SLIDE 1
CSZone.co.uk
Click to advance · Arrow keys also work
CAIE 9618 · Paper 1 · Topic 1.1.3

Floating-Point
Representation

Mantissa · Exponent · Normalisation · Precision vs Range

CSZone Cambridge International AS & A Level Computer Science 9618
Learning Objectives

By the end of this video you will be able to...

Explain the structure of a floating-point number: mantissa and exponent
Convert between denary and binary floating-point representation
Normalise a floating-point binary number
Explain the trade-off between precision and range
Why Floating Point?

Representing Very Large or Very Small Numbers

"Fixed-point binary cannot efficiently represent numbers with very wide ranges. Floating point uses a mantissa × 2^exponent format — similar to scientific notation."
SCIENTIFIC NOTATION (DENARY)
3.14 × 10² = 314
6.02 × 10²³ (Avogadro)
1.6 × 10⁻¹⁹ (electron charge)
FLOATING POINT (BINARY)
Value = mantissa × 2^exponent
Mantissa: significant digits
Exponent: magnitude (how big/small)
Floating-Point Structure

Mantissa and Exponent

EXAMPLE: 8-BIT MANTISSA + 4-BIT EXPONENT (two's complement)
Mantissa: 0.1011010    Exponent: 0011
Value = 0.1011010₂ × 2^3
Shift binary point 3 places right: 101.1010₂
= 4 + 1 + 0.5 + 0.125 = 5.625₁₀
MANTISSA
Contains the significant digits
More bits = more precision
EXPONENT
Scales the mantissa by a power of 2
More bits = greater range
Normalisation

Maximising Precision

A floating-point number is normalised when the mantissa is in its most precise form — no wasted leading bits.
POSITIVE NORMALISED
Mantissa starts with 0.1...
e.g. 0.1011010
The bit after the binary point must be 1
NEGATIVE NORMALISED
Mantissa starts with 1.0...
e.g. 1.0100110
The bit after the binary point must be 0
Not normalised: 0.0011010 — leading zero after point wastes precision. Shift left and adjust exponent.
Precision vs Range

The Fundamental Trade-Off

Given a fixed total number of bits, allocating more bits to the mantissa increases precision but reduces range, and vice versa.
MORE MANTISSA BITS
Greater precision — more decimal places
Fewer exponent bits → smaller range
Example: scientific measurements need precision
MORE EXPONENT BITS
Greater range — very large or small values
Fewer mantissa bits → less precise
Example: astronomical distances
Exam Practice

Cambridge-style questions

Question 1
A floating-point number is stored with a 6-bit two's complement mantissa of 0.10110 and a 4-bit two's complement exponent of 0010. Calculate the denary value of this number. Show your working.
3 marks
✓ MARK SCHEME
1 mark
Exponent = 0010 = +2 in two's complement
1 mark
Shift mantissa 2 places right → 010.110
1 mark
= 2 + 0.5 + 0.25 = 2.75
Exam Practice
Question 2
Explain what is meant by normalisation of a floating-point number, and state the benefit of using normalised form.
3 marks
✓ MARK SCHEME
1 mark
Normalisation adjusts the mantissa and exponent so the mantissa has no unnecessary leading bits
1 mark
Positive normalised: mantissa starts with 0.1; negative: starts with 1.0
1 mark
Benefit: maximum precision is achieved for the given number of bits
Question 3
State one effect of increasing the number of bits in the exponent while keeping total bit length constant.
1 mark
1 mark
The range of numbers that can be represented increases, but the precision (number of significant digits) decreases
Common Mistakes

Don't lose easy marks

1
Confusing the direction of the shift — a positive exponent of +n means shift the binary point n places to the right (multiply by 2ⁿ). A negative exponent means shift left (divide).
2
Normalised form for negative numbers — many students write 1.1... instead of 1.0... for a normalised negative two's complement mantissa. The bit after the point must be 0 for a negative number.
3
Saying "more mantissa bits gives more range" — it gives more precision. More exponent bits give more range. Don't swap these terms.
4
Forgetting the exponent is also in two's complement — a negative exponent shifts the binary point left, giving a fraction smaller than 1. Don't assume the exponent is always positive.
Topic Summary — 1.1.3

What You Need to Know

STRUCTURE
Value = mantissa × 2^exponent
Both mantissa and exponent stored in two's complement
Mantissa: significant digits
Exponent: powers of 2 scale factor
NORMALISATION
Positive: mantissa starts 0.1...
Negative: mantissa starts 1.0...
Maximises precision for given bit count
Adjust exponent when shifting mantissa
PRECISION vs RANGE
More mantissa bits → more precision
More exponent bits → greater range
Total bits fixed → trade-off applies
Rounding errors occur when exact value can't be stored
CSZone

Next Video

1.1.4
Character Encoding
ASCII · Unicode · UTF-8 · Why Unicode?
Head to CSZone.co.uk for the complete worksheet, quiz, and interactive tools