Spec & Goals 3 min
AQA Spec 3.3.2 · Converting between binary and denary
By the end of this lesson you can:
- Convert an 8-bit binary number to denary using place values.
- Convert a denary number (0–255) to 8-bit binary by subtracting place values.
- Identify the most and least significant bit in an 8-bit number.
Warm-Up 5 min
Last lesson you met three bases. Binary uses column values that are powers of two — that is the key to converting.
Quick starter
Write the eight place values for an 8-bit number, largest first.
Reveal the answer
128 64 32 16 8 4 2 1 — each column is double the one to its right. These add up to 255, the largest 8-bit value.
Key Concept — the place-value method 14 min
An 8-bit number has eight columns. Each column has a fixed place value, doubling from right to left.
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
|---|---|---|---|---|---|---|---|
| — | — | — | — | — | — | — | — |
Binary → denary
Write the bits under the place values. Add the place values where there is a 1 — ignore the columns with a 0.
Denary → binary
Work left to right. Subtract the largest place value that fits and put a 1 there; if a place value is too big, put a 0. Repeat until you reach 0.
Most and least significant bit
The largest 8-bit value is 11111111 = 255; the smallest is 00000000 = 0. So 8 bits store 256 different values (0–255).
Worked Example — both directions 12 min
Binary → denary: convert 01101101
Write each bit under its place value, then add the columns that hold a 1.
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 |
Add the 1-columns: 64 + 32 + 8 + 4 + 1 = 109.
So 01101101 = 109.
Denary → binary: convert 75
Take place values largest first; subtract each one that fits.
| Place value | Fits in remainder? | Bit | Remainder left |
|---|---|---|---|
| 128 | No (128 > 75) | 0 | 75 |
| 64 | Yes (75 − 64) | 1 | 11 |
| 32 | No (32 > 11) | 0 | 11 |
| 16 | No (16 > 11) | 0 | 11 |
| 8 | Yes (11 − 8) | 1 | 3 |
| 4 | No (4 > 3) | 0 | 3 |
| 2 | Yes (3 − 2) | 1 | 1 |
| 1 | Yes (1 − 1) | 1 | 0 |
Reading the Bit column top to bottom: 01001011.
So 75 = 01001011 (check: 64 + 8 + 2 + 1 = 75).
Try It Yourself 12 min
Goal: Convert the 8-bit binary number 00010110 to denary using a place-value table.
Hint: add only the columns that hold a 1.
Goal: Convert the denary number 100 to 8-bit binary by subtracting place values.
Hint: start at 128 — does it fit? Then try 64, and so on.
Goal: For 10110001, state the value of the MSB and the LSB, then convert the whole number to denary.
Hint: MSB is the 128 column; LSB is the 1 column.
📝 Exam Practice 10 min
Answer the way the examiner expects — the command word and the marks tell you how much to write.
Convert the 8-bit binary number 00111010 to denary. Show your working.
Mark scheme
- Correct place values added: 32 + 16 + 8 + 2 (1).
- Answer = 58 (1).
Convert the denary number 200 to 8-bit binary. Show your working.
Mark scheme
- Correct method: 128 + 64 + 8 = 200 (1).
- Answer =
11001000(1).
What is the denary value of the 8-bit binary number 00001111?
- A — 8
- B — 15
- C — 16
- D — 240
Mark scheme
- B (1) — 8 + 4 + 2 + 1 = 15.
Recap & Key Terms 3 min
To go binary → denary, add the place values where there is a 1. To go denary → binary, subtract the largest place values that fit and fill the table. The leftmost bit is the most significant; the rightmost is the least significant.
- Place value
- The value a column is worth; in 8-bit binary the columns are 128, 64, 32, 16, 8, 4, 2, 1.
- Most significant bit (MSB)
- The leftmost bit — the 128 column — worth the most in the number.
- Least significant bit (LSB)
- The rightmost bit — the 1 column — worth the least in the number.
Homework 1 min
Task (≤ 15 min): Convert 11010010 to denary, then convert the denary number 45 back to 8-bit binary. Show your working.
Model answer
11010010 → 128 + 64 + 16 + 2 = 210.
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 |
45 → 32 + 8 + 4 + 1 = 45, so 00101101.
Award marks for: correct denary value 210 (1), correct working for 45 (1), correct 8-bit answer (1).