What you'll learn
Computers store everything — numbers, text, images and sound — as binary, patterns of 1s and 0s. Data representation is the study of how those patterns are organised. This guide shows you how to convert between binary, denary and hexadecimal, how to add binary numbers and spot overflow, how binary shifts multiply and divide, how characters are stored using ASCII and Unicode, and how image and sound files are built — including how to calculate their file sizes. It finishes with compression, including run-length encoding.
This is the most calculation-heavy topic in ICT. The arithmetic itself is simple, but marks are lost through careless place values, forgetting to convert bits to bytes, and not showing working. Practise the methods until they are automatic.
Key terms and definitions
Bit — A single binary digit, 0 or 1.
Nibble — A group of 4 bits.
Byte — A group of 8 bits.
Denary — The base-10 number system we use every day.
Binary — The base-2 number system used by computers.
Hexadecimal — The base-16 number system, using digits 0–9 and letters A–F for 10–15.
Overflow — When the result of a calculation is too large to fit in the number of bits available.
Character set — A list of characters and the binary code that represents each one, such as ASCII or Unicode.
Pixel — The smallest single element of a bitmap image.
Resolution — The number of pixels in an image, given as width × height.
Colour depth (bit depth) — The number of bits used to store the colour of each pixel.
Sample rate — The number of times per second a sound wave is measured, in hertz (Hz).
Sample resolution — The number of bits used to store each sound sample.
Compression — Reducing the size of a file. Lossy compression removes some data permanently; lossless compression allows the original to be restored exactly.
Core concepts
Why binary?
A computer's circuits are built from millions of tiny switches that are either off or on. Two states map naturally to two digits, 0 and 1. Everything a computer stores, however complicated, is ultimately a pattern of these two states.
Units of data
| Unit | Size |
|---|---|
| Bit | 0 or 1 |
| Nibble | 4 bits |
| Byte | 8 bits |
| Kilobyte (KB) | 1,024 bytes |
| Megabyte (MB) | 1,024 KB |
| Gigabyte (GB) | 1,024 MB |
| Terabyte (TB) | 1,024 GB |
Some contexts use 1,000 instead of 1,024; follow whichever your question states. The key conversion you will use again and again is 8 bits = 1 byte.
Binary to denary
An 8-bit binary number has these place values:
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
|---|
Write the binary digits under the headings and add the place values where there is a 1.
Example: 10110011 → 128 + 32 + 16 + 2 + 1 = 179.
With n bits you can represent 2ⁿ different values. Eight bits give 256 values, from 0 to 255. Four bits give 16 values, from 0 to 15.
Denary to binary
Work from the largest place value down. For 45: 32 fits (remainder 13), 16 does not, 8 fits (remainder 5), 4 fits (remainder 1), 2 does not, 1 fits. So 45 = 00101101.
Hexadecimal
Hexadecimal uses sixteen digits: 0–9, then A = 10, B = 11, C = 12, D = 13, E = 14, F = 15. Each hex digit represents exactly one nibble, which is why it is used: long binary numbers become short and easier to read, write and check. Colour codes in web design and memory addresses are often written in hex.
Hex to denary: a two-digit hex number has place values 16 and 1. For 9E: (9 × 16) + 14 = 144 + 14 = 158.
Binary to hex: split the binary into nibbles and convert each one. 11101111 → 1110 | 1111 → 14 | 15 → EF.
Hex to binary: convert each digit to a 4-bit nibble. 3F → 0011 | 1111 → 00111111.
Binary addition and overflow
The rules are: 0 + 0 = 0; 0 + 1 = 1; 1 + 1 = 10 (write 0, carry 1); 1 + 1 + 1 = 11 (write 1, carry 1). Work from right to left, just as in denary.
If the answer needs more bits than are available, overflow occurs. In an 8-bit register, the largest value is 255. Adding 200 and 100 gives 300, which needs nine bits, so the ninth bit is lost and the stored result is wrong.
Binary shifts
Shifting every bit one place to the left doubles an unsigned binary number; shifting one place to the right halves it (dropping any remainder). A left shift of two places multiplies by 4, three places by 8, and so on. Bits pushed off the end are lost, which can cause errors.
Representing characters
A character set gives each character a binary code. ASCII uses 7 bits, giving 128 characters — enough for English letters, digits, punctuation and control codes. Extended ASCII uses 8 bits for 256 characters. Unicode uses more bits per character (commonly 16 or more), giving room for characters from nearly every language, as well as symbols and emoji. The trade-off is that text stored in a wider encoding takes more space.
Representing images
A bitmap image is a grid of pixels. Each pixel's colour is stored as a binary code. The number of bits per pixel is the colour depth:
- 1 bit → 2 colours
- 8 bits → 256 colours
- 16 bits → 65,536 colours
- 24 bits → about 16.7 million colours
File size (bits) = width × height × colour depth. Divide by 8 to get bytes. Increasing resolution or colour depth improves quality but increases file size. Image files also store metadata — information such as width, height and colour depth — so the computer knows how to display them.
