1. Computer Systems

2. There are 10 groups of people in the world
Those who understand binary
Those who don’t

3. What you need to know Conversion from binary to decimal and vice versa
Representation of positive numbers up to 32 bits
Representation of negative numbers (using two’s compliment) up to 8 bits
Conversion to and from bit, byte, kilobyte, megabyte, gigabyte, terabyte
Description of floating point representation

4. Data Representation

5. Computer Data
The purpose of this unit is to find out how data is stored in a computer
There are 3 types of data:
Numbers (integer (+ and -) and real)
Text (ASCII and Unicode)
Graphics (bit mapped and vector)

6. Why we use decimal numbers
Humans use the number system called base 10 (decimal).
The reason for this is that humans have ten toes and ten fingers.
This system requires the use of ten different numbers (0 to 9).

7. If we have to represent a number greater than nine then we have to introduce another column (10’s).
The place value system that works with decimal is shown below
10, , units

8. Binary numbers
Binary numbers are numbers which represent the base 2. The columns in binary are…

9. Binary Vocabulary
Here is an example of how to convert the binary number to a decimal:
= = 218

10. Binary Vocabulary - Revision
A bit is short for binary digit. A bit can either be a 1 or a 0.
A byte is a group of 8 bits. This can take on 28 = 256 possible values.
A nibble is a group of 4 bits. There are 2 nibbles in 1 byte.

11. Positive Integers
Positive integer numbers can be converted exactly in to their equivalent binary numbers.
The size of number that can be handled depends on the number of bytes allocated to each number.

12. For example, if two bytes (16 bits) are used to store each integer number then this can handle numbers in the range from up to
In decimal this is 0 to 216-1
= 0 to 65535

13. If you use 3 bytes to store an integer this can handle numbers in the range from
In decimal this would be
= 0 to

14. Storing Positive Integers
The range of numbers which can be stored is dependent on the number of bytes allocated
1 byte = 0 to 28 -1
2 bytes = 0 to
3 bytes = 0 to
4 bytes = 0 to

15. If you use 4 bytes to store an integer this can handle numbers in the range from
In decimal this would be
= 0 to

16. Written Task
Complete Qs 1 – 4 on pages 2 and 3 of How to Pass Higher Computing

17. More Revision Petabyte Exabyte Zettabyte Yottabyte 1 byte – 8 bits
1 kilobyte = 1024 bytes
1 megabyte = 1024 kilobytes
1 gigabyte = 1024 megabytes
1 terabyte = 1024 gigabytes
Petabyte
Exabyte
Zettabyte
Yottabyte

18. Conversion
You need to be able to convert to and from bits – bytes - kilobytes, etc
If you have 2 quantities in different unit values you must put them into the same unit value to perform calculations

19. Conversion example 1 A CD writer writes at 1200 Kb per second
Hard disk writes at 2 Mbs per second
300 megabytes of file size
Calculate the time the transfer will take on each device
300 Mb * 1024 = Kb
/ 1200 = 256 seconds
/2 = 150 seconds (already in same format so no need to convert)

20. Conversion Example 2 CD writer reads at a speed of 1500 Kbps
3 minutes to read a file
What is the size of the file?
3 minutes * 60 = 180 seconds
1500 kbps * 180 seconds = Kbs
/ 1024 = Mb
If the answer is greater than 1024 you should always divide again

21. Negative Integers
Negative integers can be represented using a method called Two’s Compliment
Two’s compliment involves 3 simple steps
Convert the positive number to binary
Change all the 1’s to 0’s and all the 0’s to 1’s (called inverting)
Add 1 to the result

22. Example Two’s complement of 5 (to get negative 5) +5 in binary =
Invert to change 1s to 0s
add 1
= -5 in two’s complement

23. -5 in two’s compliment to get +5
Invert to change 1s to 0s
add 1
= -5 in two’s complement

24. Sample Exam Question
Write the following numbers as 8-bit binary numbers using two’s compliment
(i) –18 (ii) –26
+18 in binary =
Invert =
Add 1 =
-18 in 2s comp =
+26 in binary =
Invert =
Add 1 =
-26 in 2s comp =

