In simple terms
A friendly intro before the formal notes — no formulas yet.
Scientific Notation for Computers
Floating-point is how computers handle numbers that are very large, very small, or have a fractional part. It's like scientific notation, splitting a number into a significant value (the mantissa) and a power-of-two scale factor (the exponent).
Imagine writing down the distance to the sun. You wouldn't write 150,000,000,000 metres; you'd write it in scientific notation as metres. This separates the core digits (1.5) from the scale (the power of 11). Floating-point representation does the exact same thing for computers, but using binary (base 2) instead of denary (base 10).
- 1
Convert the denary number into a fixed-point binary number (e.g., 9.75 becomes 1001.11).
- 2
Move the binary point so it's at the beginning, creating a fractional number. This is called normalisation.
- 3
Count how many places the binary point moved. This value becomes the exponent.
- 4
Store the normalised mantissa and the exponent as two's complement binary numbers using the specified number of bits.
Explore the concept
Use the live diagram and synced steps — play it or tap a step card to walk through.
Key formulas
Tap any symbol to reveal exactly what it means and its units.
$Value = Mantissa \times 2^{Exponent}$
Full topic notes
Formal explanation with the rigour you need for the exam.
The Structure of a Floating-Point Number
A floating-point number is not stored as a single binary value. Instead, the bits allocated to it are split into two distinct parts: the mantissa and the exponent. Both parts are typically stored as signed binary numbers using the two's complement system.
$Value = Mantissa \times 2^{Exponent}$
Mantissa: A fixed-point binary fraction that holds the significant digits of the number. Its sign bit (Most Significant Bit) indicates whether the number is positive (0) or negative (1). It determines the number's precision.
Exponent: A binary integer that represents the power of 2 by which the mantissa should be multiplied. It determines the position of the binary point and thus the number's range (magnitude).
Conversion and Normalisation
Converting a denary number to floating-point involves several steps. The most critical is normalisation, which standardises the format to maximise precision. For the Cambridge syllabus, a normalised positive number has a mantissa starting '01' (representing 0.1...), and a normalised negative number has a mantissa starting '10' (representing -0.1...). This is achieved by shifting the binary point and adjusting the exponent accordingly.
Impact of Bit Allocation: Range vs. Precision
The total number of bits for a floating-point number is fixed. How these bits are divided between the mantissa and the exponent creates a fundamental trade-off. Allocating more bits to one part means fewer bits are available for the other.
More bits for Mantissa: Increases precision. The number can store more significant figures, reducing rounding errors. However, this leaves fewer bits for the exponent, decreasing the range.
More bits for Exponent: Increases range. The system can represent much larger and smaller numbers (further from zero). However, this leaves fewer bits for the mantissa, decreasing precision.
This trade-off means a floating-point system is designed for either high precision or a wide range, but cannot maximise both simultaneously within a fixed bit length.
Exam questions frequently ask about the consequences of changing the bit allocation. For example, 'What would be the effect of increasing the number of bits used for the exponent?'. The answer should always mention the impact on both range (increases) and precision (decreases).
Worked examples
See the formulas applied — reveal one step at a time, like the exam.
Represent the denary number 13.5 in normalised floating-point format using a 10-bit mantissa and a 6-bit exponent. Both should use two's complement.
- 1
Convert to Binary:
Represent the denary number -5.25 in normalised floating-point format using an 8-bit mantissa and a 4-bit exponent. Both should use two's complement.
- 1
Convert the positive equivalent to Binary:
How it all connects
The big idea sits in the middle — tap a linked idea to explore the link.
Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.
Glossary
Key terms for this topic — skim now; the Check step will test them.
Quick check
Write your answer first, then compare it with the model one — the gap is what you would have lost.
Teach it back
If you can explain it simply, you own it — gaps here are marks you’d lose.
Teach it back
Explain this topic as if teaching a friend. We name the gaps an examiner would still dock.
Revision flashcards
Guess first, then flip — retrieval beats re-reading.
Key takeaways
Review these before you close the topic — retrieval beats re-reading.
Mantissa: A fixed-point binary fraction that holds the significant digits of the number. Its sign bit (Most Significant Bit) indicates whether the number is positive (0) or negative (1). It determines the number's precision.
Exponent: A binary integer that represents the power of 2 by which the mantissa should be multiplied. It determines the position of the binary point and thus the number's range (magnitude).
Practice — then mark it
The whole point: a real Cambridge question, marked mark-by-mark.
Test Your Knowledge on Floating-Point Numbers
Test Your Knowledge on Floating-Point Numbers
Extra simulations & links
PhET, GeoGebra and other curated tools — open in a new tab.
Frequently asked
Checkpoint
One marked question is worth ten re-reads — close the loop before you move on.
Reading it isn’t knowing it — prove it.
Before you move on: do Test Your Knowledge on Floating-Point Numbers on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.
Discuss Floating-point numbers, representation and manipulation
Ask, share and discuss with other Computer Science students