1. Data Representation
Conversion
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2. Learning Objectives:
Express numbers in binary, binary coded decimal (BCD), octal and hexadecimal.
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3. The Binary System
Computers store information (data of all types – numbers, characters, sound, pictures, …) in Binary format i.e. base 2.
i.e. 0 or 1
Used because computers can only store and understand 2 states:
i.e. whether a circuit has current flowing or not / circuit is closed or open / voltage is high or low. 1
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4. Bits and Bytes
A binary digit (1 or 0) is known as a ‘bit’, short for BInary digiT.
In modern computers bits are grouped in 8 bit bytes.
A Nibble is 4 bits (half a byte).
A Word is the number of bits that the CPU can process simultaneously.
Determines the speed of the computer.
Processors can have 8-, 16-, 32-(standard) or 64-(fast) bit word sizes (or more).
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5. Character set
The symbols that a computer (software) can recognise which are represented by binary codes that the computer understands.
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6. Character Representation
Over the years different computer designers have used different sets of binary codes for representing characters in a character set.
This has led to great difficulty in transferring information from one computer to another.
i.e. which binary code represents each character
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7. ASCII (American Standard Code for Information Interchange)
Represents each character in a standard character set as a single byte binary code.
The standard code form that most PCs use to allow for communication between systems.
Usually uses a 7 bit binary code so can store 128 different characters and simple communications protocols.
Sufficient for all characters on a standard keyboard plus control codes.
Can be extended (extended ASCII) to use 8 bits (so can store 256 characters) to encode Latin language characters.
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8. ASCII code
The first 32 ASCII codes are used for simple communications protocols, not characters.
e.g. ACK – acknowledge and would be sent by a device to acknowledge receipt of data.
– 2
– 1
…..
– A
– B
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9. Representing Characters and Numbers
e.g. If the ‘A’ key is pressed ‘ ’ is sent to the CPU.
If the 1 key is pressed then ‘ ’ is sent to the CPU.
If the user wants to print ‘123’ the codes for 1, 2 & 3 are sent to the printer.
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10. Binary Arithmetic Rules
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 (carry 1)
1+1+1 = 1 (carry 1)
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11. Arithmetic
ASCII coding is fine for input and output but useless for arithmetic:
i.e. not 1
There is no easy way to perform calculations on the numbers stored in this way.
Numbers which are to be used in calculations are therefore held in binary format.
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12. Decimal or Denary system
134 =
Each column is worth 10X as much as the last i.e. base 10 (10 fingers!).
100 10 1 3 4
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13. Binary system
134 =
Each column is worth 2X as much as the last i.e. base 2.
128 64 32 16 8 4 2 1
Most Significant Bit
(MSB)
Least Significant Bit
(LSB)
Increasing Bit Status
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14. Binary – Decimal Spreadsheet Converter 1
Try using it to ‘play’ with binary numbers.
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15. Denary -> Binary e.g. 117
Always use the column headings for a byte (8 bits).
117 < 128 so put a 0 and repeat.
128 64 32 16 8 4 2 1
117 > 64 so put a 1.
128 64 32 16 8 4 2 1
= 53 , 53 > 32, so put a 1.
128 64 32 16 8 4 2 1
= 21 , 21 > 16, so put a 1.
Continue this until:
128 64 32 16 8 4 2 1

16. Binary -> Denary e.g. 10110110 So 10110110 = 128 + 32 + 16 + 4 + 2
Put the column headings above the binary number and add up all the columns with a 1 in them.
128 64 32 16 8 4 2 1
So =
= 182 (denary)

17. Questions 1. Convert the following binary numbers to decimal. 00113
0110 6
1010 10 65 69
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18. 8 bit patterns
Because in modern computers bits are grouped in 8 bit bytes numbers in binary format are usually written in 8 bit patterns even if there are unnecessary left leading 0’s.
e.g. 11(binary) = 3 (decimal)
But you will usually find it written as
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19. Decimal -> Binary Questions
2. Convert the following decimal numbers to binary: 5 7 1 26 68
137
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20. Size of number
Using only one byte to hold a number of places a restriction on the size of number the computer can hold.
Therefore four or more consecutive bytes are commonly used to store numbers
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21. Binary -> Decimal Questions
3. What is the largest decimal number that can be held in (hint: 2^no. of bits):
1 byte
255 (2^8 - 1)
2 bytes
65535 (2^16 - 1)
3 bytes
(2^24 - 1)
4 bytes
(2^32-1)
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22. Hexadecimal Counts in 16’s. Uses the digits 0 – 9 and letters A – F.
We need symbols going further than 0 to 9 (only 10 symbols and we need 16!).
We could invent 6 more symbols but we would have to learn them, so we use 6 that we already know, the letters A to F.
Each digit is worth 16X as much as the one to the right.
Each hex bit = 4 binary bits.
e.g. F (decimal 15) = 1111
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23. Hexadecimal Denary / Decimal Binary Hexadecimal 1 00000001 2 00000010
etc …. 9 10 A 11 B 15 F 16
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24. Decimal -> Hexadecimal e.g. 75
Use the column headings 1, 16, 256, … (16^n)
75 < 4096 & 75 < 256 so put a 0.
4096
256 16 1
75 > 16 & 75 / 16 = 4 r 11
11= B (hexadecimal)
4096
256 16 1 4 B
So 75 = 4B (Hexadecimal)

