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DATA REPRESENTATION (Binary Shifts (:star: Use to multiply or divide by 2…
DATA REPRESENTATION
Data Storage Capacities
:
:star:
1
Bi
nary Digi
t
is a
1
or
0
known as a
bit
:star:
4
Bit
s is a
Nibble
8
Bit
s is a
Byte
:star:
1024
Byte
s = 1
KiloByte
1024
KiloByte
= 1
MegaByte
1024
MegaByte
= 1
GigaByte
1024
GigaByte
= 1
TeraByte
1024
TeraByte
=
??
:!!:
In the exam you can use
1000
instead of 1024
RULE
:check:
To convert between Data Capacities:
From Higher Unit to Lower Unit
Multiply
From Lower Unit to Higher Unit
Divide
:pen:
i.e.
Convert GB to MB
250
GB
X
1024 = 256,000
MB
Convert GB to TB
250
GB
/
1024 = 0.244
TB
:<3:
BBC Bitesize GCSE Bits and Bytes
Binary Overflow
"An error that occurs when the computer attempts to handle a number that is too large for it."
:<3:
BBC Bitesize GCSE Binary Overflow
Binary arithmetic can lead to an overflow condition i.e.
:pen:
The addition of 2 8bit binary numbers
can result in a 9bit number being generated. This is overflow
:star:
Computers can
store
the extra bit(s) elsewhere in the processor
:star:
An overflow
flag
can be set to show that and overflow error/condition has occurred
Binary Shifts
:star:
Use to
multiply
or
divide
by
2
:star:
Left
shift ->
Multiply
. Doubling the value for each shift, or
2 to the power n
where n is the number of shifts
:star:
Right
shift ->
Divide
. Halving the value for each shift, or
2 to the power n
where n is the number of shifts
:star:
Fill and
leading
or
trailing
spaces with
'0'
:star:
Left
shift(s) or
multiplication
can cause
Overflow
:<3:
Binary Shift Video
Parity Bits
or
Check Digits
:star:
Used to
check for errors
:star:
If
one bit
is
read incorrectly
then an
error
will be identified
:star:
If
two bits
in the same binary string are
read incorrectly
then an
error
will be
NOT
be identified
:star:
Check digits
are a way of checking that data
entered, read or transmitted
is correct
:star:
check digits
are
appended
to the end of the number
:star:
check digits
are
calculated
using the other digits in the number
:star:
Binary
number
check digits
are called
parity bits
Even Parity
:star:
An
even
parity bit is
appended
to the binary string to give the final binary string an
even
number of
1s
Odd Parity
:star:
An
odd
parity bit is
appended
to the binary string to give the final binary string an
odd
number of
1s
:<3:
Parity Bit Video
Number Systems
Binary Numbers
What is Binary?
Computers processors
are made of
electrical (logic) circuits
containing
switches
switches
can only be
on
or
off
The
switch state
in
logic circuits
is represented by
1
and
0
1
=
on
0
=
off
Binary
is a
Base2
number system
Placeholders
are based on
2^n
An
8bit
number
place holder values
:
2^8 2^7 2^6 2^5 2^4 2^3 2^2 2^1 2^0
or
128 64 32 16 8 4 2 1
Highest value
represented by
8bits
=
255
128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255
Maximum number range
in
8bits
=
256
0
is a value, thus
0 - 255 represents 256 unique numbers
Only
2
characters are used in the
Binary
number system to represent
all numbers
:
1
and
0
:<3:
BBC Bitesize Introduction to Binary
Converting
Denary
to
Binary
:pen:
Division by 2
method
:pen:
Subtraction
method
Binary
to
Denary
:pen:
Multiply
the
Binary value
by its
place holder value
and note the
Denary value
Add
all the
resultant Denary values
to give the
final Denary value
i.e. Binary 1 0 0 0
Place holder values: 8 4 2 1
1x8 = 8
0x4 = 0
0x2 = 0
0x1 = 0
8+0+0+0 = 8
Binary
number
1000
is equivalent to
Denary
value
8
:<3:
Binary Numbers
Addition
RULES: :check:
0 + 0 = 0
0 + 1 = 1
1 + 1 = 10
1 + 1 + 1 = 11
10 equates to
0 carry 1
11 equates to
1 carry 1
Hexadecimal(Hex) Numbers
What is Hexadecimal
Hexadecimal
is a
Base16
number system
Place holder values
:
16^n......16^5 16^4 16^3 16^2 16^1 16^0
or
.....1,048,576, 65,536, 4,096, 256,16, 1
Uses
16 Characters
to represent
all numbers
:
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
where
A-F
represent
Denary values 10-15
One
Hexadecimal value
is equivalent to a
Binary Nibble
Converting
Denary
to
Hexadecimal
:pen:
3 Step Method:
Covert to Binary
Split Binary
number into 4 bit
Nibbles
Convert
each Nibble
into its Hex equivalent
:<3:
Denary to Hexadecimal video
Hexadecimal
to
Denary
:pen:
16x method
Convert the Hexadecimal number(s) to Denary equivalent:
Working right to left,
Denary value multiplied by 16^0(1),
next
Denary value multiplied by 16^1(16),
next
Denary value multiplied by 16^2(256),
etc
Add all the resultant values
for the multiplications together to give the Denary equivalent of the starting Hexadecimal.
:pen:
Convert 7C to its Denary equivalent.
7C
7 = 7, C = 12
7 x 16 = 112, 12 x 1 = 12
112 + 12 = 124
7C = 124
:pen:
Via Binary
7C
7 = 0111 C = 1100
8 bit Binary = 01111100
01111100 = 64 + 32 + 16 + 8 + 4
7C = 124
Benefits
Simpler
to remember than large binary numbers
Less
chance of an
error
than with
Binary
Easier to
convert
between
Binary
and
Hex
than between
Binary
and
Denary
Quicker
to write
Hex
than
Binary
:<3:
BBC Bitesize Hexadecimal
Denary
or
Decimal
Denary
is a
Base10
number system
Uses
10 Characters
to represent
all numbers
:
0,1,2,3,4,5,6,7,8,9
Place holder values
:
10^n......10^5 10^4 10^3 10^2 10^1 10^0
or
.....100,000 10,000, 1,000 100,10, 1
Exam Board Specification
Learners should have studied the following:
Units
• bit, nibble, byte, kilobyte, megabyte, gigabyte, terabyte, petabyte
• how data needs to be converted into a binary format to be processed by a computer
Numbers
• how to convert positive denary whole numbers (0–255) into 8 bit binary numbers and vice versa
• how to add two 8 bit binary integers and explain overflow errors which may occur
• binary shifts
• how to convert positive denary whole numbers (0–255) into 2 digit hexadecimal numbers and vice versa
• how to convert from binary to hexadecimal equivalents and vice versa
• check digits
Characters
• the use of binary codes to represent characters
• the term ‘character-set’
• the relationship between the number of bits per character in a character set and the number of
characters which can be represented (for example ASCII, extended ASCII and Unicode)
Images
• how an image is represented as a series of pixels represented in binary
• metadata included in the file
• the effect of colour depth and resolution on the size of an image file
Sound
• how sound can be sampled and stored in digital form
• how sampling intervals and other factors affect the size of a sound file and the quality of its playback:
sample size
bit rate
sampling frequency
Compression
• need for compression
• types of compression:
lossy
lossless
:<3:
BBC Bitesize KS4 Binary and data representation
Characters
:star::
Character Set
A character set is
a collection of characters
that a
computer recognises
from their
Binary representation
:star:
ASCII: American Standard Code for Information Interchange
Unicode
:<3:
BBC Bitesize Character Sets