# Boolean Algebra. 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.

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Boolean Algebra

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...

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

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

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

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

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

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

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

Base 2

1286432168421

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

Base 2 1286432168421 10010110

150

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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?

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

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

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

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

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

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

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

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?

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

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)

ASCII Character Set

Words In order to write the word Hello

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

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

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

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

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

Abstract a little Hollerith Punch Card

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Count down, and reset the clock

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Shifting 2 Types Logical Shift Arithmetic Shift

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

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

Arithmetic Shift Similar to the Logical Shift except:

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

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

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

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

Shifting Summary Shifting is useful because?

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

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

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

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

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

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

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

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

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

This is like saying 01010000 x 2 00000110

Task 1 Can you represent 24 as a binary number?

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

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

-14

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

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...

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.

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

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

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

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

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

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...

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

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

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

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 ……?

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

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

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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