AQA A-Level Computer Science: Fundamentals of Data Representation — Revision Notes

Fundamentals of data representation (section 4.5 of AQA A-Level Computer Science 7517) explains how numbers, characters, images and sound are stored as bits, how data is checked for errors, compressed and encrypted. Paper 2 tests it with hand calculations – two’s complement, floating point, file sizes – and short explanations, so learn the methods until they are automatic.

Condensed for the final weeks. For the full explanation, use the Fundamentals of data representation study guide. Practise with the Fundamentals of data representation practice questions.

Syllabus coverage

AQA 7517 specification (Version 1.6, 31 July 2023), section 4.5, assessed on Paper 2:

• 4.5.1 Number systems – natural, integer, rational, irrational, real, ordinal; counting and measurement
• 4.5.2 Number bases – decimal, binary, hexadecimal
• 4.5.3 Units of information – bits, bytes, binary and decimal prefixes
• 4.5.4 Binary number system – unsigned binary and arithmetic, two’s complement, fixed and floating point, rounding, absolute and relative errors, range and precision, normalisation, underflow and overflow
• 4.5.5 Information coding systems – character form of a digit, ASCII and Unicode, error checking
• 4.5.6 Representing images, sound and other data – ADC/DAC, bitmaps, vectors, sound, MIDI, compression, encryption

Number sets and units

Set	Symbol	Meaning
Natural	ℕ	{0, 1, 2, 3, …} – includes zero; used for counting
Integer	ℤ	{…, -2, -1, 0, 1, 2, …}
Rational	ℚ	Can be written as a fraction of integers; includes the integers (7 = 7/1)
Irrational	–	Cannot be written as a fraction, e.g. √2
Real	ℝ	All of the above; used for measurement
Ordinal	–	Gives position: 1st, 2nd, 3rd

n bits give 2ⁿ different values. Binary prefixes are powers of 2: kibi (Ki) 2¹⁰, mebi (Mi) 2²⁰, gibi (Gi) 2³⁰, tebi (Ti) 2⁴⁰. Decimal prefixes are powers of 10: kilo 10³, mega 10⁶, giga 10⁹, tera 10¹². So 1 KiB = 1,024 bytes but 1 kB = 1,000 bytes.

Hexadecimal is a shorthand for binary: one hex digit = 4 bits, so it is shorter to write and less error-prone to read. Write the base as a subscript: 67₁₀, 10011011₂, AE₁₆.

Unsigned binary and two’s complement

• Unsigned n bits: range 0 to 2ⁿ - 1.
• Two’s complement n bits: range -2ⁿ⁻¹ to 2ⁿ⁻¹ - 1 (8 bits: -128 to 127). The most significant bit has a negative place value (-128 in 8 bits).
• To negate: invert all bits and add 1. -45: 45 = 00101101 -> 11010010 -> 11010011.
• Subtraction = add the two’s complement: 23 - 45 = 00010111 + 11010011 = 11101010 = -22.
• Two’s complement is the only signed representation AQA examines.

Fixed point and floating point

Fixed point: the binary point is in a fixed position. With 4 bits after the point, 0110.1010 = 4 + 2 + 0.5 + 0.125 = 6.625.

Floating point (AQA format): mantissa × 2^exponent, with both mantissa and exponent in two’s complement. You do not need the IEEE standard.

Worked example: floating point to decimal

8-bit mantissa 1.0110000, 4-bit exponent 0010 (= 2).

Mantissa 1.0110000 = -1 + 0.25 + 0.125 = -0.625
Value = -0.625 × 2² = -2.5
(Check: move the point 2 places right: 101.10000 = -4 + 1 + 0.5 = -2.5)

Normalisation

A normalised mantissa starts 0.1 (positive) or 1.0 (negative). Normalising gives the maximum precision for the bits available and a unique representation of each number.

Unnormalised 0.0011000 × 2^0101 (5): shift the mantissa 2 places left -> 0.1100000, subtract 2 from the exponent -> 0011 (3). Value is unchanged: 0.11₂ × 2³ = 110₂ = 6.

Range, precision and errors

	Fixed point	Floating point
Range	Smaller for the same bits	Much larger
Precision	Constant across the range	Varies; more bits in the mantissa = more precision
Speed	Faster to process	Slower (needs normalising and exponent handling)

• Rounding errors: a value is exact only if it is a binary fraction that fits the bits; 0.1₁₀ can never be stored exactly.
• Absolute error = |actual - stored|. Relative error = absolute error ÷ actual value. Storing 6.7 with 4 fractional bits gives 6.6875: absolute error 0.0125, relative error ≈ 0.0019 (0.19%). The same absolute error is far more serious for a small number than a large one.
• Overflow: the result is too large to represent. Underflow: the result is too small (too close to zero) to represent, e.g. dividing a very small number by a very large one.

