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# Binary

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## L Stevens

on 29 January 2014

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#### Transcript of Binary

010000100110010100100000011100110111010101110010011001
010010000001110100011011110010000001100100011100100110
100101101110011010110010000001111001011011110111010101
110010001000000100111101110110011000010110110001110100
01101001011011100110010100101110 Binary to Decimal Binary in 60s Binary Hexadecimal http://forums.cisco.com/CertCom/game/binary_game_page.htm Binary Coded Decimal * Two's Complement * Adding Binary Binary Subtraction Problems with Binary How old are you in denary?

How many bedrooms in your house?
When was liverpool fc founded?
When was Everton FC founded?
When was Lemmings released?

What are these in BCD? BCD: Activity Add the following binary numbers

0001+0001
1000+0101
1111+1001
1001+1000 Activity: Binary Addition In pairs create 3 binary addition questions for each other
Attempt the questions you have been given then give them back and get your original ones back
Mark the answers and see if you understand how to do binary addition Make 3 Binary Addition Questions Subtract the following binary numbers

0001-0001
1000-0101
1111-1001
1001-1000 Activity: Binary Subtraction In pairs create 3 binary subtraction questions for each other
Attempt the questions you have been given then give them back and get your original ones back
Mark the answers and see if you understand how to do binary subtraction Make 3 Binary Subtraction Questions Represent the following numbers using sign and magnitude

-4
-1
-73
-127 Activity: Sign and Magnitude 11110000
10110000
01111111
10101011
11111101

Can you test your friend again? What are the following sign
and magnitude numbers in denary? Represent the following numbers using two's complement
-4
-1
-73
-127 Activity: Two's Complement 11110000
10110000
01111111
10101011
11111101

Can you test your friend again? What are the following
two's complement numbers in denary? How old are you in denary?

How many bedrooms in your house?

What are these in Octal? Octal Activity How old are you in denary?

How many bedrooms in your house?
When was liverpool fc founded?
When was Everton FC founded?
When was Lemmings released?

What are these in Octal? Octal Activity How old are you in denary?

How many bedrooms in your house?
When was liverpool fc founded?
When was Everton FC founded?
When was Lemmings released?

What are these in Hexadecimal Hexadecimal Activity What is this below? What is a byte? What is a bit? What is a nibble? Find Out Octal Numbers Know of different forms of number
Apply techniques to convert numbers from denary to another form
Apply techniques to add and substract binary numbers Objectives Identify different forms of number (e)
convert numbers from denary to: Binary, Binary coded Decimal, Octal, Hexadecimal (c)
convert negative numbers into binary using sign and magnitude (e)
convert negative numbers into binary using two's complement (c)
Use techniques to Add and subtract binary numbers (c) What about letters? ABCDEFGHIJKLM
NOPQRSTUVWXYZ It does appear to be more difficult using twos complement.

However it is essential as it makes arithmetic easier

This is especially true for subtracting a number More Difficult Two's Complement This is another way to represent a negative number. Two's Complement The most significant bit stays as a number but is made negative. This means the column headings are -128 64 32 16 8 4 2 1 + 117 doesn't need to use the MSB so it stays as
01110101 - 117 is
- 117 = -128+(8+2+1)
This then becomes 10001011 Sign and Magnitude If a computer systems uses a byte to store a number in the way suggested before there are three problems

The biggest number that can be represented is 255 because there aren't enough bits to store bigger numbers.
This can be resolved by using more than one byte to store a number.
Most computer systems use either two or four bytes to store a number.

How do we represent fractions? This is looked at in Unit 3 A2

Negative numbers cannot be stored
This can be easily solved using two different methods
Sign and Magnitude
Twos Complement * Problems with Sign and Magnitude * There are two types of data in the same byte. IE the MSB is a sign and therefore must be treated differently

This makes arithmetic algorithms very complex
Also extending from 1 byte to 2 or more bytes is also very difficult

Can you see why? * Twos Complement * Try to write a step by step guide of how to calculate two's complement as described in the video

NB. You may get asked a question on this * 1 - work out the positive binary value
* 2 - change all the digits

Try the previous activity using this method It does appear to be more difficult using twos complement.

However it is essential as it makes arithmetic easier

This is true when you compare it to Sign and Magnitude More Difficult This is where twos complement is useful

To take away one number from another simply write the number to be subtracted as a two's complement negative number and then add them up

Try

91 - 18 Subtraction Using Two's Complement Did you work it out? 91 = 01011011
-18 as a twos complement number is -128 + 110
= -128 + (+64 + 32 +8 +4 +2)

=11101110

01011011+
11101110
=
1/01001001

Notice when you do this you get an extra 1 to carry at the end. This gets cancelled out as the number can only be 8 bits

Try it, is it correct?

Try these and test the above theory (represent them as a byte/8 bits)
3 - 2
32 - 12
10 - 6 Octal and Binary Relationship Can you explain the relationship between binary and octal representation? Binary and Octal Groups of three binary bits from the right
Give octal digits when converted into decimal values Binary and Octal So say we had to work out 90 in octal

We would work it out as a binary number first

01011010

split it into 3 bits from the right

001 011 010

then work out each of those groups of three bits

1 3 2
so 90 as an OCTAL number is 132 Lets Check 132 octal in Denary

64 8 1
1 3 2

(1*64) + (3*8) + (2*1) = 90 In pairs create 3 Octal questions for each other
Attempt the questions you have been given then give them back and get your original ones back
Mark the answers and see if you both understand how to do Octal Make 3 Octal Questions Hexadecimal What is the base of Hex?

Can you find out? Hexadecimal so Hex is Base 16

As we have to use 16 numbers we do not want to have to use base ten numbers 10/11/12 ... etc. again as it will get confusing with the single digits 1/2/3 etc..

Therefore we use ABCDEF for 10, 11, 12, 13, 14, 15

But How do we find out the representation of a denary number as a Hex number?

Well as before we have the powers of 16 at the top

4096 256 16 1

If we want to get the number 15 then it would just be
F
as 1 F is 15.