KS3 Computing — Study Pack

Topic: Binary & Hexadecimal

Year 7–9 | Data Representation | UK National Curriculum

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Overview

Every piece of data inside a computer — text, images, videos, programs — is ultimately stored and processed as binary: sequences of 0s and 1s. Understanding why computers use binary, how to convert between number systems, and how binary arithmetic works is a fundamental skill in computing.

Hexadecimal (base 16) is a shorthand used by professionals because it compresses long binary strings into a much more readable form. It appears everywhere from HTML colour codes (#FF5733) to memory addresses in programming.

By the end of this pack you will be able to:

• Explain why computers use binary
• Convert between denary, binary, and hexadecimal
• Perform binary addition including carrying
• Explain overflow and why it matters

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Section 1: Binary (Base 2)

Why Binary?

Modern computers are built from billions of tiny electronic switches called transistors. Each transistor has exactly two states:

• Off → represented as 0
• On → represented as 1

Because hardware can only reliably represent two states, all data must be encoded in base 2 (binary), using only the digits 0 and 1.

Key Terms

Term	Definition
Bit	A single binary digit — either 0 or 1; the smallest unit of data
Nibble	4 bits grouped together
Byte	8 bits grouped together
Kilobyte (KB)	1,024 bytes
Megabyte (MB)	1,024 KB
Gigabyte (GB)	1,024 MB
Terabyte (TB)	1,024 GB

8-Bit Column Values

In an 8-bit binary number, each position (column) represents a power of 2:

| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
|  2⁷ | 2⁶ | 2⁵ | 2⁴ | 2³| 2²| 2¹| 2⁰|

The leftmost bit is the most significant bit (MSB); the rightmost is the least significant bit (LSB).

The maximum value an 8-bit number can hold: 128+64+32+16+8+4+2+1 = 255

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Converting Denary to Binary

Method: Find the largest column value that fits into the number. Write a 1 in that column. Subtract and repeat with the remainder. Write 0 in any column that does not fit.

Worked Example: Convert 177 to 8-bit binary

Column values:  | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

Step 1: Does 128 fit into 177? YES (177 - 128 = 49)   → write 1
Step 2: Does  64 fit into  49? NO                      → write 0
Step 3: Does  32 fit into  49? YES (49 - 32 = 17)      → write 1
Step 4: Does  16 fit into  17? YES (17 - 16 = 1)       → write 1
Step 5: Does   8 fit into   1? NO                      → write 0
Step 6: Does   4 fit into   1? NO                      → write 0
Step 7: Does   2 fit into   1? NO                      → write 0
Step 8: Does   1 fit into   1? YES (1 - 1 = 0)         → write 1

Result: 1 0 1 1 0 0 0 1

Answer: 177 in 8-bit binary is 10110001

Verification (Binary to Denary check):

128 + 32 + 16 + 1 = 177 ✓

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Converting Binary to Denary

Method: Write out the column values. For each 1 in the binary number, add that column's value.

Example: Convert 01001110 to denary

| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
|   0 |  1 |  0 |  0 | 1 | 1 | 1 | 0 |

Values where digit = 1:  64 + 8 + 4 + 2 = 78

Answer: 01001110 = 78 in denary

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Binary Addition

The four rules of binary addition:

Operation	Result	Notes
0 + 0	0	No carry
0 + 1	1	No carry
1 + 0	1	No carry
1 + 1	10	Write 0, carry 1 to next column
1 + 1 + 1	11	Write 1, carry 1 to next column (three 1s: two inputs plus carry)

Worked Example: Add 01101101 + 00110110

    Carry:   1  1  1  1  0  0  0
             0  1  1  0  1  1  0  1
           + 0  0  1  1  0  1  1  0
           ─────────────────────────
             1  0  1  0  0  0  1  1

Column-by-column (right to left):

Position 1 (1s):    1 + 0 = 1              → write 1, carry 0
Position 2 (2s):    0 + 1 = 1              → write 1, carry 0
Position 3 (4s):    1 + 1 = 10             → write 0, carry 1
Position 4 (8s):    1 + 0 + carry 1 = 10   → write 0, carry 1
Position 5 (16s):   0 + 1 + carry 1 = 10   → write 0, carry 1
Position 6 (32s):   1 + 1 + carry 1 = 11   → write 1, carry 1
Position 7 (64s):   1 + 0 + carry 1 = 10   → write 0, carry 1
Position 8 (128s):  0 + 0 + carry 1 = 1    → write 1, carry 0

Result: 1 0 1 0 0 0 1 1

Answer: 01101101 + 00110110 = 10100011

Denary check: 109 + 54 = 163 → 10100011 = 128+32+2+1 = 163 ✓

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Overflow

Overflow occurs when the result of a binary calculation is too large to be stored in the available number of bits.