25. Hexadecimal-Denary Converter
Try using it to ‘play’ with hexadecimal numbers.
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26. Denary -> Hexadecimal Question:
Translate 101 (denary) into hexadecimal. 65
Translate 64 (denary) into hexadecimal. 40
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27. Hexadecimal -> Denary e.g. BD
4096
256 16 1 B D 11 13
B = 11 , D = 13
BD = (11 * 16) + 13 =
= 189 ( in denary)
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28. Hexadecimal -> Denary Question:
Translate 96 (hexadecimal) into denary.
150
Translate the 75 (hexadecimal) into denary.
117
Translate the 30 (hexadecimal) into denary. 48
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29. Binary -> Hexadecimal
Translate each group of 4 bits into denary and then into hexadecimal.
e.g (binary) = (1001)(0010) = (9)(2) = 92 (hexadecimal)
An alternative way to convert from denary to hexadecimal is to convert to binary and then do the above.
Subsequently, vice versa for hexadecimal to denary.
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30. Hexadecimal
In order to ease the task of examining the contents of memory or a computer file, binary numbers are commonly put into groups of 4 bits and displayed in the form of hexadecimal numbers, base 16.
Used as a shorthand for binary.
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31. Questions:
The ASCII code for the letter N is Show how to represent this in
a) denary 78
b) hexadecimal
= (0100) (1110) = (4 + 14) = 4E
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32. Octal
Counts in 8’s.
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33. Decimal -> Octal e.g. 117 So 117 = 165 (Octal)
Use the column headings 1, 8, 64, 512 … (8^n)
117 < 512 so put a 0.
512 64 8 1
117 > 64 & 117 / 64 = 1 r 53
512 64 8 1
53 > 8 & 53 / 8 = 6 r 5
512 64 8 1 6 5
So 117 = 165 (Octal)

34. Decimal -> Octal Question:
Translate 101 (denary) into octal.
145
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35. Octal -> Denary e.g. 76 76 (octal) = (7 * 8) + 6 = 56 + 6
512 64 8 1 7 6
76 (octal) = (7 * 8) + 6 =
= 62 ( in denary)
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36. Octal -> Decimal Question:
Translate 101 (octal) into denary. 65
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37. Binary -> Octal
(binary)
Arrange in 3’s from the right.
Add leading 0
Each group of three bits converted denary but the final result is octal.
165 (octal)
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38. BCD (Binary Coded Decimal)
Represents numbers only by representing each decimal digit by a 4 bit binary code.
Decimal
Binary
0000 1
0001 2
0010
etc.... 9
1001
e.g. 19 is shown by:
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39. Denary -> BCD Question
Convert (denary) into BCD.
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40. BCD -> Denary e.g. 001001110110 Split into groups of 4.
(0010) (0111) (0110)
Translate each group of 4 into decimal in the same fashion as binary -> decimal.
So = 276 (denary)
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41. BCD -> Denary Question
Convert (BCD) into denary.
835
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42. BCD (Binary Coded Decimal)
Advantages:
Easy to convert from BCD to decimal and vice versa:
as a BCD number is split into groups of four bits and each group converted directly to the corresponding decimal digit
Used in some pocket calculators
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43. . Fixed Point Binary 8 4 2 1 ½ ¼ 1/8 1/16 6.75 = 0110.1100
A number with a decimal point is known (strangely!) as a real number as opposed to an integer which is a whole number.
We can extend the binary system to represent real numbers by reserving some bits for the real or fractional part. 8 4 2 1
1/8
1/16 .
6.75 =
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44. Fixed Point Binary Precision
110.1 = 6.5
= 6.75
We have missed out 6.51 to 6.74!
This means accuracy is poor.
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45. BCD (Binary Coded Decimal)
Advantages:
When storing fractional numbers in BCD no ‘rounding’ occurs:
As each digit is encoded separately so as many bits as necessary are used to represent the complete number
Used in business applications where every significant digit has to be retained in a result.
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46. BCD (Binary Coded Decimal)
Disadvantage
More bits are required to store each number than pure binary e.g.
19 in BCD = (8 bits)
19 in normal binary format = (5 bits)
Difficult to calculate with:
e.g
‘10’ is wrong
invalid
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47. BCD (Binary Coded Decimal)
Only the first ten out of 16 combinations of 4 digits are used to encode the decimal digits ‘0’ to ‘9’
Therefore whenever the sum of two binary digits is >9, 6 has to be added to skip over the 6 unused codes.
e.g
i.e. 20 which is correct
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48. Unicode
16 bit code so can store characters and codes and simple communications protocols.
Used to allow coding of languages that do not use Western characters.
Currently supports 24 language scripts.
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49. Plenary Express the denary value 109 as
A binary number using an 8-bit byte.
An octal number.
A hexadecimal number.     
A number in binary coded decimal (BCD).
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50. Plenary
155 6D
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