O Level Computer Science: Data Representation — Revision Notes

Condensed for the final weeks. For the full explanation, use the Data Representation study guide.

Number bases

Denary to binary: repeatedly divide by 2 and read the remainders upwards. Binary to denary: add the place values where a 1 appears. Binary to hex: split into groups of four bits from the right and convert each.

Why hexadecimal is used: it is far more compact than binary, and each hex digit maps to exactly four binary digits, so conversion is trivial and human error is much less likely. Used for memory addresses, MAC addresses, colour codes and error codes.

Worked example. Convert denary 205 to hexadecimal.

Denary 205  ->  Binary 11001101  ->  Hex CD

128 64 32 16  8  4  2  1
  1  1  0  0  1  1  0  1     = 128+64+8+4+1 = 205

Split into nibbles: 1100 = 12 = C,  1101 = 13 = D  ->  CD

Binary addition and overflow

0+0=0    0+1=1    1+1=10 (carry)   1+1+1=11 (carry)

Overflow occurs when the result needs more bits than the register holds. In 8 bits the largest unsigned value is 255, so adding beyond that gives an incorrect result. The phrasing that scores is “the result is too large for the number of bits allocated”.

Binary shifts

• Left shift by n → multiply by 2ⁿ.
• Right shift by n → divide by 2ⁿ, discarding the remainder.

Bits shifted out are lost, so shifting can lose data — which is why repeated shifting is not reversible.

Character sets

ASCII — 7 bits, 128 characters; extended ASCII uses 8 bits for 256. Unicode — up to 32 bits in a fixed-width encoding, covering a very wide range of the world’s writing systems.

Unicode was needed because ASCII could not represent non-Latin scripts such as Arabic, Chinese or Urdu. The trade-off is that Unicode text can require more storage per character.

Note that character codes are ordered, so 'A' (65) is less than 'a' (97), which is why case-sensitive sorting behaves as it does.

Images

Bitmap images are grids of pixels.

file size = width x height x colour depth

• Resolution — number of pixels.
• Colour depth — bits per pixel; n bits give 2ⁿ colours.

Higher resolution and colour depth give better quality but larger files. That trade-off answers most image questions.

Vector graphics store the properties of objects rather than pixels, so they scale without loss of quality and are usually smaller for line art — but they are unsuitable for photographs.

Worked example. An image is 800 by 600 pixels with a colour depth of 24 bits. Find the file size in megabytes.

bits  = 800 x 600 x 24 = 11 520 000 bits
bytes = 11 520 000 / 8 = 1 440 000 bytes
KB    = 1 440 000 / 1024 = 1406.25 KB
MB    = 1406.25 / 1024   = 1.37 MB

Dividing by 8 to reach bytes, then by 1024 twice to reach MB, is where most marks are lost.

Sound

file size = sample rate x bit depth x duration

• Sample rate — samples per second (Hz).
• Bit depth (resolution) — bits per sample.

Sampling is an approximation of a continuous wave. A higher sample rate captures the waveform more faithfully and a higher bit depth records each amplitude more precisely — so both improve accuracy at the cost of file size.

Compression

	Lossless	Lossy
Data	Fully recoverable	Permanently removed
Use	Text, program files, spreadsheets	Images, audio, video
Method	Run-length encoding	JPEG, MP3

Lossless must be used where every bit matters — a compressed program or spreadsheet missing data would be corrupt. Lossy is acceptable where the removed data is imperceptible to human senses.

Run-length encoding replaces runs of identical values with a value and a count. It works well on images with large uniform areas but can produce a larger file on noisy or photographic data — so compression does not always reduce size.

Exam traps

• Reading binary remainders downwards.
• Saying hex is used because computers work in it — they work in binary.
• Confusing resolution with colour depth.
• Saying lossy compression is always worse.
• Assuming RLE always shrinks a file.
• Forgetting that shifting can lose bits.
• Dividing by 1024 only once when converting bits to megabytes, instead of twice (bits to bytes, then bytes to KB, then KB to MB).
• Splitting a binary number into nibbles from the wrong end when converting to hex.

Self-test

1. Why is hexadecimal used rather than binary?
2. What is overflow, and how should it be described?
3. Why was Unicode introduced?
4. Give the two file-size formulae for images and sound.
5. When is run-length encoding a poor choice?
6. Convert denary 205 to hexadecimal, showing the binary intermediate step.
7. An image is 800 × 600 pixels with a colour depth of 24 bits. Calculate its file size in megabytes.

Answers: 1. It is far more compact and each hex digit corresponds to exactly four binary digits, making conversion straightforward and reducing human error. 2. When the result of a calculation requires more bits than the register can hold, so the stored value is incorrect. 3. ASCII could not represent non-Latin writing systems, so a larger character set was needed to cover a very wide range of the world’s scripts. 4. Image: width × height × colour depth. Sound: sample rate × bit depth × duration. 5. On noisy or photographic data with few repeated adjacent values, where it can produce a file larger than the original. 6. 205 = 11001101 in binary, split into nibbles 1100 (C) and 1101 (D), giving CD in hexadecimal. 7. 800 × 600 × 24 = 11,520,000 bits ÷ 8 = 1,440,000 bytes ÷ 1024 = 1406.25 KB ÷ 1024 ≈ 1.37 MB.