Representing sound
Sound is an analogue wave. To store it digitally, the wave's amplitude is measured (sampled) at regular intervals. A higher sample rate captures the wave more accurately, and a higher sample resolution records each measurement more precisely. Both improve quality and both increase file size.
File size (bits) = sample rate × sample resolution × duration in seconds (× number of channels, if stereo).
Compression
Compression reduces file size so files take less storage and transfer more quickly.
Lossy compression permanently removes data that people are unlikely to notice, such as subtle colour differences or sounds outside normal hearing. JPEG images and MP3 audio use it. Files become much smaller, but the original can never be restored.
Lossless compression shrinks files without losing anything. It is essential for text, program files and anything where every bit matters.
Run-length encoding (RLE) is a simple lossless method. It replaces runs of repeated values with a count and the value. The row RRRRRGGGRR becomes 5R 3G 2R. It works well where long runs are common (simple graphics) and poorly where values change constantly (photographs).
Common file formats
The way data is represented also depends on the file format, which tells software how the bits are organised. You should recognise the common formats and when to use them:
- JPEG (.jpg) — lossy compressed images; ideal for photographs where small file size matters, such as on websites.
- PNG (.png) — lossless compressed images that support transparency; good for logos, diagrams and screenshots.
- GIF (.gif) — limited to 256 colours; used for simple graphics and short animations.
- MP3 (.mp3) — lossy compressed audio; small files for music players and streaming.
- WAV (.wav) — uncompressed audio; high quality but large, used in recording and editing.
- MP4 (.mp4) — compressed video with sound; the standard for sharing and streaming video.
- TXT (.txt) — plain text with no formatting; tiny and readable by almost any program.
- CSV (.csv) — comma-separated values; plain-text tables used to move data between spreadsheets and databases.
- PDF (.pdf) — a fixed-layout document format that looks the same on any device; good for forms and documents that should not be edited.
Choosing a format is a trade-off between quality, file size and compatibility. A photographer archiving the only copies of images should keep a lossless or original version, while a web designer will usually choose a compressed format so pages load quickly. A business sending a finished contract will choose PDF so the layout cannot shift, while a colleague who needs to edit the text needs the original word-processed file.
Worked examples
Example 1: Converting binary to hexadecimal
Convert 11010110 to hexadecimal.
Split into nibbles: 1101 | 0110. 1101 = 8 + 4 + 1 = 13 = D. 0110 = 4 + 2 = 6. Answer: D6.
Example 2: Image file size
An image is 800 × 600 pixels with a colour depth of 8 bits. Calculate its size in bytes, ignoring metadata.
Pixels = 800 × 600 = 480,000. Bits = 480,000 × 8 = 3,840,000. Bytes = 3,840,000 ÷ 8 = 480,000 bytes (about 469 KB).
Example 3: Sound file size
A mono recording lasts 4 minutes, with a sample rate of 8,000 Hz and a sample resolution of 8 bits. Calculate its size in bytes.
Duration = 240 s. Bits = 8,000 × 8 × 240 = 15,360,000. Bytes = 15,360,000 ÷ 8 = 1,920,000 bytes.
Example 4: Binary addition with overflow
Add 11001000 and 01100100 in an 8-bit register.
11001000 = 200 and 01100100 = 100. The total, 300, is greater than 255, so it cannot be stored in 8 bits. Overflow occurs; the ninth bit is lost and the stored result (00101100 = 44) is incorrect.
Common mistakes and how to avoid them
- Forgetting to divide by 8. File-size formulas give bits. If the question asks for bytes, divide by 8.
- Mixing up 255 and 256. Eight bits give 256 values, but the largest value is 255 because counting starts at 0.
- Writing the hex letters as numbers. 1F is not "115"; F is 15, so 1F = 16 + 15 = 31.
- Reading place values right to left incorrectly. Always write the headings out before converting.
- Saying lossy compression "doesn't lose quality". It does remove data; the point is that the loss may not be noticeable.
- Confusing sample rate with sample resolution. Rate is how often; resolution is how precisely.
Exam technique for "Data Representation"
- Show your working for every conversion and calculation. Method marks are available even if the final answer slips.
- Write units on every answer: bits, bytes, KB, MB.
- For "explain why hexadecimal is used" answers, give two reasons: shorter to write and read, and fewer errors when copying, with each digit mapping to exactly four bits.
- For quality versus file size questions, always mention both: higher resolution or sample rate improves quality and increases file size.
Quick revision summary
- 8 bits = 1 byte; n bits give 2ⁿ values.
- Binary place values: 128, 64, 32, 16, 8, 4, 2, 1. Hex digits 0–9 and A–F (10–15); one hex digit = one nibble.
- Overflow happens when a result needs more bits than are available.
- A left shift doubles; a right shift halves.
- ASCII: 7 bits, 128 characters. Unicode: more bits, characters from nearly every language.
- Image size = width × height × colour depth. Sound size = sample rate × resolution × seconds.
- Lossy compression removes data permanently; lossless (such as RLE) does not.