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AQA 7517 · Paper 2 · 4.5.3

Floating-Point
Representation

Mantissa · Exponent · Normalisation · Overflow/Underflow · Section 4.5

WHAT YOU'LL LEARN
Mantissa × 2^exponent · Normalisation · Precision · Range · Overflow · Underflow
AQA SPEC LINK
4.5.3 — Floating-point numbers: mantissa, exponent, normalisation
Floating-Point Concept

What is Floating-Point?

value = mantissa × 2^exponent
Mantissa — holds the significant digits (precision); binary fraction starting with 0.1…
Exponent — scales the mantissa by a power of 2 (determines magnitude/range)
Similar to scientific notation: 3.14 × 10⁶ = mantissa=3.14, exponent=6
Floating-point allows both very large and very small numbers in the same format
Format

IEEE 754 Single Precision (32-bit)

Sign
1 bit
Exponent
8 bits
Mantissa
23 bits
AQA uses a simplified version in exams — typically 8–16 bit examples with defined mantissa and exponent sizes.
Normalisation

Normalised Floating-Point

A floating-point number is normalised when the mantissa starts with 0.1 (for positive) or 1.0 (for negative — in two's complement). This ensures maximum precision.
NORMALISED (good)
Positive: 0.1xxxxxx
Negative: 1.0xxxxxx
NOT NORMALISED (wastes precision)
Positive: 0.01xxxx (shift needed)
Positive: 0.001xxx (shift needed)
Worked Example

Floating-Point Example

8-bit format: 4-bit mantissa, 4-bit exponent (two's complement). Value: mantissa × 2^exponent.
Example: mantissa=0.101, exponent=0011
mantissa = 0.101₂ = 0.625
exponent = 0011₂ = 3
value = 0.625 × 2³ = 0.625 × 8 = 5.0
Normalised check: mantissa starts with 0.1 ✓
Precision vs Range

Mantissa Bits vs Exponent Bits

MORE MANTISSA BITS
Greater precision (more significant figures)
Fewer exponent bits → smaller range
MORE EXPONENT BITS
Larger range (bigger/smaller numbers)
Fewer mantissa bits → less precision
Errors

Overflow, Underflow & Rounding

OVERFLOW
Result too large for exponent to represent. Causes incorrect (often infinite) result.
UNDERFLOW
Result too small (too close to zero) for exponent to represent. May be rounded to zero.
ROUNDING ERRORS
Some numbers cannot be represented exactly (e.g. 0.1 in binary). Accumulated errors can affect calculations.
Floating vs Fixed

Floating-Point vs Fixed-Point

FLOATING-POINT
Wider range
Better for scientific calculations
Slower, more complex hardware
FIXED-POINT
Faster, simpler
Good for financial/embedded systems
Limited range
AQA Exam Style

Practice Question

AQA 7517 — Paper 2 Style
A floating-point number is stored using 8 bits: 5 bits for mantissa (two's complement), 3 bits for exponent (two's complement).
(a) Explain what is meant by a normalised floating-point number. [2]
(b) For mantissa=01010, exponent=010, calculate the denary value. [3]
(c) Explain what is meant by "underflow" in floating-point arithmetic. [1]
[6 marks]
2 marks
(a) The leading bits of the mantissa are 0.1 (positive) or 1.0 (negative) — ensures maximum precision with no leading redundant zeros
3 marks
(b) Mantissa = 0.1010₂ = 0.625; Exponent = 010₂ = 2; Value = 0.625 × 2² = 0.625 × 4 = 2.5
1 mark
(c) The result is too small (too close to zero) to be represented — exponent cannot be negative enough
Summary

Key Points to Remember

Floating-point = mantissa × 2^exponent; similar to scientific notation
Normalised — mantissa starts 0.1 (positive) or 1.0 (negative); maximises precision
More mantissa bits = more precision; more exponent bits = larger range
Overflow = too large; underflow = too close to zero
Rounding errors: cannot represent all real numbers exactly in binary
🎉 Lesson complete — move to the quiz!