Spec & Goals 3 min
AQA Spec 3.3.1 · Number bases
By the end of this lesson you can:
- State the base and digit set of denary, binary and hexadecimal.
- Explain why computers use binary and why people use hexadecimal.
- State that 4 bits represent 16 different values and map them to one hex digit.
Warm-Up 5 min
You count in tens every day without thinking about it. A computer cannot — it only knows two states.
Quick starter
In the everyday number 4096, what is the value of the digit 4?
Reveal the answer
It is worth 4000 — the place values are 1000, 100, 10, 1, so the 4 sits in the thousands column. Each column is ten times the one to its right. That is what "base 10" means.
Key Concept — three number bases 14 min
A base is how many different digits a number system uses. The base also sets the value of each column.
Denary — base 10
Denary uses ten digits, 0–9. Column values are powers of ten: 1000, 100, 10, 1. This is the system we use every day.
Binary — base 2
Binary uses just two digits, 0 and 1. Column values are powers of two: 8, 4, 2, 1. Each binary digit is called a bit.
Hexadecimal — base 16
Hexadecimal uses sixteen digits: 0–9, then A, B, C, D, E, F for ten to fifteen. Column values are powers of sixteen: 256, 16, 1.
Why people use hexadecimal
Long binary numbers are hard for people to read and easy to copy wrongly. Hex is a short way to write binary.
You will meet hex in colour codes (e.g. #FF0000 for red), MAC addresses and memory addresses.
Place values for each base
| Base | Column values (right to left) |
|---|---|
| Denary (10) | … 1000, 100, 10, 1 |
| Binary (2) | … 16, 8, 4, 2, 1 |
| Hexadecimal (16) | … 256, 16, 1 |
Worked Example — counting 0 to 15 12 min
Task: count from 0 to 15 in all three bases. Notice how 4 bits of binary map to exactly one hex digit.
| 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 |
What to notice: 4 bits run from 0000 to 1111 — that is exactly 16 different patterns (0 to 15). Each pattern has one matching hex digit.
Try It Yourself 12 min
Goal: State the base and the digit set of denary, binary and hexadecimal.
Hint: base = how many different digits; list them for each.
Goal: Using the 0–15 table, write the hex digit for the binary patterns 1010, 1100 and 1111.
Hint: read each 4-bit row straight off the worked-example table.
Goal: Explain, in two sentences, why a programmer might write a colour as #FF0000 rather than as 24 binary digits.
Hint: think about reading and copying long binary by hand.
📝 Exam Practice 10 min
Answer the way the examiner expects — the command word and the marks tell you how much to write.
State the base of the hexadecimal number system.
Mark scheme
- 16 / base 16 (1).
Give one reason why hexadecimal is used instead of binary.
Mark scheme
- It is shorter / easier for people to read / less likely to be copied wrongly / one hex digit represents 4 bits (1). Accept any one.
State how many different values 4 bits can represent.
Mark scheme
- 16 (1).
Which row lists a valid digit for each base?
- A — Binary: 2 · Denary: 9 · Hex: F
- B — Binary: 1 · Denary: 9 · Hex: G
- C — Binary: 1 · Denary: 9 · Hex: F
- D — Binary: 0 · Denary: A · Hex: F
Mark scheme
- C (1) — binary digits are 0–1, denary 0–9, hex 0–9 then A–F. (A uses binary 2; B uses hex G; D uses denary A — all invalid.)
Recap & Key Terms 3 min
The base is how many digits a system uses. Denary is base 10, binary is base 2, hex is base 16. Computers use binary because components have two states; people use hex as a short way to write binary, with one hex digit per 4 bits.
- Denary
- The base-10 system using digits 0–9; the everyday number system.
- Binary
- The base-2 system using only the digits 0 and 1; each digit is a bit.
- Hexadecimal
- The base-16 system using digits 0–9 then A–F (A=10 … F=15); a short way to write binary.
- Base
- The number of different digits in a number system; it sets the value of each column.
Homework 1 min
Task (≤ 15 min): Without copying the lesson table, write the binary (4-bit) and hex values for denary 0, 5, 10 and 15.
Model answer
| Denary | Binary (4-bit) | Hex |
|---|---|---|
| 0 | 0000 | 0 |
| 5 | 0101 | 5 |
| 10 | 1010 | A |
| 15 | 1111 | F |
Award marks for: correct binary patterns (1) and correct hex digits, including A and F (1).