1. Can a finite binary fraction represent every real number exactly?
- Yes
- No
- Only with a sign bit
Check the answer and reasoning
No
Finite bit patterns cannot encode infinitely many distinct real values.
CS / AS level
Advanced data representation
Signed binary formats and floating-point representations encode negative and fractional values within finite storage. Range and precision trade off against each other.
Learning objectives
- Explain advanced data representation using a concrete example.
- Apply the worked method and justify each step.
- Recognise the limitation described in the final check.
The big idea
Signed binary formats and floating-point representations encode negative and fractional values within finite storage. Range and precision trade off against each other.
Bitmap/vector graphics, sampled sound and compression involve explicit models. An overflow means the mathematical result cannot fit the chosen range; rounding loses low-order information even without overflow.
Equations and relationships
8-bit two’s-complement range: −128 to 127
The most significant bit has weight −128.
Worked example
What does 11111111 represent in 8-bit two’s complement?
Follow the solution
Its value is −128+64+32+16+8+4+2+1=−1.
Investigate and explain
Predict the worked result before calculating. Change one input, record the new result, and explain which assumptions still hold.
Use the prediction activity below; a local model appears only where it supports this topic.
Check your understanding
2. What does 11111111 represent in 8-bit two’s complement?
- 255
- −127
- −1
Check the answer and reasoning
−1
Its value is −128+64+32+16+8+4+2+1=−1.
3. Does adding more mantissa bits mainly improve range or precision?
- Precision
- Range only
- Neither
Check the answer and reasoning
Precision
More significant bits allow finer spacing; exponent width primarily controls range.