Characters and error checking

• The character code for a digit is not its binary value: ‘7’ is 0110111 (55) in ASCII, but 7 in pure binary is 0111.
• ASCII uses 7 bits (128 characters, English only). Unicode was introduced to represent the characters of all the world’s writing systems and symbols; the first 128 Unicode codes match ASCII.
• Parity bit: added so the number of 1s is even (or odd); detects a single-bit error but not which bit.
• Majority voting: each bit is sent (e.g.) three times and the receiver takes the majority – this can correct an error, at the cost of extra data.
• Checksum: a value calculated from the data and sent with it; the receiver recalculates and compares.
• Check digit: a digit calculated from the other digits of a code (e.g. ISBN, barcode) to catch entry errors.

Images and sound

• ADC: samples an analogue signal and converts each sample to a binary number; used with analogue sensors. DAC: converts digital to analogue; most commonly digital audio to an analogue signal.
• Bitmap: a grid of pixels. Size in pixels = width × height; colour depth = bits per pixel; resolution = dots (pixels) per inch. Storage (ignoring metadata) = width × height × colour depth. Metadata includes width, height and colour depth.
• Vector graphic: a list of objects (e.g. circle, rectangle) with properties (position, radius, line colour, fill colour). Scales without losing quality and is usually smaller for simple drawings; bitmaps suit photographs.
• Sound: sampling rate = samples per second (Hz); sample resolution = bits per sample. Size in bytes = rate × resolution × seconds ÷ 8. Nyquist theorem: sample at least twice the highest frequency to reproduce the sound accurately.
• MIDI: stores event messages (instructions such as which note to play and when) rather than sampled sound. Files are small and the music is easy to edit, e.g. changing tempo or instrument.

Worked example: file sizes

Bitmap: 1920 × 1080 pixels, 24-bit colour
  1920 × 1080 × 24 = 49,766,400 bits ÷ 8 = 6,220,800 bytes ≈ 6.2 MB

Sound: 30 s, 44,100 Hz, 16-bit, one channel
  44,100 × 16 × 30 = 21,168,000 bits ÷ 8 = 2,646,000 bytes ≈ 2.6 MB

Compression and encryption

• Lossless compression keeps all the data (needed for text and program files); lossy removes data permanently for much smaller files (acceptable for images and sound).
• Run length encoding: replace runs of the same value with a count and the value.
• Dictionary-based: store repeated patterns once in a dictionary and replace each occurrence with a reference to its entry.
• Encryption: turning plaintext into ciphertext using a cipher and key so it is unintelligible without the key.
• Caesar cipher: shift each letter by a fixed amount (shift 3: HELLO -> KHOOR). Easily cracked: only 25 possible shifts to try, and letter frequencies survive.
• Vernam cipher (one-time pad): XOR each character with a key that is truly random, at least as long as the message and used once only. It is the only cipher proven to offer perfect security, because the ciphertext is uniformly distributed. All other ciphers rely on computational security – they could in theory be broken given enough ciphertext and time.

Common mistakes / exam traps

• Forgetting the negative place value of the mantissa’s first bit in floating point.
• Normalising by shifting the mantissa but not adjusting the exponent (or adjusting it the wrong way).
• Mixing KiB (1,024) and kB (1,000) in a calculation.
• Dividing by 8 twice, or not at all, when converting bits to bytes.
• Saying a parity bit can correct an error – it only detects one; majority voting can correct.
• Describing Vernam as secure without stating all three key conditions: random, as long as the message, used once.

Frequently asked questions

Why are floating point numbers normalised? Normalising puts the most significant bits in the mantissa, which gives the greatest precision for the number of bits and a single unique representation for each value.

Why can 0.1 not be stored exactly in binary? 0.1 in decimal is a recurring fraction in binary, so it cannot be written exactly as a sum of 1/2, 1/4, 1/8 and so on in a finite number of bits. A rounding error always results.

What is the difference between absolute and relative error? Absolute error is the difference between the true and stored value. Relative error divides that by the true value, so it shows how significant the error is compared with the size of the number.

Related resources

Fundamentals of data representation study guide | Fundamentals of data representation practice questions

Official syllabus

AQA AS and A-level Computer Science (7516/7517) specification, Version 1.6 (31 July 2023), section 4.5 – aqa.org.uk/7517.