🔢 Chapter 1 · Topic 1 · Paper 1 & 2

Denary to Binary
Conversion

How computers think in 0s and 1s — and why it powers everything

🔥 01 · Did You Know?

Every photograph you've ever taken, every song you've streamed, every text message you've sent — all of it is stored as a sequence of just two digits: 0 and 1. When NASA's Voyager 1 probe sends signals from 24 billion kilometres away, those signals arrive as binary. When your bank processes a £5,000 transfer, the amount is stored in binary. The entire digital world — from TikTok's recommendation algorithm to the NHS patient database — runs on a number system invented in the 17th century by Gottfried Leibniz, inspired by Chinese hexagrams. Today you learn the language every computer on Earth speaks.

02 · Core Concept

What is Binary?

Denary (base-10) is the number system humans use — it has 10 digits (0–9). Binary (base-2) is what computers use — it has only 2 digits: 0 and 1. Each binary digit is called a bit. A group of 8 bits is called a byte.

💡 Why binary? Computer circuits are built from transistors that are either ON (1) or OFF (0). Binary maps perfectly onto the physics of the machine.
🔟
Base 10
Denary — digits 0–9
🔢
Base 2
Binary — digits 0 and 1
💻
1 bit
One binary digit (0 or 1)
📦
8 bits
= 1 byte

The Place Value System

Just like denary has place values of 1, 10, 100, 1000 — binary has place values that are powers of 2. For an 8-bit number, the columns are:

2⁷
128
2⁶
64
2⁵
32
2⁴
16
2³
8
2²
4
2¹
2
2⁰
1

To convert denary to binary: work from left to right. Ask "Does this number fit in column 128?" If yes → write 1 and subtract. If no → write 0. Repeat for each column.

🎯 Max value in 8 bits: 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255. That's why IP addresses go up to 255 in each section!

Two Methods — Same Answer

📊 Place Value Method
➗ Division by 2 Method
1

Write out the 8 place values: 128 · 64 · 32 · 16 · 8 · 4 · 2 · 1

2

Starting from 128: if your number ≥ column value → write 1 and subtract. If not → write 0 and move on.

3

Repeat for each column until you reach the 1s column. The remainder must be 0 at the end.

1

Divide your denary number by 2. Record the remainder (0 or 1).

2

Take the quotient and divide by 2 again. Record the remainder. Repeat until the quotient is 0.

3

Read the remainders bottom to top — that is your binary number. Pad with leading zeros to make 8 bits.

⚠️ Common Exam Mistakes

❌

Reading division remainders top to bottom — you must read them bottom to top. The last remainder is the most significant bit (leftmost).

❌

Forgetting to pad to 8 bits — if your binary is only 5 bits long, add leading zeros: 00101010, not 101010.

❌

Skipping a column in the place value method — always write a 0 if the number doesn't fit. Never leave a column blank.

🏆
Cambridge Exam Tip: Always show your working — write the place value row, then the binary row underneath. Even if your final answer is wrong, you can earn method marks. Questions are typically 2–4 marks for a single conversion.
⚡ 03 · Interactive Tool
Denary → Binary Converter
// Enter any number 1–255 · See the full step-by-step working
Quick examples:
42
75
100
128
200
255
Place value grid:
128
—
64
—
32
—
16
—
8
—
4
—
2
—
1
—
Division by 2 working:
Number ÷ 2 = Quotient Remainder
Enter a number above and click Convert
Result — read remainders bottom to top:
04 · Worked Example

Converting 185 to Binary

Click each step to reveal the working — try to predict first!

📋 Question: Convert the denary number 185 to an 8-bit binary number. Show your working. [2]
1
📊 Set Up Place Values
Write out the 8 column headings. Which is the largest power of 2 that fits into 185?
▶ Click to reveal
128 · 64 · 32 · 16 · 8 · 4 · 2 · 1 185 ≥ 128 ✓ → Write 1 in the 128 column. Subtract: 185 − 128 = 57 remaining.
2
🔢 Work Through Each Column
57 remaining — does it fit in 64?
▶ Click to reveal
57 < 64 → Write 0 in the 64 column. 57 ≥ 32 ✓ → Write 1 in the 32 column. Remaining: 57 − 32 = 25. 25 ≥ 16 ✓ → Write 1 in the 16 column. Remaining: 25 − 16 = 9. 9 ≥ 8 ✓ → Write 1 in the 8 column. Remaining: 9 − 8 = 1.
3
🏁 Final Columns
1 remaining — what goes in the 4, 2, and 1 columns?
▶ Click to reveal
1 < 4 → Write 0 in the 4 column. 1 < 2 → Write 0 in the 2 column. 1 = 1 ✓ → Write 1 in the 1 column. Remaining: 0. ✓
4
✅ Final Answer
Read the bits from left to right — what is the 8-bit binary result?
▶ Click to reveal
128 64 32 16 8 4 2 1 1 0 1 1 1 0 0 1 185 in 8-bit binary = 10111001 ✓ Verification: 128 + 32 + 16 + 8 + 1 = 185 ✓
✍️
Marks strategy: Write the place value row AND the bit row in your answer. The marker needs to see your method. If you write "10111001" alone with no working, you risk losing a method mark if the answer is wrong.
05 · Exam-Style Questions

Cambridge-Style Practice

Write your answers in the lines below, then reveal the marking scheme.

Question 1
2 marks
Convert the denary number 47 into an 8-bit binary number. Show your working.
✓Place value row: 128 · 64 · 32 · 16 · 8 · 4 · 2 · 1 shown or implied[1]
✓Correct binary: 00101111 — accept with or without leading zeros if 6-bit version shown (101111)[1]
ℹWorking: 47 − 32 = 15 · 15 − 8 = 7 · 7 − 4 = 3 · 3 − 2 = 1 · 1 − 1 = 0
Question 2
2 marks
A student states: "The largest number that can be stored in 8 bits is 256." State whether this is correct and justify your answer.
✓The statement is incorrect[1]
✓The largest value is 255 — because 8 bits can represent 2⁸ = 256 different values, from 0 to 255 (not 1 to 256)[1]
Question 3
3 marks
A file stores the colour value 173 for the red channel of a pixel. Convert this value to 8-bit binary using the division by 2 method. Show all steps of your working.
✓Correct division working shown (at least 4 steps of divide-by-2 with remainders recorded)[1]
✓Remainders correctly identified as 1,0,1,1,0,1,0,1 (from first to last division)[1]
✓Final answer 10101101 (read bottom to top) — must be 8 bits[1]
ℹ173÷2=86r1 · 86÷2=43r0 · 43÷2=21r1 · 21÷2=10r1 · 10÷2=5r0 · 5÷2=2r1 · 2÷2=1r0 · 1÷2=0r1 → read up: 10101101
Question 4
1 mark
State the term used to describe a group of 8 bits.
✓Byte[1]
06 · Live Scored Quiz

5-Question Challenge

Answer all 5 questions to earn your XP badge. Instant feedback on every question.

Score:
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🏅
Topic 1 Complete — Binary Converter Unlocked!
+50 XP · Chapter 1 · Data Representation