Data representation (1.1)

Humans count using the denary (or decimal) system, which is base-10. It uses ten distinct digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). The position of a digit determines its value, based on powers of 10. For example, the number 123 is (1 × 10²) + (2 × 10¹) + (3 × 10⁰). Computers, however, are built from electronic circuits that can only be in one of two states: on or off. These states are represented by the binary system, which is base-2. It uses only two digits: 1 (on) and 0 (off). These binary digits are called 'bits'. All data inside a computer, from numbers to text to images, is ultimately stored as a sequence of bits. The value of a binary number is determined by powers of 2.

Being able to convert between denary and binary is a fundamental skill. To convert a binary number to denary, you sum the place values of all the '1's. A place value grid is very helpful for this. To convert a denary number to binary, you use repeated division by 2. You divide the denary number by 2, write down the remainder (which will be 0 or 1), and use the result for the next division. You repeat this until the result is 0. The binary number is formed by reading the remainders from the bottom up.

While computers use binary, long strings of 1s and 0s are difficult for humans to read and write without making errors. The hexadecimal system (or 'hex') is a base-16 system used as a more compact, human-friendly way to represent binary data. It uses 16 symbols: the digits 0-9 and the letters A-F to represent the denary values 10-15. The key relationship is that one hexadecimal digit represents exactly four binary bits (a nibble). This makes conversion between binary and hex very simple and is why it's widely used in computing for things like memory addresses, MAC addresses, and HTML colour codes.

Converting between number systems is a key skill. Binary-to-Hex is the easiest: starting from the right, split the binary number into groups of four (nibbles), then convert each nibble into its corresponding hex digit. Hex-to-Binary is the reverse: convert each hex digit into its 4-bit binary equivalent. For Hex-to-Denary, use place values based on powers of 16. For Denary-to-Hex, use repeated division by 16, recording the remainders and converting any remainder from 10-15 into its A-F character. As with binary, read the remainders from the bottom up.

Data in a computer is measured in specific units. The smallest unit is a bit (0 or 1). Bits are grouped together to form larger units. A group of 4 bits is a nibble, and a group of 8 bits is a byte. The byte is the fundamental unit for measuring memory and storage size. For larger amounts of data, metric prefixes are used: Kilo (K), Mega (M), Giga (G), and Tera (T). In computing, these prefixes can have two meanings. Historically, they were based on powers of 2 (e.g., 1 Kilobyte = 1024 bytes). However, it is now common practice, especially for storage and data transfer speeds, to use the standard metric meaning based on powers of 10 (e.g., 1 Kilobyte = 1000 bytes). For exams, unless told otherwise, it is often acceptable to use the 1000-based units as it simplifies calculations. Always state your assumption.

Every character on your keyboard needs a unique binary code so the computer can process and store it. Character sets are standards that define these codes. The most common early standard was ASCII (American Standard Code for Information Interchange). Standard ASCII uses 7 bits, allowing it to represent 128 different characters (2⁷ = 128), which is enough for all uppercase and lowercase English letters, numbers, and common punctuation. Extended ASCII uses 8 bits (1 byte) to represent 256 characters. However, ASCII is insufficient for global use. Unicode is a modern, universal standard that aims to represent every character from every language. It uses 16 or even 32 bits per character, allowing for millions of unique codes, covering everything from ancient scripts to modern emojis.

A bitmap image is a digital picture made up of a grid of tiny dots called pixels (picture elements). The computer stores information about the colour of each individual pixel. The two key properties that determine a bitmap's quality and file size are its resolution and colour depth. Resolution is the number of pixels in the grid, usually expressed as width × height (e.g., 1920 × 1080). Colour depth is the number of bits used to store the colour of each pixel. A higher colour depth allows for more colours (e.g., 8 bits = 2⁸ = 256 colours; 24 bits = 2²⁴ ≈ 16.7 million colours) but increases the file size. The total file size can be calculated by multiplying the number of pixels by the colour depth.

Unlike bitmaps, which store a map of pixels, vector graphics store an image as a set of mathematical instructions. Instead of saving every pixel of a circle, a vector graphic stores the properties of the circle: the coordinates of its centre, its radius, its line colour, and its fill colour. This approach has two major advantages. Firstly, vector graphics are scalable; they can be resized to any dimension, from a tiny icon to a giant billboard, without any loss of quality or 'pixelation'. Secondly, for images made of simple shapes and lines (like logos, charts, and diagrams), the file size is often much smaller than a corresponding high-resolution bitmap. However, they are not suitable for representing photorealistic images.