25. Express -10 using 2’s compliment

26. Express -10 using 2’s compliment

27. Express -6 in binary using two’s compliment

28. Express -6 in binary using two’s compliment

29. Express -25 in binary using two’s compliment

30. Express -25 in binary using two’s compliment

31. Express -92 in binary using two’s compliment

32. Practice for you
Chose 5 numbers between 1 and 100 and show the binary negative number using two’s compliment – to be done in your jotter
Check your answers using the Scholar website

34. Real Numbers
We can represent real numbers in decimal by extending the power column further to the left as show below:

35. The same method can be used to represent fractional numbers in binary:
/2 1/4 1/8

36. How does the binary number 1001.111 represent in decimal ?
/2 1/4 1/8
Add up /2 + 1/4 + 1/8 = 9.875
( )

37. This system is not usually used as there is no way to find out where the binary point is stored.
The solution to this is to keep the binary point in a fixed position and move the bits about.

38. Floating Point Representation
Any decimal number can be represented by using a mantissa, an exponent and a base.
exponent
0.23 X 1056
mantissa base

39. The same system can be used in binary as long as the binary point is assumed to be at the start of a floating point number.
Example
The fractional number would be stored as (mantissa) 100 (exponent).

40. 1001.111 1001111 (mantissa) - the number being stored
100 (exponent) – the number of places to move the binary point (remember the exponent is also stored in binary so 100 means 4!!!)

41.
The exponent (100) tells the computer to place the binary point four places from the start of the mantissa.
As the binary point is assumed to be at the start of any floating point number, only the mantissa and exponent need be stored.

42. To change in to floating point in binary is similar except the point is moved in front of the first 1 and we are in binary.
So would be written as * 24 where the 4 is the number of places that the point has moved and the 2 is the base (because we are working in binary)

43.
The full answer is *2100 because the 4 is converted into binary as well.
The computer would store the mantissa and exponent 100
Because we are always working in binary with a base of 2 we don’t need to store the base

44. ANOTHER EXAMPLE 11100101 =11100101.0 =0.11100101 x 28 (because we have
moved the point 8 places)
MANTISSA
BASE
EXPONENT
The exponent would be store as

45. The normal way is to use four bytes for the mantissa and one byte for the exponent
This gives nine figure accuracy when converted to decimal) and a range of 20 to 2255 which is approximately 100 to 1076.

46. Accuracy - Mantissa 1/3 = 0.33333333333333 ………….. 1/3 = 0.333
The first method is more accurate because there are more decimal places
In floating point the number of bits in the Mantissa determines the accuracy – the more bits the more accuracy.

47. Range - Exponent
Increasing the number of bits allocated to the exponent increases the range of numbers that can be stored.
Exponent with 4 bits = 24 = 16 numbers
Exponent with 8 bits = numbers

48. Written Task
Complete Q 11 on page 5 of How to Pass Higher Computing

49. Representing Text
ASCII and Unicode

50. What you should already know
Every character is stored as a unique binary code
A character is a letter, number or symbol from the keyboard
Character Set is all the characters that the computer can store and process

51. How is Text stored? Two common methods of storing text are ASCII
Unicode

52. ASCII American Standard Code for Information Interchange.
Each character has a unique ASCII code
Allows text to be transmitted between computers using a common standard

53. ASCII codes can be used to represent the following:
Digits 0 to 9
Upper and Lower case letters
Symbols such as (!,#$?)
Control characters

54. Control Characters
ASCII codes 32 – 47 are for symbols and control characters
Return
Break
Escape
Caps Lock
48 – 57 represent 0 – 9
65 – 122 represent upper and lower case letters

55. More… On all computers the ASCII codes from 0 to 127 are fixed
Those from 128 to 255 can be used for anything the computer manufacturer desires

56. How many characters can be stored?
7 bits are used to store each character
The 8th bit is used as a parity check digit (explained later)
If 7 Bits are used:
= 27
= 128 characters
If 8 Bits are used:
= 28
= 256 characters

57. Unicode
Developed to enable representation of all languages not just English/American/European
Uses 16 bits to represent each character
216 = characters

