1. Boolean Algebra

2. Starter Task State 5 basic data types together with one operation that can be performed on each. We have looked at 4 in the past few weeks...

4. Answers Variable Array – Insert a value into a certain position in the array Stack – Pop, Push Queue – enQueue, Serve Binary Tree – Search through and add values / Delete values Linked List – Add items to the list, remove items from the list

5. Over the next Few Weeks... Working through the booklets... Binary Numbers Hexadecimal Numbers Characters ASCII – Strings and Character sets Negative Numbers Two’s complement, representation sign, magnitude Two’s complementsign, magnitude Shifting Fixed Point, Floating Point – Conversion + Benefits Rounding + Truncation effects on accuracy Overflow, Underflow

6. We use Base 10... Because we have 10 fingers

7. What if we had 16?

8. This gives us the structure of... Every 10 th value we add an extra number... – 2 – 12 – 23 – 178 – Etc...

9. Base 10 looks like... The number 1583 means 1 'thousand', 5 'hundreds', 8 'tens' and 3 'units'...

10. Where each column is 10^X 10^1 = 10 10^2 = 100 10^3 = 1000 10^4 = 10,000

11. Consider this... Computers only use base 2... How would this look in a table? What will a base 2 set of numbers look like?

12. Base 2

13. 1286432168421

14. Representing numbers... What number do you think this will represent? 10010110

15. Base 2 1286432168421 10010110

16. 150

17. What do the following numbers = 10000101 01000001 11111111 10110101 00000000 10101111 00101010 00111011 10111010

18. Answers 10000101 = 128 + 0 + 0 + 0 + 0 + 4 + 0 + 1 = 133 65 255 181 0 175 42 59 186

19. What if I wanted to go the other way... How would I convert 22 into binary? 1286432168421

20. Actions 22 / 2 = 11 r 0 _ _ _ _0

21. Actions 22 / 2 = 11 r 0 11 / 2 = 5 r 1 _ _ _10

22. Actions 22 / 2 = 11 r 0 11 / 2 = 5 r 1 5 / 2 = 2 r 1 _ _110

23. Actions 22 / 2 = 11 r 0 11 / 2 = 5 r 1 5 / 2 = 2 r 1 2 / 2 = 1 r 0 _ 0110

24. Actions 22 / 2 = 11 r 0 11 / 2 = 5 r 1 5 / 2 = 2 r 1 2 / 2 = 1 r 0 1 / 2 = 0 R 1 1 0110

25. What if I wanted to go the other way... How would I convert 22 into binary? 1286432168421 00010110

26. Actions Divide each value into largest number and put a one in the table...

27. What are the following binary numbers? 156 45 78 97 123 245 253 7 184 111

28. What are the following binary numbers? 156 = 10011100 45 = 00101101 78 = 01001110 97 = 01100001 123 = 01111011 245 = 11110101 253 = 11111101 7 = 00000111 184 = 10111000 111 = 01101111

29. Bytes So far everything has consisted of once byte... 8 bits... Byte 4bits... Nybble (rarely used now) 1101=13

30. Standard Computers are 32bit... What would be the maximum value that a 32bit computer can hold? You might want to use a calculator...

31. 32 bit = 4,294,967,295 This is the biggest value for a 32bit computer... However it doesn’t exist in many operating systems...

32. 64Bit... What about 64bit?

33. Again... 18,446,744,073,709,552,000 This number will not be found in 64bit operating systems... Can you look down the list of contents in the booklet to see why?

34. Summary Video’s http://www.youtube.com/watch?v=qdFmSlFoj Iw http://www.youtube.com/watch?v=qdFmSlFoj Iw http://courses.cs.vt.edu/csonline/NumberSyst ems/Lessons/DecimalToBinaryConversion/ind ex.html http://courses.cs.vt.edu/csonline/NumberSyst ems/Lessons/DecimalToBinaryConversion/ind ex.html

35. Convertor http://mistupid.com/computers/binaryconv.ht m http://mistupid.com/computers/binaryconv.ht m

36. Lesson 2 Hexadecimal notation... What do you think hexadecimal notation looks like?

37. Hexadecimal notation 12345678910111213141516 0123456789ABCDEF

38. What’s it used for Give a more readable notation for people to use. Decimal =10,995 Binary =10101011110011 Hexadecimal= 2AF3

39. How it’s used... Have you ever seen... #33FD56 In HTML coding... Gives you a colour #33FD56

40. Each part = nybble #33FD56 33=51=0011 0011 FD=253=1111 1101 56=86=0101 0110

41. Task 2 Fill in the table: You will have to remember how to complete binary numbers...

42. Now we have 3 ways to represent numbers... What is the point? We can represent 0-255 numbers using 1byte or 1 hexadecimal code Can you think of why we would use this?

43. Character Sets All the symbols, letters, numbers have a binary representation There are 128 different characters that we call ASCII This is a Standard!

