Decode two’s complement and check its range
Interpret an 8-bit pattern as signed, convert negative values back to denary and identify out-of-range values.
Exam scope and assessment notes
This sequence follows Cambridge IGCSE Computer Science 0984, 2026–2028 syllabus Version 4. It covers subject content sections 1.1–1.3 only. The separately published 2029–2031 syllabus is not mapped here. Practice is original, self-checked and does not award an official Cambridge grade. Full timed papers and human subject-teacher review remain outstanding. Cambridge states that 0984 has the same 2026–2028 subject content as 0478, with 9–1 grading. In June and November, Cambridge lists 0984 in Administrative zone 3 and for centres in Oman and the UAE. Confirm availability for your centre and exam series with Cambridge International. The separately published 2029–2031 syllabus changes to Python 3 only and removes pseudocode; this 2026–2028 route does not cover those changes.
Before you begin
- No programming language is required. Be comfortable with place value and multiplication; each lesson introduces the computer-science notation it uses.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does a leading 1 mean in an 8-bit two’s complement pattern?
What you will learn
- Decode positive and negative 8-bit patterns
- Convert a negative two’s complement value to denary
- Identify values outside the signed 8-bit range
Read the most significant bit first
If the most significant bit is 0, read the pattern as a positive binary integer. If it is 1, the value is negative. To find its magnitude, invert all eight bits, add 1, and then apply a negative sign.
Apply and check the rule
The range is −128 to +127. For example, 10000000 is −128. The positive value +128 cannot fit because its eight-bit unsigned pattern starts with 1, which would signal a negative value in this representation.
Pause and explain
Which signed 8-bit value cannot be represented?
Worked example
Convert 11101010₂ to denary using two’s complement.
Show the worked solution
- The leading bit is 1, so the value is negative.
- Invert: 00010101; add 1: 00010110, which is 22.
- Apply the negative sign.
Answer −22
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Convert 11111101₂ from 8-bit two’s complement to denary.
Give me a hint
Invert, add one, then apply the sign.
Compare my reasoning
- Leading bit 1 means negative.
- Invert: 00000010; add one: 00000011=3.
- Apply the negative sign.
−3
Look for these in your work
- Negative sign identified
- Magnitude equals 3
- Final value −3
Convert 01011011₂ to denary in 8-bit two’s complement.
Give me a hint
A leading zero indicates an ordinary positive value.
Compare my reasoning
- The set places are 64, 16, 8, 2 and 1.
- Add 64+16+8+2+1.
+91
Look for these in your work
- Positive sign identified
- Place values correct
- Sum equals 91
A learner encodes +130 in an 8-bit two’s complement register and obtains 10000010. Explain the mistake.
Give me a hint
Check the range before encoding, then interpret the leading bit.
Compare my reasoning
- The signed range ends at +127.
- +130 is out of range.
- 10000010 is interpreted as a negative two’s complement value, not +130.
+130 cannot be represented in 8-bit two’s complement. The pattern 10000010 represents −126, so the learner has mistaken an unsigned-like conversion for a valid signed encoding.
Look for these in your work
- Range checked
- Pattern interpreted as signed
- Mistake explained
Common mistakes
- Reading a leading-one two’s complement pattern as an unsigned value
- Negating after inversion but before adding one
What can you explain now?
What is 10000000₂ in 8-bit two’s complement?
Compare with the explanation
−128
It is the minimum signed 8-bit value; its magnitude is not obtained by a simple positive 8-bit pattern.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
What two steps find the magnitude of a negative 8-bit two’s complement pattern?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.