Fundamentals of Data Representation

• Integers (ℤ): The set of numbers with no fractional part. The natural numbers are a subset of the integers. ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}
• Rational Numbers (ℚ): The set of numbers that can be expressed as the ratio of two integers. The integers are a subset of the rational numbers since all integers can be expressed as a ratio with 1. ℚ = {0, ½, 0.75, 0.111111…, 300.5, -42, ...}
• Real Numbers (ℝ): The set of numbers that can represent real world quantities and have an imaginary part of 0. Rational and irrational numbers are all members of the real numbers. ℝ ={𝜋, 1.5, -7, ¾, 2, 100000000, -11.3432, ...}
• Ordinal Numbers: Natural numbers used to describe numerical position or order of objects.
• Binary: A number system that only uses ones and zeros to represent numbers (a base 2 system).
• Decimal: A number system that only uses 10 characters (0 to 9) to represent numbers (a base 10 system).
• Hexadecimal: A number system that only uses 16 characters (0 to 9 and A to F) to represent numbers (a base 16 system).
• Number Base: The number of unique digits used by a particular number system to represent numbers.
• Bit: A binary digit used by computers as the fundamental unit of information.
• Binary Prefix: A prefix to a unit representing a power of 2.
• Decimal Prefix: A prefix to a unit representing a power of 10.
• Signed Binary: A binary number system that can represent both positive and negative numbers.
• Unsigned Binary: A binary number system that can only represent positive numbers
• Two’s Complement: A coding scheme used in signed binary to represent negative as well as positive numbers. A negative number is represented by flipping all its digits and adding 1 to the most significant bit.
• Fixed Point Form: A form used to represent numbers with a fractional part in any number system. Digits after the fixed point are multiplied by the base raised to a negative power.
• ASCII: A character set used to represent alphanumeric characters or symbols as a set of 8 bits.
• Unicode: A character set that is a superset of ASCII. It is used to represent alphanumeric characters or symbols as an integer code point which is equal to the character’s ASCII code.
• Check Digits: A method of checking codes for errors during data transmission by adding an extra digit to the end (usually calculated/processed from digits in the code itself) that checks whether the data is accurate.
• Majority Voting: A method of checking binary codes for errors during data transmission by sending each bit multiple times, in a set. The receiver takes the value with most occurrences in a set as the value for that bit.
• Parity Bits: A method of checking binary codes for errors during data transmission by counting the number of ones and zeroes present.
• Analogue Data: Data whose values can vary continuously and take on any value between two extremes.
• Analogue Signals: A transmission of a set of analogue data structures, that varies with time, between computational processes.
• Analogue to Digital Converter (ADC): An integrated circuit capable of converting continuous analogue data to discrete digital data for a computer.
• Digital to Analogue Converter (DAC): An integrated circuit capable of converting discrete digital data from a computer to continuous analogue data.
• Bitmapped Graphics: An image composed of an array of pixels each with an allocated number of bits, arranged to form an image. Also known as raster graphics.
• Colour Depth: A measure of the amount of colour used in an image, expressed in terms of the number of bits per pixel.
• Image Size: The total number of pixels in an image expressed in terms of its dimensions: (width in pixels) × (height in pixels).
• Metadata: Data related to the image file data itself. This includes image properties such as width, height and colour depth.
• Resolution: A measure of the total number of pixels in an image, typically expressed in terms of the number of dots/pixels per inch.
• Nyquist theorem: A sufficiently accurate digital waveform of an analogue signal would require a sampling rate of at least twice the highest frequency that appears in the original analogue signal.
• Sample Resolution: The number of bits used to represent a single sample.
• Sampling Rate: The number of samples taken per second.
• Sound Sampling: The process of converting analogue sound waves to a digital waveform, by storing a finite number of readings in binary.
• Event Messages: Binary data transmitted between the MIDI device and computer processor that carries properties controlling when and how sounds are produced.
• Lossless Compression: A compression algorithm that retains all the data in the file by only storing the instructions needed to reconstruct the original file. No data is lost.
• Lossy Compression: A compression algorithm that removes non-essential data from a file leading to a noticeable decrease in accuracy of the data. Data lost is non-recoverable.
• Ren-Length Encoding: A type of lossless compression where repeated occurrences of the same data (like several pixels of the same colour in an image) are stored as single data values with their counts.
• Encryption: The process of converting the original data (plaintext) into a form which cannot be understood by unauthorised users (ciphertext) using an encryption algorithm (cipher).
• Caesar Cipher: A substitution cipher where each letter of plaintext is substituted for another that is a fixed number of letters ahead in the alphabet, which becomes the ciphertext.
• Vernam Cipher: A cipher that uses a one-time pad (a secret random key) to convert each character to cipher text by modularly adding it with the corresponding character of the key. This is impossible to decrypt without a key.