Data
Computers store and manipulate all data — numbers, text, images, sound — using binary. Understand how data is represented, stored, compressed and encrypted.
All data inside a computer is ultimately stored and processed as binary (base 2) — sequences of 0s and 1s. Students must understand how numbers, characters, images and sound are represented in binary, how file sizes are calculated, and how data can be protected through encryption.
Binary Representation of All Data (3.1.1)
Every piece of data stored and processed by a computer is represented in binary (base 2) using only the digits 0 and 1, called bits. This includes numbers, text characters, images, sound, and even program instructions themselves. The reason computers use binary is that electronic circuits can reliably represent two states: on (1) and off (0).
An 8-bit binary number (one byte) — the position values are powers of 2:
The value above is 128+32+16+4+1 = 181 in denary.
Representing Integers (3.1.2)
- Unsigned integers — represent positive numbers only. An 8-bit unsigned integer can store values 0 to 255 (2⁸ − 1).
- Sign and magnitude — leftmost bit = sign (0=positive,1=negative). Two representations of zero. Range for 8-bit: −127 to +127.
- Two’s complement — MSB has negative weight. To negate: invert bits then add 1. Single zero. Range for 8-bit: −128 to +127.
Binary ↔ Denary Conversion (3.1.3)
Must convert between binary and denary for values 0–255 (8 bits).
Binary to denary: add place values of positions with 1.
E.g. 1010 0110 = 128+32+4+2 = 166
Denary to binary: repeatedly divide by 2, record remainders bottom-up.
E.g. 200 = 128+64+8 = 1100 1000
Binary Arithmetic (3.1.4)
Binary addition: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (0 carry 1), 1+1+1=11.
Logical shift left: moves bits left, fills with 0s → multiplies by 2ⁿ.
Logical shift right: moves bits right, fills with 0s → divides by 2ⁿ.
Overflow: result too large for available bits — carry bit lost.
Hexadecimal (3.1.5)
Hexadecimal (base 16) uses digits 0–9 and letters A–F (A=10, B=11, C=12, D=13, E=14, F=15). Each hex digit = 4 binary bits (a nibble). Hex is compact and human-readable.
- Binary to hex: split into groups of 4 from right, convert each group. E.g. 1010 1100 → A C = AC₁₆
- Hex to binary: replace each hex digit with its 4-bit binary equivalent. E.g. 3F → 0011 1111
- Uses: colour codes (#FF5733), memory addresses, MAC addresses, IPv6.
Character Encoding — ASCII and Unicode (3.2.1)
- ASCII — 7-bit, 128 characters (English letters, digits, punctuation). Extended ASCII uses 8 bits for 256 characters.
- Unicode — represents characters from all world writing systems. UTF-8 uses 1–4 bytes per character. Backwards compatible with ASCII.
- Why Unicode? ASCII cannot represent Chinese, Arabic, Hindi, emoji, or thousands of other characters.
- File size: Unicode files are larger than ASCII files.
Bitmap Images (3.2.2)
- Pixel — smallest element. Each pixel stores a colour as binary.
- Resolution — width × height in pixels. Higher resolution = larger file.
- Colour depth — bits per pixel. 1-bit = 2 colours, 8-bit = 256 colours, 24-bit = 16.7M colours (true colour).
- File size (bits) = width × height × colour depth
- E.g. 100×100 image, 24-bit depth = 100×100×24 = 240,000 bits = 30,000 bytes
Sound Representation (3.2.3)
Sound is analogue — must be sampled for digital storage. Measurements of amplitude are taken at regular intervals and stored as binary values.
- Sampling frequency (Hz) — samples per second. CD quality = 44,100 Hz.
- Bit depth — bits per sample. CD quality = 16 bits.
- File size (bits) = frequency × bit depth × duration × channels
Limitations of Binary Representation (3.2.4)
- Low sample rate → missing high frequencies, aliasing.
- Low bit depth → quantisation noise, distortion.
- Low colour depth → banding, loss of gradient detail.
- Trade-off: higher quality = larger file size.
Data Units (3.3.1)
| Unit | Binary (IEC) | Exact value |
|---|---|---|
| Bit | – | 0 or 1 |
| Nibble | – | 4 bits |
| Byte | – | 8 bits |
| Kibibyte (KiB) | 2¹⁰ | 1,024 bytes |
| Mebibyte (MiB) | 2²⁰ | 1,048,576 bytes |
| Gibibyte (GiB) | 2³⁰ | ~1.07 billion bytes |
| Unit | Decimal (SI) | Exact value |
|---|---|---|
| Kilobyte (kB) | 10³ | 1,000 bytes |
| Megabyte (MB) | 10⁶ | 1,000,000 bytes |
| Gigabyte (GB) | 10⁹ | 1,000,000,000 bytes |
| Terabyte (TB) | 10¹² | 1,000,000,000,000 bytes |
Compression (3.3.2)
- Lossless — original perfectly reconstructed. Used for text, code, ZIP. Algorithms: RLE, Huffman.
- Lossy — removes imperceptible data. Original cannot be exactly restored. Used for images (JPEG), audio (MP3), video.
Run-Length Encoding (RLE) (3.3.3)
RLE is a lossless algorithm. Replaces consecutive runs of the same value with (count, value).
- Original: A A A A A B B C C C C = 11 chars
- RLE: 5A 2B 4C = 6 chars (45% smaller)
Most effective when data has long repeated runs (simple graphics, fax). Ineffective for random data.
Calculating file sizes — Image size (bits) = width × height × colour depth. Sound size (bits) = sample rate × bit depth × duration × channels. Divide by 8 for bytes, then by 1024 for KiB.
Why Encrypt Data? (3.4.1)
Encryption converts plaintext to unreadable ciphertext using an algorithm and a key. Only those with the correct key can decrypt.
- Protects data in transit (HTTPS, online banking, email)
- Protects data at rest (encrypted drives, password databases)
- Encryption does NOT prevent theft — it makes stolen data useless
Caesar Cipher
Each letter shifted by fixed number (key). Shift 3: A→D, B→E. Easy to break by trying all 25 shifts (brute force).
Vigenère Cipher
Polyalphabetic — uses keyword repeated to match message length. Each letter shifted by corresponding keyword letter. Much harder to crack than Caesar.
Pigpen Cipher
Substitution cipher replacing letters with symbols from 2×2 grids and X-shapes. Visual — no letter shifting.
Rail Fence Cipher
Transposition cipher — rearranges letters without changing them. Message written in zigzag across “rails” then read row by row. Key = number of rails.
Substitution vs Transposition Ciphers
Substitution ciphers — replace characters with different characters. Letters stay in same positions but are replaced. Examples: Caesar, Vigenère, Pigpen.
Transposition ciphers — rearrange characters without changing them. Same letters, different order. Example: Rail Fence.
Modern encryption (AES, RSA) combines multiple rounds of both substitution and transposition.