Describe advanced data representation: floating-point numbers, normalisation, fixed-point, and the limits of representing real numbers.

What this dot point is asking

WJEC wants you to represent real numbers in floating-point form, to normalise them, to understand fixed-point representation, and to explain the precision and range limits of representing real numbers in a fixed number of bits. This deepens the AS data-representation topic into the harder territory of real numbers, and the floating-point and normalisation questions are reliably examined in Unit 4, rewarding careful working and clear explanation of the precision-versus-range trade-off.

The answer

Floating-point representation

The "floating" point means the binary point can move, set by the exponent, which lets a fixed number of bits represent both very large and very small numbers, unlike a fixed-point scheme.

Mantissa, exponent and the trade-off

This trade-off is the central insight of the topic: you cannot improve both precision and range at once within a fixed word length, so a design chooses the split that suits its purpose.

Normalisation

Normalising a floating-point number adjusts the mantissa to a standard range (for a positive number the bits after the binary point begin 0.1...), changing the exponent to compensate so the value is unchanged. It ensures each number has a single unique representation and that the mantissa carries the maximum significant figures, wasting no bits on leading zeros.

Fixed-point and the limits of real numbers

Fixed-point representation places the binary point at a fixed position, simpler but with a fixed range and precision. Because any fixed word length can store only finitely many values, real numbers are generally stored approximately, which is the source of rounding errors.

Examples in context

Example 1. Why scientific calculations use floating point

Computations spanning from atomic to astronomical scales need both very small and very large magnitudes, which the exponent provides. Fixed point could not cover the range without enormous word lengths, so floating point's movable binary point is what makes large-scale scientific computing practical.

Example 2. Rounding error accumulating in a loop

Adding 0.1 a thousand times in floating point does not give exactly 100, because 0.1 has no exact binary floating-point representation, and the tiny errors accumulate. This concrete behaviour is why programmers avoid testing floating-point values for exact equality, and it traces directly to the finite-bit limit on representing real numbers.

Example 3. Choosing precision over range for currency

A system handling money may prefer a fixed-point or integer representation (storing pence as whole numbers) to avoid floating-point rounding errors on exact decimal amounts. This shows the design choice the trade-off forces: the right representation depends on whether range or exactness matters more for the data.

Try this

Q1. State which part of a floating-point number determines its precision. [1 mark]

• Cue. The mantissa (significand): more mantissa bits give more significant figures.

Q2. State what happens to the exponent when the mantissa is shifted one place to the left during normalisation. [1 mark]

• Cue. The exponent is reduced by one, to keep the represented value unchanged.