🔢 Chapter 1 · Topic 2 · Paper 1 & 2

Binary to Denary
Conversion

Reading the language of machines — decode 0s and 1s back to human numbers

🔥 01 · Did You Know?

When a computer stores your age, your exam score, or the temperature setting on a smart thermostat, it writes those values in binary. But to display them to you on screen, it must convert them back to denary. This happens billions of times per second inside every device you own. Understanding this conversion isn't just exam knowledge — it's how security researchers reverse-engineer malware, how network engineers read IP addresses, and how game developers decode colour values from image files. In 2003, a NASA engineer manually decoded binary telemetry from the Mars Exploration Rover to diagnose a memory fault from 300 million kilometres away. Same skill. Same method. You're learning it today.

02 · Core Concept

How Binary to Denary Works

Each bit in a binary number has a place value — a power of 2 based on its position. To convert to denary, simply multiply each bit by its place value and add all the results together. Only bits that are 1 contribute to the total.

🔑 The rule: Bit × Place Value — if the bit is 1, add that column's value. If 0, add nothing.
2⁷
128
2⁶
64
2⁵
32
2⁴
16
2³
8
2²
4
2¹
2
2⁰
1

Memorise this row. It appears on almost every data representation question in Cambridge exams.

👆
Bit = 1
Add that column's place value to the total
☐
Bit = 0
Add nothing — skip that column
➕
Sum
Add all contributing values for the final answer
✅
Check
Total must be between 0 and 255 for 8-bit numbers

The 3-Step Method

1

Write the place value row above the binary number: 128 · 64 · 32 · 16 · 8 · 4 · 2 · 1

2

Circle or tick every column where the bit is 1. Ignore all columns where the bit is 0.

3

Add together the place values of all circled columns. That sum is your denary answer.

⚠️ Common Exam Mistakes

❌

Starting from the wrong end — the leftmost bit is always 128 (2⁷), not 1. Students sometimes write the place values in reverse order and get a completely wrong answer.

❌

Adding all place values regardless of the bit — only add the value if the bit underneath is 1. A 0 bit contributes absolutely nothing.

❌

Misaligning bits with place values — if given fewer than 8 bits (e.g. 1011), align from the right: that's a 4-bit number where the leftmost bit = 8, not 128.

🏆
Cambridge Exam Tip: Always write the place value row in your answer even if not asked. It earns a method mark if your addition goes wrong. Most binary-to-denary questions are worth 1–2 marks and expect working to be shown.
⚡ 03 · Interactive Tool
Binary → Denary Decoder
// Click bits to toggle · Or type a binary number · Watch the maths unfold
Quick examples:
00101010
01001011
11000100
10110111
11111111
10000001
⚠ Please enter exactly 8 bits using only 0s and 1s.
Or click bits to toggle:
Step-by-step calculation — each bit × its place value:
04 · Worked Example

Converting 10110110 to Denary

Try to work out each step before clicking to reveal — then check your working.

📋 Question: Convert the 8-bit binary number 10110110 to denary. Show your working. [2]
1
📊 Write the Place Value Row
Position the 8 place values above the binary digits. Which bits are 1?
▶ Click to reveal
128 64 32 16 8 4 2 1 1 0 1 1 0 1 1 0 Bits that are 1: positions 128, 32, 16, 4, 2
2
➕ Add the Active Columns
Take only the place values under a 1 bit. What do they add up to?
▶ Click to reveal
128 + 32 + 16 + 4 + 2 = 128 + 32 → 160 = 160 + 16 → 176 = 176 + 4 → 180 = 180 + 2 → 182
3
✅ Final Answer & Verification
State the denary value and verify your answer is in the valid 8-bit range.
▶ Click to reveal
10110110 in denary = 182 ✓ Verification check: — All bits accounted for: ✓ — Result is between 0 and 255: ✓ (182 is valid) — Bits NOT added: 64 (bit=0), 8 (bit=0), 1 (bit=0) ✓
✍️
Show it clearly: Write the place value row, then the binary row directly beneath it. Then list the values you're adding. Examiners award a mark for method even if you make an arithmetic slip on the final sum.
05 · Exam-Style Questions

Cambridge-Style Practice

Attempt each question, then reveal the marking scheme to check your answer.

Question 1
2 marks
Convert the binary number 01101001 to denary. Show your working.
✓Correct place value method shown — identifying bits at 64, 32, 8, 1[1]
✓Correct answer: 105  (64 + 32 + 8 + 1 = 105)[1]
Question 2
2 marks
A sensor records a temperature reading stored as 11010011. Convert this to denary to find the reading in degrees.
✓Method: 128 + 64 + 16 + 2 + 1 identified (bits at positions 128, 64, 16, 2, 1 are set to 1)[1]
✓Correct answer: 211  (128 + 64 + 16 + 2 + 1 = 211)[1]
Question 3
3 marks
A student says: "The binary number 10000000 has the denary value 8."
State whether the student is correct. Give the correct denary value and explain what mistake the student likely made. [3]
✓The student is incorrect[1]
✓The correct value is 128 — the single 1 bit is in the 2⁷ (128) column, not the 2³ (8) column[1]
✓The student likely counted the position of the bit from the right (position 7 + 1 = 8) instead of using the correct place value of 2⁷ = 128[1]
Question 4
1 mark
What is the denary value of the binary number 00000001?
✓1 — only the rightmost bit (2⁰ = 1) is set to 1[1]
06 · Live Scored Quiz

5-Question Challenge

Answer all 5 to earn your Topic 2 XP badge. No going back once you select an answer!

Score:
0 / 5
🥈
Topic 2 Complete — Binary Decoder Unlocked!
+50 XP · Chapter 1 · Data Representation