Spec & Goals 3 min
AQA Spec 3.3.2 · Hexadecimal
By the end of this lesson you can:
- State why hexadecimal is used and what the digits A–F mean.
- Convert between binary and hexadecimal by splitting into nibbles.
- Convert between denary and hexadecimal, via binary or by dividing by 16.
Warm-Up 5 min
Last lesson you converted between binary and denary. Hexadecimal is just a shorter way to write the same binary.
Quick starter
Using a place-value table, convert the 4-bit binary number 1011 to denary.
Reveal the answer
The columns are 8 4 2 1. So 1011 = 8 + 0 + 2 + 1 = 11 denary. Hold on to that 11 — in hexadecimal it becomes a single digit.
Key Concept — base 16 and nibbles 14 min
Denary is base 10 (digits 0–9). Binary is base 2 (digits 0–1). Hexadecimal is base 16, so it needs sixteen digits.
Programmers like hexadecimal because it is shorter than binary and easier to read. A long binary value like 11010110 becomes just two hex digits, D6. Fewer digits means fewer copying mistakes.
The nibble — the key idea
A nibble is a group of 4 bits. With 4 bits you can make 16 patterns (0000 to 1111), which is exactly 0 to 15 — the same range as one hex digit.
The conversion table
Learn this table. Every nibble is one hex digit and one denary value.
| Denary | Binary (4-bit) | Hex |
|---|---|---|
| 0 | 0000 | 0 |
| 1 | 0001 | 1 |
| 2 | 0010 | 2 |
| 3 | 0011 | 3 |
| 4 | 0100 | 4 |
| 5 | 0101 | 5 |
| 6 | 0110 | 6 |
| 7 | 0111 | 7 |
| 8 | 1000 | 8 |
| 9 | 1001 | 9 |
| 10 | 1010 | A |
| 11 | 1011 | B |
| 12 | 1100 | C |
| 13 | 1101 | D |
| 14 | 1110 | E |
| 15 | 1111 | F |
Worked Example — three conversions 12 min
1. Binary to hex: 11010110
Step 1 — split into nibbles (from the right): 1101 0110.
Step 2 — convert each nibble using the place values 8 4 2 1:
| Nibble | 8 4 2 1 | Denary | Hex digit |
|---|---|---|---|
1101 | 8 + 4 + 0 + 1 | 13 | D |
0110 | 0 + 4 + 2 + 0 | 6 | 6 |
Answer: 11010110 = D6 in hexadecimal.
2. Hex to denary: 2C
Step 1 — each hex digit becomes a nibble: 2 = 0010, C = 1100. So 2C = 0010 1100.
Step 2 — read the 8-bit binary in a place-value table:
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 |
Step 3 — add the on values: 32 + 8 + 4 = 44.
Answer: 2C = 44 in denary.
3. Denary to hex: 200
Method — divide by 16. 200 ÷ 16 = 12 remainder 8.
| Step | Working | Result |
|---|---|---|
| 200 DIV 16 | whole part | 12 → hex digit C |
| 200 MOD 16 | remainder | 8 → hex digit 8 |
The first digit (12 = C) is the most significant. Answer: 200 = C8 in hexadecimal.
Check: C8 = 1100 1000 = 128 + 64 + 8 = 200. Correct.
Try It Yourself 12 min
Goal: Convert the binary number 1010 to a single hexadecimal digit.
Hint: use the place values 8 4 2 1, find the denary value, then read it off the conversion table.
Goal: Convert the binary number 10010111 to hexadecimal.
Hint: split into two nibbles 1001 and 0111, then convert each.
Goal: Convert the denary number 175 to hexadecimal by dividing by 16.
Hint: 175 DIV 16 gives the first digit; 175 MOD 16 gives the second.
📝 Exam Practice 10 min
Answer the way the examiner expects — the command word and the marks tell you how much to write.
Convert the binary number 10111110 to hexadecimal. Show your working.
Mark scheme
- Split into nibbles
1011and1110→ 11 and 14 (1). - Answer BE (1).
Convert the hexadecimal number 4F to denary. Show your working.
Mark scheme
4F=0100 1111in binary (1).- 64 + 8 + 4 + 2 + 1 = 79 (1).
How many binary bits does one hexadecimal digit represent?
A) 2 B) 4 C) 8 D) 16
Mark scheme
- B — 4 bits, one nibble (1).
Convert the denary number 140 to hexadecimal. Show your working.
Mark scheme
- 140 DIV 16 = 8, 140 MOD 16 = 12 (1).
- 12 = C, so answer 8C (1).
Recap & Key Terms 3 min
Hexadecimal is base 16, using 0–9 then A–F for 10–15. To convert binary ↔ hex, split into nibbles of 4 bits — each nibble is one hex digit. For denary ↔ hex, go via binary or divide by 16.
- Hexadecimal
- A base-16 number system using digits 0–9 and A–F, where A–F represent 10–15.
- Nibble
- A group of 4 bits; half a byte. One nibble maps to exactly one hexadecimal digit.
Homework 1 min
Task (≤ 15 min): Convert the binary number 11100101 to hexadecimal, and convert the hexadecimal number 3A to denary. Show your working for both.
Model answer
11100101 → split 1110 0101 → 14 and 5 → E5.
3A → 0011 1010 → 32 + 16 + 8 + 2 = 58 denary.
Award marks for: correct nibble split (1), correct hex answer E5 (1), correct denary answer 58 with working (1).