Data Representation

natural number

a positive whole number, including zero

rational number

can be expressed as a fraction or ratio of integers

irrational number

cannot be represented as a fraction or ratio as the decimal form will contain infinite repeating values

real number

a value that represents any quantity along an infinite number line

ordinal number

a number used to identify position relative to other numbers

cardinal number

a number used to identify the size of something

well-ordered set

a group of related numbers with a defined order

bit

fundamental unit of information, can only take two values, 1 and 0, which can be represented by computers using high or low currents.

advantages of hexadecimal

• most compact representation- it can represent the same number as binary or denary while using far fewer digits

• saves the programmer time writing data

• easier to read/write/remember

• lowers likelihood of errors when typing data

• can be displayed in less space

fixed point

a specified number of bits are placed before a binary point and the remaining bits fall behind the binary point

floating point

comparable to scientific notation in that a number is represented as a mantissa and an exponent

advantages of fixed point

• processing of numbers is faster as there is no processing required to move the binary point

• represents all numbers to a constant level of precision/accuracy

advantages of floating point

• much wider range of numbers can be produced with same number of bits

• able to represent values with varying degrees of precision

advantages of normalisation

• gives a standard representation

• maximises precision/accuracy for given number of bits

• unique representation of each number means its simpler to test for equality of numbers

absolute error

the amount by which a value is inaccurate, the difference between the given value and the actual value

relative error

measure of uncertainty in a given value compared to the actual value which is relative to the size of the actual value- absolute error / actual value

character code

a unique code to represent each different character

ASCII

a standard binary coding system for characters and numbers, using seven bits

limitations of ASCII

• not enough bits to represent all of the possible characters, numbers and symbols

• initially developed in English, therefore did not represent all of the other languages and scripts in the world

Unicode

a standard binary coding system that has superseded ASCII, ASCII codes have been subsumed within Unicode

why should there be a universal coding system

if different encoding systems are used, it means that data can be corrupted when used on any system other than that on which it was created

limitations of parity bit

• two errors would cancel each other out and go undetected

• impossible to know where the error happened

advantages of parity bit

• straightforward to implement and require minimal hardware or software resources

• add only one extra bit to the data, making them a cost-effective method for basic error checking

majority voting

each bit is sent several times, when there is a discrepancy, you see which bit occurs the most frequently, assuming that the majority is more likely to be accurate than any single value

limitations of majority voting

• requires redundant systems, meaning you need multiple copies of the same data or multiple components performing the same calculations- increases hardware costs, processing time, memory usage, and power consumption

• only effective against random, independent errors - if all components share a common error source, all values may be identical but incorrect

• voting adds latency because it requires multiple copies of data to be produced, processed, and compared before reaching a decision

advantages of majority voting

• able to detect and correct errors, meaning there is no need for retransmission

• can detect multiple errors as check each individual bit

check digit

the check digit is created by taking the digits that make up the number itself and using them in some way to create a single digit

when the data is being processed, the check digit is recalculated and compared with the digit that has been transmitted

where the check digit is the same then it is assumed that the data is correct

bit-mapped graphic

digital image composed of a matrix of pixels (picture elements)

colour depth

the number of bits allocated to represent the colour of a single pixel

vector graphic

an image made up of objects and co-ordinates

how do you re-scale a vector graphic

adjust the coordinates- graphics are being controlled mathematically rather than being completely re-generated

advantages of vector graphics

• less storage space likely to be needed for geometric images as contains the mathematical description required to create the image rather than storing the actual image

• facilitates perfect re-scaling without distortion

• individual elements of the image can be manipulated independently

disadvantages of vector graphics

• unsuitable if colour of each pixel is likely to vary as it would become way too complicated

• vector files require specific software to open and edit, making them less accessible

advantages of bit-mapped graphics

• allows for precise control over individual pixels, making them suitable for photographs and complex designs

• widely used in online environments and most software and hardware platforms can easily handle them- this makes them accessible

uses of hexadecimal

• representing colour codes

• representing MAC addresses

examples of primitives (simple vector objects)

• line

• curve

• polygon

• point

vector object

individual element within a vector graphic

uses of vector graphics

• logos- requires scalability

• text rendering- for clarity at any size

Analogue-Digital Converter

• microphone inputs sound in the form of changes in air pressure, and then converts them into electrical signals

• ADC takes samples at regular intervals of the analogue signal

• amplitude of each sample is approximated to an integer value

• each sample is assigned a binary value

MIDI device

transmits a set of instructions that tell a connected device how to generate sound, what to play, and how to manipulate it- uses event messages to control various properties of the sound in real time

advantages of MIDI

• midi files tend to be much smaller, meaning they require less memory and load faster

• files completely editable as individual instruments can be selected and modified

• no data lost through sampling

• no background noise

disadvantages of MIDI

• not compatible with every instrument

• the final sound's quality depends on the sound source and playback device, meaning that the sound can vary greatly

sampling resolution

number of bits allocated to representing the sound

Nyquist’s Theorem

to faithfully recreate the analogue signal, you should sample at least twice the highest frequency present

double the frequency to ensure the sample covers the complete range of peaks and troughs in the analogue signal

advantages of lossless compression

• able to perfectly reconstruct the original data after decompression, ensuring no information is lost

• lossless compression ensures that no quality is compromised

• can be applied to a wide range of file types

disadvantages of lossless compression

• achieves smaller reduction in data

• may require more computational resources to compress and decompress

types of lossless compression

• run-length encoding

• dictionary-based encryption

lossy compression

a data compression technique where some original data is permanently deleted to reduce file size, resulting in a small loss of quality

advantages of lossy compression

• results in drastically smaller file sizes- saves storage space, makes it faster to upload, download and stream

• prioritise human perception, so lost data is usually imperceptible

• users can adjust compression level to balance size and quality based on their needs

lossless compression

a data compression technique where the original data is perfectly reconstructed after decompression, resulting in no information loss

disadvantages of lossy compression

• irreversible loss of data

• at high compression levels, there is distortion/ loss of clarity in sound and bluriness in images

JPEG compression

• break up image into blocks of 8×8 pixels

• in each block, data is converted into frequencies

• some frequencies considered more important than others

• high frequency data is more difficult for the human eye to perceive so changes will be less noticeable

• important to maintain contrast

encryption

the process of turning plaintext into scrambled ciphertext which can only be understood if it is decrypted with a key

what does a digital signature do

• authenticates senders identity

• detects unauthorised changes to message

polyalphabetic cipher

uses more than one alphabet- Alberti cipher

railfence transposition cipher

message split across several lines

route transposition cipher

message put into a grid

why is the Vernam cipher mathematically impossible to crack

• randomness means that each character in the ciphertext is completely independent of the others, making it impossible to deduce any information about the plaintext from the ciphertext

• different plaintext letters can map onto same ciphertext letter, therefore cannot use frequency analysis

• too many possible keys to use brute force

conditions for Vernam cipher to be unbreakable

• key only used once

• key completely random

• key kept secret

computationally secure cipher

theoretically breakable but not using current technology in a useful time frame