz
http://www.nameacolor.com/RGB.HTML
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Using hexcodes in HTML
<HTML>
<BODY BGCOLOR="#FFFF00">
<P>This is my very first web page!</P>
</BODY>
</HTML>
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