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Binary & Denary Conversion

Lesson CS1 of 10 Approx 50 min No prerequisites

Learning intentions

  • Describe what binary is and why computers use it
  • Convert between 8-bit binary and denary (base 10) numbers
  • Understand place values in binary (27, 26, …, 20)

Success criteria

  • You can convert a denary number to 8-bit binary
  • You can convert an 8-bit binary number to denary
  • You can explain why computers use binary (switches: ON/OFF = 1/0)
Warm up — what do you already know?

Answer before the lesson begins. These check prior knowledge — it's fine if you're unsure.

1. What does "8-bit" mean?

2. Why do computers use binary?

Key vocabulary

Bit
A single binary digit (0 or 1). The smallest unit of data a computer can process.
Byte
8 bits grouped together. A standard unit for storing and transferring data.
Binary
A number system using base 2 (only digits 0 and 1). The native language of computers.
Denary
Base 10 number system (digits 0–9). The decimal system we use in everyday life.
Place value
The value assigned to a digit based on its position. In binary: 27, 26, 25, etc.
Convert
Change from one number system to another (e.g., denary to binary).
Exponent
The power to which a number is raised (e.g., 23 = 8). Used in place values like 27.
Power of 2
A number multiplied by itself a specific number of times (e.g., 23 = 2 × 2 × 2 = 8).

How computers store numbers

What is binary?

Computers are made up of billions of tiny electronic switches called transistors. Each switch can only be in one of two states: ON (powered) or OFF (not powered). To represent data, computers use this simple two-state system by assigning values: ON = 1, OFF = 0. This is the basis of binary.

Binary is a number system with only two digits: 0 and 1. Unlike denary (base 10), which uses digits 0–9, binary is base 2. Every number a computer stores, every image it displays, every word in a document—all are ultimately represented as long strings of 0s and 1s.

Binary place values

In denary, each position (or "place") represents a power of 10. For example, in the number 352: the 3 is in the hundreds place (3 × 102), the 5 is in the tens place (5 × 101), and the 2 is in the ones place (2 × 100).

Binary works the same way, but each position represents a power of 2 instead of 10. For an 8-bit binary number, the place values from left to right are: 128, 64, 32, 16, 8, 4, 2, 1 (which are 27 through 20).

How to convert denary to binary

To convert a denary number to 8-bit binary, use the place-value method:

  1. Write down the 8 place values: 128, 64, 32, 16, 8, 4, 2, 1
  2. Starting from the left (128), ask: "Does this place value go into my number?"
  3. If yes, write 1 underneath and subtract that value from your number
  4. If no, write 0 underneath and move to the next place value
  5. Continue until you've processed all 8 places

This method is reliable and much faster than repeated division by 2.

How to convert binary to denary

To convert from binary back to denary, reverse the process:

  1. Write the 8-bit binary number
  2. Write the place values (128, 64, 32, 16, 8, 4, 2, 1) underneath each bit
  3. For every 1 in the binary number, note down that place value
  4. Add all the noted place values together
  5. The sum is your denary number

Worked examples

Example 1 — Convert 105 to 8-bit binary
Converting denary 105 to binary 01101001 using place values Denary to convert 105 128 64 32 16 8 4 2 1 0 1 1 0 1 0 0 1 64 32 8 1 64 + 32 + 8 + 1 = 105 ✓ Binary answer 01101001
Subtraction trace: 105 − 64 = 41  →  41 − 32 = 9  →  9 − 8 = 1  →  1 − 1 = 0
Example 2 — Convert 11010101 to denary
Converting binary 11010101 to denary 213 by adding place values Binary to convert 11010101 128 64 32 16 8 4 2 1 1 1 0 1 0 1 0 1 128 64 16 4 1 128 + 64 + 16 + 4 + 1 = 213 ✓ Denary answer 213
Reading the bits left → right: 1×128 + 1×64 + 0×32 + 1×16 + 0×8 + 1×4 + 0×2 + 1×1 = 213
Example 3 — Convert 200 to 8-bit binary
Converting denary 200 to binary 11001000 using place values Denary to convert 200 128 64 32 16 8 4 2 1 1 1 0 0 1 0 0 0 128 64 8 128 + 64 + 8 = 200 ✓ Binary answer 11001000
Subtraction trace: 200 − 128 = 72  →  72 − 64 = 8  →  8 − 8 = 0
Now you try

Convert 175 to 8-bit binary. Show your working using the place-value method.

Model answer:

128 into 175? YES → 1, remaining 47

64 into 47? NO → 0

32 into 47? YES → 1, remaining 15

16 into 15? NO → 0

8 into 15? YES → 1, remaining 7

4 into 7? YES → 1, remaining 3

2 into 3? YES → 1, remaining 1

1 into 1? YES → 1

Answer: 10101111

Check: 128 + 32 + 8 + 4 + 2 + 1 = 175 ✓

Common mistakes
✕
Counting 9 places instead of 8. Always use exactly 8 bits. Stick to: 128, 64, 32, 16, 8, 4, 2, 1.
✕
Using powers of 10 instead of powers of 2. Binary uses base 2, not base 10. Double-check: 23 = 8, NOT 1000.
✕
Forgetting leading zeros. Write 8 digits. The number 5 is 00000101, NOT just 101.
✕
Adding wrong place values when converting from binary. Write them all out and double-check: 01101001 = 64 + 32 + 8 + 1 = 105 ✓
Exam tip

Based on 2025 N5 exam report: Most candidates converted denary to binary correctly who used the place-value method.

  1. Use place values every time: Write out 128, 64, 32, 16, 8, 4, 2, 1 at the top of every conversion question.
  2. Check your work: After converting, add up your place values to verify you get the original denary number.

Example answer: "Convert 90 to 8-bit binary. 90 = 64 + 16 + 8 + 2, so the binary is 01011010."

Task Set

Questions 1–5 are auto-checked. Questions 6–8 are self-marked — write your answer, then reveal the model answer.

1. Convert 50 to 8-bit binary. TYPE 1

2. Convert 11111111 to denary. TYPE 1

3. Convert 00000000 to denary. TYPE 1

4. What is the place value of position 2 from the right in 8-bit binary? TYPE 1

5. Convert 127 to 8-bit binary. TYPE 1

6. Explain why computers use binary instead of denary. TYPE 2

Model answer: Computers are built from transistors, which are tiny switches that can only be ON (1) or OFF (0). Binary matches this two-state nature perfectly, making it the natural and most efficient way for computers to store and process information.

7. Convert 10101010 to denary and show your working. TYPE 2

Model answer: Place values where there's a 1: 128 + 32 + 8 + 2 = 170

8. A byte can store 0 to 255. Why is 255 the maximum? TYPE 3

Model answer: A byte has 8 bits. The maximum is when all 8 bits are 1: 128+64+32+16+8+4+2+1 = 255. You cannot go higher with 8 bits—you'd need more bits to store larger numbers.

Teacher notes — Shift+T to hide

Suggested timing: 50 minutes. Warm-up (5), notes (25), examples (15), tasks (30, overflow OK).

Key points: Emphasise transistor ON/OFF → binary. Build place values systematically. Repeat "128, 64, 32, 16, 8, 4, 2, 1" until automatic.

Common misconception: Pupils forget leading zeros. Reinforce: "8 bits = 8 digits always."

Extension: Why powers of 2 (not 10)? Why 8, 16, 32, 64-bit? What's the largest number in 16-bit binary?