Data representation

Worth about 18 of the 75 marks on a typical Paper 1. Almost every paper opens with this topic.

1. Why computers use binary

Everything a computer stores or processes, whether numbers, text, photos, music or video, is represented as binary: patterns of only two digits, 0 and 1. A single binary digit is called a bit. Data must be converted to binary before a computer can process it.

Computers use binary because their circuits are built from switches (transistors) with only two states, on and off. Data is processed by logic gates and stored in registers, which hold a fixed number of bits.

2. Denary, binary and hexadecimal

Hexadecimal and denary are just easier ways for people to read the same values the computer stores in binary.

System	Base	Digits used	Who uses it
Denary	10	0–9	People, everyday counting
Binary	2	0, 1	The computer, internally
Hexadecimal	16	0–9, A–F (A = 10 … F = 15)	Programmers, as a short form of binary

3. Converting binary and denary

Each binary column is worth double the one to its right.

128	64	32	16	8	4	2	1
0	1	1	0	1	0	0	1

0110 1001 = 64 + 32 + 8 + 1 = 105. The largest 8-bit value is 1111 1111 = 255; 16 bits reach 65 535, the longest binary number you will be asked to convert.

Denary to binary: start at the largest place value. If it fits, write 1 and subtract; if not, write 0. Move right until you reach 1.

4. Converting hexadecimal

One hex digit stands for exactly four bits (a nibble), so 8 bits = 2 hex digits.

1. Binary → hex: split into groups of 4 from the right, then convert each group. 1011 0110 → B 6 → B6.
2. Hex → binary: write each digit as 4 bits, keeping leading zeros. 3F → 0011 1111.
3. Hex → denary: first digit × 16, plus second digit. 3F → 3 × 16 + 15 = 63.

5. Uses of hexadecimal

Hex appears in HTML colour codes (#FF8800), MAC addresses, IPv6 addresses, error codes and memory dumps.

It is used because it is shorter and easier for people to read than binary, so fewer mistakes are made. It does not save storage: the computer still stores binary.

6. Binary addition and overflow

Sum	Result	Carry
0 + 0	0	0
0 + 1	1	0
1 + 1	0	1
1 + 1 + 1	1	1

Add column by column from the right and always show the carries.

Overflow happens when the result needs more bits than the register holds. Two 8-bit numbers adding to more than 255 leave a 9th bit that cannot be stored, so the stored answer is wrong.

7. Logical binary shifts

• Left shift by n: every bit moves n places left; 0s fill the gaps on the right. Each place multiplies by 2.
• Right shift by n: bits move right; 0s fill on the left. Each place divides by 2 (whole-number division).
• Bits pushed off the end (the most or least significant bits) are lost, so a shift can give a wrong result.

Example: 0001 0110 (22) shifted left 2 → 0101 1000 (88).

8. Two’s complement

In 8-bit two’s complement the leftmost bit is worth −128, so the range is −128 to +127. A leading 1 means the number is negative.

−128	64	32	16	8	4	2	1
1	1	1	0	1	0	1	0

To write −22: write +22 (0001 0110), flip every bit (1110 1001), then add 1 → 1110 1010. Check: −128 + 64 + 32 + 8 + 2 = −22.

9. Text: ASCII and Unicode

Character set	Bits per character	Characters	Good for
ASCII	7 (extended: 8)	128 (256)	English text
Unicode	Up to 32	Over a million	Every language and emojis

Each character has a binary code from a character set. Unicode covers far more characters, but needs more bits per character, so text files are bigger.

10. Sound

A microphone produces an analogue wave. The computer measures (samples) its height at regular intervals and stores each sample in binary.

• Sample rate (sampling rate): samples taken per second, in hertz. Higher rate = more accurate recording, bigger file.
• Sample resolution (bit depth): bits used to store each sample. More bits = more accurate amplitude, bigger file.

Sound file size (bits) = sample rate × sample resolution × length in seconds.

11. Images

A bitmap image is a grid of pixels, and each pixel’s colour is stored as a binary number.

• Resolution: the number of pixels (width × height). More pixels = more detail, bigger file.
• Colour depth: bits per pixel. With n bits a pixel can be one of 2ⁿ colours (8 bits = 256 colours).

Image file size (bits) = width × height × colour depth.

12. Storage units and file size

The syllabus uses powers of 2 (1024), not 1000.

Unit	Size
Nibble	4 bits
Byte	8 bits
Kibibyte (KiB)	1024 bytes
Mebibyte (MiB)	1024 KiB
Gibibyte (GiB)	1024 MiB
Tebibyte (TiB)	1024 GiB
Pebibyte (PiB)	1024 TiB
Exbibyte (EiB)	1024 PiB

Bits ÷ 8 = bytes, then ÷ 1024 for each step up. Example: a 1024 × 512 image with a colour depth of 8 bits = 4 194 304 bits = 524 288 bytes = 512 KiB.

13. Compression

Compression makes files smaller, so they need less storage, transfer faster and use less bandwidth.

Type	How it works	Result	Use it for
Lossy	Permanently removes data: lower resolution or colour depth, lower sample rate or resolution, sounds people cannot hear	Much smaller file, some quality lost, original cannot be restored	Photos, music, video
Lossless	No data lost, e.g. run-length encoding (RLE): AAAAABBB → 5A 3B	Smaller file, original rebuilt exactly	Text, program files

14. Examiner tips

1. Binary addition done in denary. Add in binary and show every carry, or you lose the working marks.
2. Shifting the wrong way, or filling with 1s. Read “left” or “right” carefully, and fill with 0s.
3. Two’s complement read as normal binary. A leading 1 means negative.
4. Hex converted as one big number. For hex 14, convert the 1 and the 4 separately (0001 0100).
5. Missing leading zeros. An 8-bit answer needs all 8 bits.
6. Saying hex uses less memory. It doesn’t; it is just easier for people to read.
7. Mixing up sample rate and sample resolution. Rate = per second; resolution = bits per sample.
8. Image ideas in sound questions. Sample resolution has nothing to do with pixels.
9. Naming a file type as a compression method. “Zip” or “JPEG” is not a method.
10. Ignoring the context. Mention the scenario in your answer.

15. Practice questions

More practice: Q4–Q8 (19 marks)

Five more exam-style questions on two’s complement, sound and image file sizes, Unicode and compression, each with a full mark guide, are included in Revision Hub Plus.

16. Topic test: predict your grade