44. ASCII (American Standardised Code for Information Interchange) Needed so that computers share documents together: Others include: EBCDIC (Extended Binary Coded Decimal Interchange Code) ISO 8859, for ß (German), ñ (Spanish), å (Swedish) ANSI (American National Standards Institute)

45. ASCII Character Set

46. Words In order to write the word Hello

47. Task How would hello world read? How many bits are used for each character? Letter HELLOWORLD Number Binary

48. Hello World 01001000 01100101 01101100 01101100 01101111 00100000 01010111 01101111 01110010 01101100 01100100

49. Hello World 48 65 6c 6c 6f 20 57 6f 72 6c 64

50. Practice What about the sentence: There are 10 types of people in the world: those who understand binary, and those who don't.

51. Thankful for abstraction? Imagine trying to code in binary...

52. Abstract a little Hollerith Punch Card

53. Tools http://home2.paulschou.net/tools/xlate/

54. Monkeys We don’t need to be intelligent to talk binary

55. Summary What is binary? How can we use binary? What is Hexadecimal? Why is it used?

56. Lesson 3 Negative numbers? What is the Largest and smallest numbers that you can have?

57. Negative numbers Need to put in the minus sign – Sign/Magnitude Two’s complement

58. Sign/Magnitude Representation We have 8 bits 10010101 The first represents the sign: 10010101

59. Converting Binary -26 26/2 =13 r 0 13/2 =6 r 1 6 / 2=3 r 0 3 / 2=1 r 1 1 / 2=0 r 1 11010

60. 8 bit information Take our number: 11010 It is negative: -26 Starts with a 1, then put 0’s in and then the number... 10011010

61. Practice a) -3 b) -10 c) -62 d) 62 e) 13 f) 128

62. Answers a) -3 =10000011 b) -10 =10001010 c) -62 =10111110 d) 62 =00111110 e) 13 =00001101 f) 128 =10000000

63. What are the issues with this? What is the problem with representing 128? How would we type in -128 What is the main limtation?

64. Problems Halves the amount of numbers that you can have...

65. This is why it is rarely used any more. Sign/Magnitude has/is being phased out Two’s Complement is the new thing!

66. Two’s Compliment How else could we represent negative numbers?

67. Count down, and reset the clock

68. So to work out a number: Find the number in Binary as (-)35: – 100011 Find the complement: – 011100 Add 1 to the value: – 011100 – 000001 – 011101

69. Convert the following numbers 55 111 19 27 79 88 Can you spot a trend with these number?

70. Answers -55=11001001 -111=10010001 -19=11101101 -27=11100101 -79=10110001 -88=10101000 Can you spot a trend with these number?

71. Convertor http://www.rsu.edu/faculty/PMacpherson/Pr ograms/twos.html http://www.rsu.edu/faculty/PMacpherson/Pr ograms/twos.html

72. Math with Binary Works just like denary Math... When you add one, it increase’s the value: Add 1 to 199

73. Adding 199 001 + 200

74. Same kind of thing for Binary 88 = 01011000 Complement = 10100111 00000001 1 + 1 = 0 r 1 0 1

75. Same kind of thing for Binary 88 = 01011000 Complement = 10100111 00000001 1 + 1 = 0 r 1 00 1

76. Same kind of thing for Binary 88 = 01011000 Complement = 10100111 00000001 1 + 1 = 0 r 1 000 1

77. Same kind of thing for Binary 88 = 01011000 Complement = 10100111 00000001 0 + 1 = 1 r 0 1000 0

78. Same kind of thing for Binary 88 = 01011000 Complement = 10100111 00000001 0 + 0 = 0 r 0 01000 0

79. Same kind of thing for Binary 88 = 01011000 Complement = 10100111 00000001 0 + 1 = 1 r 0 101000 0

80. Same kind of thing for Binary 88 = 01011000 Complement = 10100111 00000001 0 + 0 = 0 r 0 0101000 0

81. Same kind of thing for Binary 88 = 01011000 Complement = 10100111 00000001 1 + 0 = 0 r 0 10101000 0

82. Your Turn Try Converting these -18 -64 -19 -16 58

83. Answers Try Converting these -18 = 11101110 -64 = 11000000 -19 = 11101101 -16 = 11110000 58 = 00111010

84. Summary Sign / Magnitude Two’s Compliment Adding Binary Numbers together

85. Shifting As the name implies: Moving items along: Try the task in the booklets... 101101011011010110110 Shift LeftShift Right

86. Shifting 10001101 – Shift Left - 00011010 10010101 – Shift Right - 01001010 00010101 – Shift Left - 00101010 10010101 – Shift Left - 00101010 10010110 – Shift Right - 01001011 01101010 – Shift Left - 11010110

87. Shifting 2 Types Logical Shift Arithmetic Shift

88. Logical Shift Need a set of binary: – Carry Flag: Everything either goes left / right The item that drops off the list gets put in the Carry Flag

