Higher Grade Computing Studies
1. Data Representation
Data Representation – Why do we use binary?
• simplicity, in only having to generate and detect two
voltage levels.
• good tolerance, because a degraded 1 is still
recognisable as a 1.
• calculations are kept simple.
1
Higher Computing
Computer Systems
S. McCrossan
Higher Grade Computing Studies
1. Data Representation
Data Representation - Numbers
• Converting binary to decimal
Here is an example of how to convert the binary number
10011010 to a decimal:
128
1
64
0
32
0
16
1
8
1
4
0
2
1
= 128 + 16 + 8 + 2 = 154.
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Higher Computing
Computer Systems
S. McCrossan
1
0
Higher Grade Computing Studies
1. Data Representation
Data Representation - Numbers
• Converting decimal to binary
Here is an example
of how to convert
the decimal number
69 to a binary:
3
2
2
2
2
2
2
2
2
69
34
17
8
4
2
1
0
R1
R0
R1
R0
R0
R0
R1
giving
1000101
Higher Computing
Computer Systems
S. McCrossan
Higher Grade Computing Studies
1. Data Representation
Data Representation - Numbers
• Positive and Negative Integers
• Twos complement representation is
used to store both positive and
negative numbers.
• The leftmost bit is used to store
whether a number is positive or
negative.
4
Binary
Integer
11111101
-3
11111110
-2
11111111
-1
00000000
0
00000001
1
00000010
2
00000011
3
Higher Computing
Computer Systems
S. McCrossan
Higher Grade Computing Studies
1. Data Representation
Data Representation - Numbers
• Positive and Negative Integers
To convert 5 to –5 you change all 0s to 1s and vice versa and
then add 1.
So
0 0 0 0 0 1 0 1
becomes
1 1 1 1 1 0 1 0
+1
1 1 1 1 1 0 1 1
5
or
5
or
-5
Higher Computing
Computer Systems
S. McCrossan
Higher Grade Computing Studies
1. Data Representation
Data Representation - Numbers
• Positive and Negative Integers
The number of integers which could be stored in one byte ( 8
bits ) is 28 = 256
The range of integers which could be stored in one byte ( 8
bits ) is -128 to +127
6
Higher Computing
Computer Systems
S. McCrossan
Higher Grade Computing Studies
1. Data Representation
Data Representation - Numbers
• Real Numbers
• A real number is a number like 134.78954673. Real numbers
are stored in a computer as floating point numbers.
• In decimal base 10, 34006.8 becomes
3.40068 * 104
• The ‘4’ is because the point has moved 4 places to the left.
• The ‘3.40068’ is called the mantissa.
• The ‘4’ is called the exponent.
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Higher Computing
Computer Systems
S. McCrossan
Higher Grade Computing Studies
1. Data Representation
Data Representation - Numbers
• Real Numbers
• The binary real number 1101.0011011101 would be written
as
.11010011011101 * 24
where the 2 is because we are working in binary and the 4
is the number of places that the point has moved.
• The full answer is .11010011011101 *2100 because the 4 is
converted into binary as well.
• The computer would store the mantissa 11010011011101
and exponent 100.
Higher Computing
8
Computer Systems
S. McCrossan
Higher Grade Computing Studies
1. Data Representation
Data Representation - Numbers
• Storage of Numbers
• Increasing the size of the storage for numeric data
increases the range of numbers that can be stored.
• Increasing the size of the mantissa increases the accuracy
with which the real number can be stored.
• Increasing the size of the exponent increases the range of
numbers that can be stored.
9
Higher Computing
Computer Systems
S. McCrossan
Higher Grade Computing Studies
1. Data Representation
Data Representation - Text
• Each character is stored in the computer as a
unique binary code.
• The list of characters that a computer can handle is called
the character set.
• ASCII character set was introduced so that all
computers would use the same codes to represent the
same characters
• Control characters are included in the character set that tell
the computer to do things, for instance, take a new page or
take a newline.
10
Higher Computing
Computer Systems
S. McCrossan
Higher Grade Computing Studies
1. Data Representation
Data Representation - Text
• ASCII can only represent, at most, 256 characters.
However, other alphabets need many more characters, e.g.
Japanese and Arabic.
• The solution was Unicode which uses 16 bits to represent
each character. This means it can store 65,536 characters.
11
Higher Computing
Computer Systems
S. McCrossan
Higher Grade Computing Studies
1. Data Representation
Data Representation - Graphics
Bit mapped Graphics
• For a graphic drawn in a painting package, the computer
stores it as a two-dimensional array of pixels
• The number of pixels that makes up an image is called
the resolution.
12
Higher Computing
Computer Systems
S. McCrossan
Higher Grade Computing Studies
1. Data Representation
Data Representation - Graphics
Bit mapped Graphics
• In a black and white display, each
white pixel is represented by a 0.
• In a black and white display, each
black pixel is represented by a 1.
• Only two values, 1 and 0, need to
be stored as there are only two
colours to be used.
13
Higher Computing
Computer Systems
S. McCrossan
Higher Grade Computing Studies
1. Data Representation
Data Representation - Graphics
Bit mapped Graphics
• However, when more than two colours are used we need
more memory to store the colour value for each pixel.
• In an 8 colour display, each white
pixel is represented by a 000.
• In an 8 colour display, each black
pixel is represented by a 001.
• In an 8 colour display, each
yellow pixel is represented by a
011.
14
Higher Computing
Computer Systems
S. McCrossan
Higher Grade Computing Studies
1. Data Representation
Data Representation - Graphics
Bit mapped Graphics
• The number of bits
used to represent the
colour of the pixels is
called the bit depth.
15
Bit depth
1
2
3
Colours
2
4
8
8
256
16
65536
24
16777216
Higher Computing
Computer Systems
S. McCrossan
Higher Grade Computing Studies
1. Data Representation
Data Representation - Graphics
Bit mapped Storage Requirements
An image, 5in by 7in is stored at 600 dpi in 65536 colours.
How much memory would be required to store this image?
Pixels used to store image = 5 x 7 x 600 x 600 = 12600000
65536 colours = 16 bits = 2 bytes
Amount of memory
16
= 12600000 x 2 bytes
= 25200000 bytes
= 24Mb
Higher Computing
Computer Systems
S. McCrossan
Higher Grade Computing Studies
1. Data Representation
Data Representation - Graphics
Bit mapped Graphics
• The advantage of using bit mapped graphics is that you
have more control over the graphic as you are able to go into
detail and edit the graphic pixel by pixel.
• The disadvantage of using bit mapped graphics is that each
picture or graphic takes up a lot of memory as the colour of
each pixel has to be stored.
• Another disadvantage of using bit mapped graphics is that if
you enlarge a bitmap image it becomes ‘blocky’.
17
Higher Computing
Computer Systems
S. McCrossan
Higher Grade Computing Studies
1. Data Representation
Data Representation - Graphics
Vector Graphics
• In a CAD or drawing package, the computer stores
information about an object by its attributes i.e. a
description of how it is to be drawn.
• For a rectangle it might be:
• start x and y position
• length, breadth and angle of
rotation
• thickness and colour of the lines
• colour fill etc.
18
Higher Computing
Computer Systems
S. McCrossan
Higher Grade Computing Studies
1. Data Representation
Data Representation - Graphics
Vector Graphics
• The advantage of using vector graphics is that you edit
shapes. This allows you to scale the graphic easily. It also
means that vector graphics don’t take up a lot of memory.
• The disadvantage of using vector graphics is that you are
limited to using only the shapes that the package offers. This
can mean that only simple graphics can be created.
19
Higher Computing
Computer Systems
S. McCrossan