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Boolean Algebra

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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...

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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

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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

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

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What if we had 16?

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

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

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

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

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Base 2

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1286432168421

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

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Base 2 1286432168421 10010110

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150

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

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

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

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

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

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

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

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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

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

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

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

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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

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

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

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

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64Bit... What about 64bit?

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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?

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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

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

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

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Hexadecimal notation 12345678910111213141516 0123456789ABCDEF

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

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

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

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

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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?

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

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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)

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ASCII Character Set

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Words In order to write the word Hello

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

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

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

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

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

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Abstract a little Hollerith Punch Card

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

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

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

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

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

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

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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

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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

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

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

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

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

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

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

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Count down, and reset the clock

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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

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

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

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

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

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Adding 199 001 + 200

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

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

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

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

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

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

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

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

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

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

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

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

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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

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Shifting 2 Types Logical Shift Arithmetic Shift

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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

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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

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Arithmetic Shift Similar to the Logical Shift except:

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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

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

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

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Answers 1.62 2.124 3.248 4.31

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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

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Shifting Summary Shifting is useful because?

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

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

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

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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

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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

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

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

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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

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

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This is like saying 01010000 x 2 00000110

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Task 1 Can you represent 24 as a binary number?

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

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How about 0.0625

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

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-14

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

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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...

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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.

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

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

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Answer 19.25 104

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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

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

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Lesson 6 Start the task...

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

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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...

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

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Your Turn

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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

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

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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 ……?

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

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

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

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Computer Systems Architecture Copyright © Genetic Computer School 2008 CSA 1- 0 Lesson 1 Number System.

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