58. Advantages of Unicode over ASCII
More characters can be represented
Many languages can be represented
Unicode becoming more popular as the standard for text transfer

59. Written Task
Complete Qs 12 – 15 on page 6 of How to Pass Higher Computing

60. Disadvantage of Unicode
Requires double the storage space of ASCII to store each character

61. Representing Graphics

62. What you need to know about graphics
Features of bitmapped graphics
Relationship between bit depth and number of colours (up to 24 bit true colour)
Features of vector graphics
Advantages and disadvantages of different methods of graphic representation
Relationship between bit depth and file size
The need for data compression

63. How are graphics stored?
There are 2 ways to store a graphic:
Bit Mapped Graphic
Vector Graphic

64. Bitmapped Graphics

65. Bit Mapped Graphics
Computer stores the graphic as a 2 dimensional array of pixels
On a monochrome image each pixel is represented by 1 bit
Once drawn, each pixel can be edited individually
Every pixel is stored

66. Disadvantages of Bitmaps
Graphic cannot be resized without pixelation occurring
Graphic will always stay the same resolution that it was created in – this is called Resolution Dependent
Take up a large amount of storage space

67. Resolution Dependent A graphic is created at 300 dpi
A printer prints at 600 dpi
The graphic is sent to the printer
The graphic only prints at 300 dpi not 600 dpi

68. Storage requirements of bitmaps
Storage in bits =
area X resolution X bit depth

69. Black and White – Example 1
Graphic resolution 600 dpi
Graphic size 2” x 5”
2 X 5 X 600 X 600 x 1
= bits / 8
= bytes / 1024
=439.5 Kb

70. Black and White Example 2
A4 page (10” x 8”)
Resolution 1200 dpi
10 X 8 X 1200 X 1200 x 1
= bits / 8
= bytes / 1024
= Kb / 1024
= Mb

71. Using Colour
We have seen that each pixel of a monochrome image can be represented by 1 bit where 1 represents black and 0 represents white
If we wish to display more than just black or white then each pixel requires more than 1 bit

72. Real life isn’t just black and white – even if grey is used there are 256 different shades of grey!!!
Lots of colours are required to represent real life
Each pixel needs more than 1 bit to represent each colour or shade
This is know as the bit depth of a graphic

73. This image uses 16 colours

74. This image uses 256 colours

75. This image uses 16,777,216 colours – this is called true colour

76. Number of colours available
The number of colours or shades depend on the bit depth of each pixel
The greater the bit depth the greater the range of colours

77. No. of bits No. of possible
used colours
= 2 colours
= 4 colours
= 8 colours
= 16 colours
= 32 colours
= 64 colours
= 128 colours
= 256 colours
The more colours an image has, the greater the memory requirements.

78. Colour – example 1
Same as example 1 for black and white but with a range of 16 colours (24)
Graphic resolution 600 dpi
Graphic size 2” x 5”
Bit depth 4
Calculate the storage requirements?

79. Colour Example 1 2 x 5 x 600 x 600 x 4 = 14400000 bits / 8
= bytes / 1024
= Kb / 1024
= 1.72 Mb
This is substantially different from the 2 colour version at Kb

80. Colour example 2
Same graphic but with a bit depth of 24 bits and a resolution of 1200 dpi
2 x 5 x 1200 x 1200 x 24
= bits / 8
= bytes/1024 / 1024
= 41.2 Mb

81. Colour Example 3 Graphic size is 2” X 2” Resolution of 90 dpi is used
256 colours are used
2 X 2 X 90 X 90 = pixels
Bits per pixel = 8 (256 colours)
32400 X 8 =
/ 8 = bytes/1024 = 32Kb

82. Colour Example 4 Graphic size is 4” X 5” Resolution of 100 dpi is used
16 shades of grey are used
4 X 5 X 100 X 100= pixels
Bits per pixel = 4 (16 colours)
X 4 = bits
/ 8 = bytes/1024 =
98 Kb

