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Fundamentals of data representation (Representing images, sound and other…
Fundamentals of data representation
Number systems
Natural numbers:
positive whole numbers including zero; represented by 'N'
Integer numbers:
whole, positive or negative numbers, including zero; represented by 'Z'
Rational numbers:
numbers that can be represented as a fraction or a ratio or integers; represented by 'Q'
Irrational numbers:
a number that cannot be represented as a fraction or ratio as the decimal form will contain infinite repeating values
Real numbers:
'R' is the set of all 'possible real world quantities' : any positive or negative number with or without a fractional part
Ordinal numbers:
a number used to identify position relative to other numbers
Cardinal numbers:
a number that identifies the size of something
Counting and measurements
count using natural numbers as we only need to use positive whole numbers
measure using real numbers as the range of numbers may be positive or negative and may require a fractional part
Number bases:
the number of digits available in within a particular number system
number base
decimal
(base 10)
binary
(base 2)
hexadecimal
(base 16)
Units of information
Bit:
a single binary digit from a binary number, a fundamental unit of information
Byte:
a group of 8 bits
2^n different values can be represented by n bits
Binary prefixes
kibi
, Ki - 2^10
mebi
, Mi - 2^20
gibi
, Gi -2^30
tebi
, Ti- 2^40
Decimal prefixes
kilo
, k - 10^3
mega
, M - 10^6
giga
, G - 10^9
tera
, T - 10^12
Binary number system
Unsigned binary:
binary that represents positive numbers only
unsigned arithmetic
0 + 0 = 0
0 + 1 or 1 + 0 = 1
1 + 1 = 0 carry 1
1 + 1 + 1 =1 carry 1
Two's complement:
method of working with signed binary values
with 8 bits it is possible to represent 256 different numbers ranging from -128 to 127
Numbers with a fractional part
Fixed point:
where the decimal/ binary point is fixed with in the number
Advantages:
the values are handled in the same way as decimal values so any hardware configured to work on integers, can be used on reals. This also makes the processing of fixed point numbers faster than floating point values as there is no processing required to move the binary point
The absolute error will always be the same, whereas with floating point numbers the absolute error could vary. This means the precision is retained albeit within a more limited range than the floating point representation
It is suited to applications where speed is more important than precision
It is suited to applications where an absolute level of precision is required
Floating point:
where the decimal/binary point can move within a number
Mantissa
and
exponent
: mantissa stores the numbers and exponent moves the binary point
Advantages:
a much wider range of numbers can be produced with the same number of nits as the fixed point system
consequently, floating point lends itself to applications where a wide range of values may need to be represented
Rounding errors
Absolute and relative ideas
Range and precision
Normalisation of floating point form
Underflow and overflow
Information coding systems
Character form of a decimal digit
ASCII and Unicode
Error checking and correction
Representing images, sound and other data
Bit patterns, images, sound and other data
Analogue and digital
Analogue/digital conversions
Bitmapped graphics
Vector graphics
Vector graphics versus bitmapped graphics
Digital representation of sound
Musical Instrument Digital Interface
Data Compression
Encryption