Computer Science - 4.5 Fundamentals Of Data Representation

Natural Numbers

Counting numbers: N \= {...2,3,4,5,6...}

Integer Numbers

Any positive or negative whole number:
Z \= {...-2,-1,0,1,2...}

Rational Numbers

Includes values that can be represented as fractions or decimals:
Q \= {...2/1,2/2,2/3...}
Q \= {...2,1,0.5,0.25,0.125...}

Real Numbers

Any number that is a natural, integer or rational number

Ordinal Number

Numbers that are used to describe the position in which they appear:

Adam, Belinda, Kyle, Tyler
Adam \= 1st, Tyler \= 4th

Binary Number System

Uses the digits 0 and 1
Base 2

Decimal Number System

Uses the digits 0-9
Base 10

Hexadecimal Number System

Consists of 16 distinct symbols — 0-9 and A-F
Base 16

Advantages of Hexadecimal

- Easier to remember and write compared to binary
- Less chance of errors
- Easy to convert into binary

What do we use Hexadecimal for?

- MAC address
- Define Colours
- Machine Code
- Assembly Language

Unsigned Integer

A data type that stores positive integer values as ordinary binary numbers; its value is always assumed to be positive.

Binary multiplication

0x0\=0
0x1\=0
1x0\=0
1x1\=1

Two's Complement

A method of representing negative numbers in the binary system. Found by flipping the postive binary representation of the number, flipping all the bits and adding one.

2 bits

4 values

3 bits

8 values

4 bits

16 values

5 bits

32 values

6 bits

64 values

7 bits

128 values

8 bits

256 values

KB Storage

10^3 \= 1,000

MB Storage

10^6 \= 1,000,000

GB Storage

10^9 \= 1,000,000,000

TB Storage

10^12 \= 1,000,000,000,000

KiB Storage

2^10 \= 1,024

MiB Storage

2^20 \= 1,048,576

GiB Storage

2^30 \= 1,073,741,824

TiB Storage

2^40 \= 1,099,511,627,776

Floating Point Numbers

Numbers where the decimal point can float because there is no fixed number of digits before and after the decimal point.

Mantissa

The bits used to represent the actual number

Exponent

Number that dictates the position of the floating point

Fixed Point Binary

In what representation of binary is the decimal point kept in the same position.

Rounding Errors

The precision of an arithmetic operation is greater than that of the floating-point number format, you can only get so close to representing true value

Absolute Error

The difference between the actual number and the nearest representable value.

Relative Error

The ratio of absolute error to actual measure.

Precision

how close a group of measurements are to each other

Normalization

Moving the binary point of a floating point number to provide the maximum level of precision for a given number of bits
Positive Value: 01
Negative Value: 10

Underflow

The result of a calculation is too small to be represented using the available number of bits.

Overflow

The result of a calculation is too large to be represented using the available number of bits.

Transmission Errors

Errors caused by external or internal factors that cause data to arrive in a different form to how it was originally sent.

Parity Bit

A binary digit appended to a group of binary digits to make the sum of all the digits, including the appended binary digit, either odd or even as established beforehand.

Even Parity

The number of 1s in the sequence add up to an even number.

Odd Parity

The number of 1s in the sequence add up to an odd number.

Majority Voting

Each bit is sent three times, and the receiving computer infers the data from what the majority of bits in the bit transmission show.

Problem With Majority Voting

If the majority of the sent bits are errors, the computer will idenfity the error bit as the correct one.

Checksums

An error detection method using an algorithm to calculate the sum of bytes in a transmission, and this calculated sum is also sent with the transmission. The receiving computer recalculates the sum with the transmission it received, and if the values of the sums don't match with the one appended to the transmission, an error has occured or data has been altered.

Character

Letter, number, symbol or control (e.g. spacebar)

Character Code

A binary representation of a particular letter, number or special character.

ASCII

American Standard Code for Information Interchange

How ASCII uses binary

Each character has its own binary representation, 7 Bit Character set, can represent 0-127 (128) characters

Unicode

A character code that enables most of the languages of the world to be symbolized with a special character identification. Can use 16 or 32 bits

Benefits of Unicode

- Can represent characters from different languages
- Can represent a lot more characters

16 Bits

65536 Values

24 Bits

16777216 Values

Digital Data

Data that changes discreetly through a finite set of possible values, patterns of 1s and 0s
Breaks analogue signal down into pieces and represents them through binary

Pixels

small squares that combine to create an image on a computer monitor. (picture elements)

Analogue to Digital Converter

Device that changes analogue data to digital data so that it can be stored on a computer system. e.g. microphone

Digital To Analogue Converter

A device used to convert digital signals into analogue signals (so the computer can control another device such as a pump, motor, etc.).

Bitmap Resolution

The number of pixels used to make up a bitmap image (Width x Height)

Megapixel

One million pixels, used in reference to the resolution of a graphics device

Image file size

The total number of binary digits in an image. The formula to calculate this is height (px) x width (px) x colour depth.

More Bits Means...

More colors that can be represented

How does Bit depth affect file size

Larger bit depth \= large file size

Bit Depth

Color depth, the number of bits devoted to each pixel in a color display.

Units of computer storage and memory

Bit
Nibble
Byte
Kilobyte
Megabyte
Gigabyte
Terabyte

Metadata

Data about data e.g. Date it was created, colour depth, geolocation, width and height in pixels

Analogue Data

Continuous Representation, is the actual information being represented

Analogue to Digital Conversion Process

- Sound signals recorded via amplifier
- Each sample is quantised to measure its wave height and translate this into an integer value
- integer value is converted and stored digitally as binary

Sample Resolution

Number of bits used to record each measurement

Resolution Decrease

Quality of sound decreases

Sample Rate

the number of samples taken per second, often measured in Kilohertz (kHz)

Sample Rate Increase

More accurate representation of the sound is created

Sound file size

Sample Rate x resolution x length in seconds

Nyquist Theorem

Samples should be at least double the highest frequency to replicate the original sound wave

Human Hearing Range

20-22,000 Hz

Lossy Compression

Non essential data is permanently removed e.g. different shades of the same colour or sound outside human hearing range

Lossless Compression

Patterns in data are spotted and summarized in a short format without permanently removing them

Run Length Encoding (RLE)

Lossless data compression that summarizes consecutive patterns of the same data, works well with images that have long runs of the same value

RLE of sound

Records one example of the sample and how many times it consecutively repeats

Dictionary Compression

Spots regularly occurring data and stores it separately in a dictionary

MIDI (Musical Instrument Digital Interface)

A set of standards that are used to represent music in digital form.

Encryption

Process of converting readable data into unreadable characters to prevent unauthorized access.

Plaintext

Normal text that has not been encrypted

Caesar Cipher

An algorithm that substitutes each letter in the plaintext with a letter a certain number of steps down the alphabet

Vernam Cipher

A method of encryption that uses a one-time pad (key) to create ciphertext that is mathematically impossible to decrypt without the key. Encryption and decryption of a message is peformed bit by bit using an XOR operation with the one-time pad (key).

One-Time Pad

A truly random sequence used to encrypt and decrypt data. It is only to be used once, and cannot be generated from computers, as they are not truly random. Sources include:
- Atmospheric Noise
- Radioactive Decay
- Movement Of Animals
- Snapshots Of A Lava Lamp