83. Colour Example 5 Graphic size is 4” X 2” Resolution of 200 dpi is used
256 colours are used
4 X 2 X 200 X 200= pixels
8 Bits per pixel = 256 colours
X 8 = bits
/ 8 = bytes/1024 =
312.5 Kb

84. Colour Example 6 Graphic size is 3” X 6” Resolution of 80 dpi is used
64 colours are used
3 X 6 X 80 X 80= pixels
Bits per pixel = 6 (64 colours)
X 6 = bits
/ 8 = bytes/1024
= 84 Kb

85. Points to remember about Bitmaps
If a graphic is made larger, then it appears blocky (pixelated)
2. Bit-mapped graphics which have a high resolution or use many colours have a high storage requirement
3. The storage requirement for a screen stays constant even if you add more detail to it

86. 4. A blank screen takes up the same storage requirements as a complex screen.
5. Examples of commands which are unique to bit-mapped applications are those that directly alter a pixel or a group of pixels, for example, set pixel, fill area.
6. Bit-mapped graphics are usually created in paint packages like Paint Shop Pro
7. You can’t edit a whole object, move whole objects or layer objects

87. Compression
Because of their large file size many bitmapped graphics are compressed
Colours are compressed
Extraneous data is deleted

88. Vector Graphics
A vector graphic is a graphic stored as a set of commands of how to draw the object (similar to a program)
Each object is made up out of different attributes (Properties)

89. Circle has the following attributes
Centre X, Centre Y, radius, line thickness, fill pattern, pen colour, fill colour
Centre (X,Y)
Radius

90. Rectangle has the following attributes
Start X, Start Y, End X, End Y,
line thickness, fill pattern, pen colour, fill colour
Start (X,Y)
End (X,Y)

91. Layering
It is possible to combine 2 or more objects together to make another shape
These objects can be moved individually and one can be sent behind the other – this is called layering

92. The following example uses a circle in the background, a rectangle in the middle and text in the foreground
No Entry
No Entry

93. Commands are not dependent on the number of pixels on the screen or on how many colours can be displayed
Objects are resolution independent
When a vector graphic is sent to print the attributes/program for each shape are sent
This allows the graphic to print at the resolution of the printer

94. Points to know about vector graphics
If the image is resized then the quality of the image should stay constant
Allows layers of objects
(good for DTP and CAD)
Eg allows you to move a circle in front of a square without destroying the square.

95. A blank screen takes up very little storage compared to the bit-mapped version
As more objects are placed on screen, the storage requirements increase
The vector graphic take advantage of any high resolution device they are used on (resolution independence)

96. Past paper questions on graphics
1. A graphic of resolution 72 DPI is printed on a 300 DPI printer, the graphic will still only print at a resolution of 72 DPI. Explain why this might happen?
2. Describe 2 ways of finding out if a graphic was created using a bitmap or a vector graphics package.

97. Vector Actions
Examples of actions which can be carried out in vector applications are:
Order of objects could be changed
Individual Objects can be resized
Individual Objects can be moved or edited
Objects can be resized with no loss of quality

98. Written Task
Complete questions 16 – 21 on pages 8 and 9 of How to Pass Higher Computing

99. Written Task
In your jotter copy and complete the table below showing the advantages and disadvantages of each type of graphic
Bitmap Graphic
Vector Graphic
Advantages
Disadvantages

100. Graphics Practical Exercise
You should now complete the practical exercises on vector graphics

101. What you should have learned in about storing numbers
Conversion from binary to decimal and vice versa
Representation of positive numbers up to 32 bits
Representation of negative numbers (using two’s compliment) up to 8 bits
Conversion to and from bit, byte, kilobyte, megabyte, gigabyte, terabyte
Description of floating point representation

102. What you should have learned about representing text
Description of ASCII
Description of Unicode
Advantages and disadvantages of each method of representing text

103. What you should have learned about storing graphics
Features of bitmapped graphics
Relationship between bit depth and number of colours (up to 24 bit true colour)
Features of vector graphics
Advantages and disadvantages of different methods of graphic representation
Relationship between bit depth and file size
The need for data compression

104. What you should do now
Complete the end of topic confidence evaluation sheet
Complete the Activote Assessment
Complete the End of Topic Checkup