Information Representation

At its most fundamental level, a computer only understands two states: on and off, represented by the digits 1 and 0. Each of these digits is called a bit (binary digit), and it is the smallest unit of data. All information—from numbers and text to complex images and sounds—must be converted into a binary format to be processed and stored. Bits are typically grouped into an 8-bit unit called a byte.

### Number Systems

To represent numbers, we use different bases. Humans use denary (base-10), but computers use binary (base-2).

* Binary (Base-2): This system uses only two digits, 0 and 1. Each position in a binary number represents a power of 2, starting from the right (2⁰, 2¹, 2², etc.). For example, the binary number 1011 is converted to denary as: (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰) = 8 + 0 + 2 + 1 = 11.

* Hexadecimal (Base-16): Binary numbers can be long and difficult for humans to read. Hexadecimal is used as a compact, human-friendly representation of binary. It uses digits 0-9 and letters A-F (representing 10-15). Each hexadecimal digit corresponds to a 4-bit binary sequence (a nibble). For example, the binary number `1110 0101` can be grouped into `1110` (E in hex) and `0101` (5 in hex), making it E5 in hexadecimal. This system is commonly used in memory addressing and defining colours in web design (e.g., #FF0000 for red).

* Two's Complement: To represent negative integers, computers use two's complement. The Most Significant Bit (MSB) indicates the sign (1 for negative, 0 for positive). The process to find the two's complement of a negative number is:

For example, to represent -6 in 8 bits: Positive 6 is `00000110`. Invert it to get `11111001`. Add 1 to get `11111010`.

### Text Representation

* ASCII (American Standard Code for Information Interchange): An early standard using 7 bits (later 8 bits in Extended ASCII) to represent 128 (or 256) different characters, including English letters, numbers, and common symbols. Its main limitation is its inability to represent characters from other languages.

* Unicode: A modern, universal character encoding standard that can represent almost every character from every writing system in the world. UTF-8 is the most common Unicode encoding, using a variable number of bytes per character, making it space-efficient and backward-compatible with ASCII.

### Image Representation

* Bitmap Images: A bitmap image is stored as a grid of tiny dots called pixels. For each pixel, a binary value is stored that represents its colour. The number of bits used per pixel is called the colour depth. A 1-bit colour depth can only represent 2 colours (e.g., black and white), while a 24-bit colour depth (True Colour) can represent over 16.7 million colours (2²⁴).

* File Size Calculation: The size of a bitmap image can be calculated with the formula:

File Size (bits) = Image Width (pixels) × Image Height (pixels) × Colour Depth (bits). This does not include metadata (additional data like creation date or camera settings) stored in the file header.

### Sound Representation

Sound is an analogue wave. To be stored on a computer, it must be digitised through a process called sampling. An Analogue-to-Digital Converter (ADC) takes measurements (samples) of the sound wave's amplitude at regular intervals.

* Sampling Rate: The number of samples taken per second, measured in Hertz (Hz). A higher sampling rate (e.g., 44,100 Hz for CD quality) results in a more accurate digital representation and better sound quality.

* Sample Resolution (Bit Depth): The number of bits used to store each sample. Higher sample resolution allows for a more precise representation of the wave's amplitude.

* File Size Calculation: The size of a sound file is calculated by:

File Size (bits) = Sampling Rate (Hz) × Sample Resolution (bits) × Duration (seconds) × Number of Channels (e.g., 1 for mono, 2 for stereo).

### Data Compression and Encryption

* Compression: This is the process of reducing file size.

* Lossless Compression: Reduces file size without losing any original data. When the file is decompressed, it is an exact replica of the original. This is essential for text files and program code. A simple method is Run-Length Encoding (RLE), which replaces repeated sequences of data with a count and a single data value (e.g., `AAAAA` becomes `5A`). Common formats include PNG and ZIP.

* Lossy Compression: Reduces file size by permanently removing data that is considered non-essential to human perception. This offers much higher compression ratios but with a loss of quality. It is widely used for images (JPEG), audio (MP3), and video (MPEG).

* Encryption: This is the process of converting plaintext (readable data) into ciphertext (unreadable data) using an algorithm and a key. This ensures data confidentiality, preventing unauthorised access. Symmetric encryption uses the same key for both encryption and decryption, while asymmetric encryption uses a public key to encrypt and a private key to decrypt.