89. Logical Shifting 10001101 – Shift Left - 00011010 10010101 – Shift Right - 01001010 00010101 – Shift Left - 00101010 10010101 – Shift Left - 00101010 10010110 – Shift Right - 01001011 01101010 – Shift Left - 01101010 Carry Flag 1 1 0 1 0 0

90. Arithmetic Shift Similar to the Logical Shift except:

91. Arithmetic Shifting 10001101 – Shift Left - 00011011 10010101 – Shift Right - 11001010 00010100 – Shift Left - 00101000 10010101 – Shift Left - 00101011 10010110 – Shift Right - 11001011 01101010 – Shift Left - 11010100 Carry Flag 1 1 0 1 0 0

92. Look at the shifting task... 00110101 =01101010 10110010 =01011001 00101110 =00010111 11001011 =10010111 Carry Flag 0 0 0 1

93. Next Task Perform the following steps, converting the number to see what happens...

94. Answers 1.62 2.124 3.248 4.31

95. Practice Using an Array of 1’s or 0’s can you code an arithmetic shift to the left or the right? Create an array : binary(10) Populate it with numbers – For i as integer 1 to binary.length-1 Shift to the left = – For i as integer 1 to binary.length-1 Binary(binary.length – i) = binary(binary.length – i+1) – Next

96. Shifting Summary Shifting is useful because?

97. Lesson 5 Complete the binary task in the booklets 11 15 -95 185 -34 -15

98. Answers 11=00001011 15=00001111 -95=10100001 125=01111101 -34=11011110 -15=11110001

99. Lesson 5 Floating Point Using “binary” places to represent numbers...

100. In Decimal In denary the integer 25000 can be written as 2.5 x 10 4 This has the structure of: – Unit.Decimal x 10 ^ X – Mantissa x 10 Exponent

101. This can also be applied to binary Mantissa x 10 Exponent First step – we need to get the number in its lowest form... This is achieved through halving = 20 x 2 1 = 10 x 2 2 = 5 x 2 3 = 2.5 x 2 4 = 1.25 x 2 5 = 0.625 x 2 6

102. Golden Rule Need to get the mantissa to a value between 0.5 and 1 = 0.625 x 2 6

103. A floating point number system uses 16-bit numbers. 8 bits for the (signed)mantissa, and 8 bits for the (signed) exponent. 01010001 00000101

104. How this works... 0.625 x 2 6 Means – 0.625 arithmetically shifted right 6 times... To get back to the number we must shift it left 6 times

105. 01010000 00000110 Sign1/21/41/81/161/321/641/128 01010000 = 0.625 = 6 = Mantissa = Exponent Sign421 0110

106. This is like saying 01010000 x 2 00000110

107. Task 1 Can you represent 24 as a binary number?

108. 24 Sign1/21/41/81/161/321/641/128 01100000 = 0.75 = 5 = Mantissa = Exponent Sign421 0101

109. How about 0.0625

110. 0.0625 Sign1/21/41/81/161/321/641/128 01000000 = 0.5 = -3 = Mantissa = Exponent Sign421 1011

111. -14

112. Sign1/21/41/81/161/321/641/128 11110000 = -0.875 Sign421 0100 = 4 = Mantissa = Exponent

113. Whats the difference with these numbers? Can you see how these would be used instead of previous numbers that we have looked at? You have used these before...

114. Real Numbers Whenever you declare a double then you are going to be using floating point numbers. The Mantissa and the exponent can change in length depending on the definitions placed upon it.

115. Conversion Techniques You start with the Mantissa If the Exponent is +ve – Shift to the left If it’s negative – Shift to the right

116. Can you use these examples to convert back? 01010001 0101 011010000 0110

117. Answer 19.25 104

118. Floating Point Numbers Pro’s – A much wider range of numbers can be declared Con’s – More Space is required to use them – Slower Processing Times – Lack of Precision some real numbers can only be represented approximately

119. Range To increase the range you increase the mantissa To increase the precision you increase the mantissa

120. Lesson 6 Start the task...

121. Answers 12 = 01100000 0100 9.76 – Can this be done? Can you convert 0.1 into a decimal?

122. Rounding For example, 1.36 rounded to one decimal place is 1.4 giving a rounding error of 0.04. This can be done in binary too however there are many different ways that this can be done...

123. If we round the following binary number... 10101 4 b.p. 2 b.p. What do we get?

124. Your Turn

125. Truncation For example, 1.36 truncated to one decimal place is 1.3 giving a truncation error of 0.6. Shift the number to the right 3 places..., then back to the left 3 places 01101001

126. Your Turn Truncate the numbers, Find a rule for the amount of error within truncation.

127. Error Accumulation If you round 1.356 to 2 decimal places the rounding errors is ………? If you then multiply the result by 1000 the rounding error becomes ……?

128. Overflow Using 8bit integer perform the following calculation... 150+130 Represent the answer as binary

129. Overflow Convert the following numbers... Add the following numbers together and then convert the answer... Is this right?

130. Summary There are issues with binary representation Good programming languages will warn you of this... Needs to be handled by the developer.