Convert between denary and binary
Use powers of two and place-value subtraction to convert positive integers accurately.
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 is the binary place value immediately to the left of 8?
What you will learn
- Convert positive binary integers to denary
- Convert positive denary integers to binary
- Check a conversion by rebuilding its place-value total
Binary is base two
From right to left, binary place values are 1, 2, 4, 8, 16 and so on. Add the place values that have a 1 to convert to denary.
For denary-to-binary conversion, start with the largest place value that fits. Write 1 and subtract it; write 0 when the value does not fit. Continue down to the units place.
Keep the requested width
Leading zeros do not change the value, but they can be required by a question. If a question requests eight bits, show eight digits.
Pause and explain
Which binary pattern represents denary 19?
Worked example
Convert denary 53 to binary, then check the result.
Show the worked solution
- The largest power of two no greater than 53 is 32; remainder 21.
- Use 16, then 4, then 1; the remainder becomes zero.
- At places 32,16,8,4,2,1 the pattern is 110101; add two leading zeros for eight bits.
Answer 53₁₀ = 00110101₂; check: 32+16+4+1=53.
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 45 to binary.
Give me a hint
Subtract powers of two from largest to smallest.
Compare my reasoning
- Use 32, leaving 13.
- Use 8, leaving 5; then 4, leaving 1; then 1.
- The pattern at 32,16,8,4,2,1 is 101101.
101101₂
Look for these in your work
- Largest place chosen
- Remainders correct
- Bit pattern correct
Convert 11001010₂ to denary.
Give me a hint
Add the place values for the 1 bits.
Compare my reasoning
- The 1 bits are at 128, 64, 8 and 2.
- Add 128+64+8+2.
202
Look for these in your work
- Set-bit positions correct
- Sum equals 202
A pixel identifier is 173. Convert it to eight-bit binary for a sensor packet.
Give me a hint
Use 128, 32, 8, 4 and 1.
Compare my reasoning
- 173−128=45.
- 45−32=13; 13−8=5; 5−4=1; 1−1=0.
- Write bits at 128,64,32,16,8,4,2,1.
10101101₂
Look for these in your work
- Eight bits shown
- Place values sum to 173
Common mistakes
- Reversing the place values
- Stopping at the first 1 instead of recording every place
- Forgetting requested leading zeros
What can you explain now?
What is 101101₂ in denary?
Compare with the explanation
45
The 1 bits represent 32+8+4+1=45.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How can you check a denary-to-binary answer without repeating the conversion method?
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.