How are very large and very small real numbers stored in a fixed number of bits, and what does normalisation achieve?
Floating-point representation of real numbers using a mantissa and an exponent (both in two's complement), normalisation of a floating-point number, and the trade-off between range and precision.
An OCR H446 answer on floating-point representation: storing real numbers with a mantissa and an exponent in two's complement, how to normalise a floating-point number, and the trade-off between range and precision when bits are divided between mantissa and exponent.
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What this dot point is asking
OCR wants you to read and write floating-point numbers stored as a two's complement mantissa and exponent, normalise a floating-point number, and explain the trade-off between range and precision. This is a calculation dot point: expect to evaluate a stored number and to normalise one.
The answer
How a floating-point number is stored
Normalisation
Range versus precision
Examples in context
Scientific and graphics computation uses floating point because magnitudes vary enormously (from subatomic to astronomical). The familiar real or float type in programming is normalised floating point, which is why can print as , a rounding artefact of finite binary precision. Choosing single versus double precision is exactly the range-and-precision trade-off in practice. OCR links this to two's complement (the encoding of mantissa and exponent) and to the mathematical-skills content on accuracy.
Try this
Q1. State what the mantissa and the exponent each contribute to a floating-point number. [2 marks]
- Cue. The mantissa holds the significant digits (precision); the exponent gives the power of two that positions the binary point (range).
Q2. State the leading bits of a normalised positive mantissa and a normalised negative mantissa. [2 marks]
- Cue. Positive starts ; negative starts (the first bit after the point differs from the sign bit).
Q3. State the effect on a floating-point format of moving two bits from the exponent to the mantissa. [1 mark]
- Cue. Greater precision (more significant figures) but a smaller range of representable magnitudes.
Exam-style practice questions
Practice questions written in the style of OCR exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
OCR 20205 marksA floating-point number uses an 8-bit two's complement mantissa with the binary point after the first bit, and a 4-bit two's complement exponent. The stored value is mantissa and exponent . Calculate the denary value it represents.Show worked answer →
Read the exponent (1 mark): in two's complement is .
Apply the exponent (up to 2): move the binary point right by 3 places. The mantissa becomes .
Convert to denary (up to 2): . So the value is . Markers reward reading the exponent correctly, shifting the point the right way and distance, and the correct denary conversion. A common error is shifting the point the wrong way for a positive exponent.
OCR 20226 marksExplain what it means for a floating-point number to be normalised, why normalisation is used, and how increasing the number of bits in the mantissa rather than the exponent affects the number.Show worked answer →
Normalisation (up to 3): a normalised positive mantissa starts and a normalised negative mantissa starts , so the first bit after the point differs from the sign bit. This uses the available bits efficiently, removing leading zeros so the maximum precision is kept, and gives each value a single unique representation.
Why used (up to 1): it maximises precision for a given number of mantissa bits and avoids multiple representations of the same number.
Mantissa versus exponent bits (up to 2): more mantissa bits increase precision (more significant figures), while more exponent bits increase range (larger and smaller magnitudes). With a fixed total, increasing the mantissa improves precision at the cost of range. Markers reward the normalised bit pattern, the precision/uniqueness reason, and the precision-versus-range trade-off.
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