Example: Adding two 8-bit numbers where the result exceeds 255:

  11111111  (255)
+ 00000001  (1)
───────────
 100000000  (256) — requires 9 bits!

The 9th bit cannot be stored in an 8-bit register. The computer discards it and stores 00000000 instead — giving the wrong answer of 0. This is a critical error that can cause program crashes or incorrect calculations.

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Section 2: Hexadecimal (Base 16)

What is Hexadecimal?

Hexadecimal uses base 16: sixteen symbols instead of ten.

Because we only have ten digit characters (0–9), letters are borrowed for the remaining six values:

Denary	Hex	Binary
0	0	0000
1	1	0001
2	2	0010
3	3	0011
4	4	0100
5	5	0101
6	6	0110
7	7	0111
8	8	1000
9	9	1001
10	A	1010
11	B	1011
12	C	1100
13	D	1101
14	E	1110
15	F	1111

Why Do We Use Hexadecimal?

1. Shorter than binary: One hex digit represents exactly 4 binary digits (a nibble). So 8-bit binary 11001010 becomes just CA — far easier to read and write.
2. Colour codes: Web colours use 6 hex digits (3 bytes) — e.g. #FF5733 means Red=FF(255), Green=57(87), Blue=33(51).
3. Memory addresses: RAM addresses are written in hex (e.g. 0x1A3F).
4. Error checking: MAC addresses, error codes — all use hex.

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Converting Binary to Hexadecimal (via Nibbles)

Method: Split the binary number into groups of 4 bits (nibbles) from the right. Convert each nibble independently to its hex digit.

Worked Example: Convert `10110100` to hex

Step 1: Split into nibbles from the right:
        1011  |  0100

Step 2: Convert each nibble:
        1011  =  8 + 2 + 1 = 11  =  B
        0100  =  4              =  4

Step 3: Combine the hex digits (left to right):
        B4

Answer: 10110100 in hex is B4

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Converting Hexadecimal to Binary

Method: Replace each hex digit with its 4-bit binary nibble equivalent.

Example: Convert 3F to binary

3 → 0011
F → 1111

Result: 0011 1111

Answer: 3F in binary is 00111111

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Converting Hexadecimal to Denary

Method: Multiply each hex digit by 16 raised to the power of its position (rightmost position = 16⁰ = 1).

Example: Convert B4 to denary

B = 11,  position 1  →  11 × 16¹ = 11 × 16 = 176
4 =  4,  position 0  →   4 × 16⁰ =  4 ×  1 =   4

Total: 176 + 4 = 180

Answer: B4 in hex = 180 in denary

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Converting Denary to Hexadecimal

Method: Divide by 16, recording the remainder. Remainders above 9 become letters (10→A, 11→B, etc.).

Example: Convert 255 to hex

255 ÷ 16 = 15 remainder 15  →  F
 15 ÷ 16 =  0 remainder 15  →  F

Read remainders bottom to top: FF

Answer: 255 in hex is FF (this is why #FFFFFF is white — all colours at maximum)

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Key Vocabulary

Term	Definition
Binary	Base-2 number system using only digits 0 and 1
Denary	Base-10 number system (standard counting system using digits 0–9)
Hexadecimal	Base-16 number system using digits 0–9 and letters A–F
Bit	A single binary digit (0 or 1); smallest unit of digital data
Nibble	4 bits; one hexadecimal digit
Byte	8 bits; the standard unit for storing a single character
MSB	Most Significant Bit — leftmost bit in a binary number (highest value)
LSB	Least Significant Bit — rightmost bit (lowest value)
Carry	A value transferred to the next column during binary addition
Overflow	Error when an arithmetic result exceeds the available number of bits
Transistor	Microscopic electronic switch; the physical basis of binary in hardware
Colour depth	Number of bits used to represent colour; affects file size and quality

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Common Misconceptions

Misconception	Correction
"Binary uses digits 0 through 9"	Binary uses ONLY 0 and 1. That is the whole point — two states match transistor on/off.
"A byte is 4 bits"	A byte is 8 bits. Four bits is a nibble.
"1 + 1 = 2 in binary"	In binary, 1 + 1 = 10 (write 0, carry 1). There is no digit 2 in binary.
"Hexadecimal is harder to convert than binary"	Hex is actually easier to convert to/from binary — just split into nibbles and look up each one.
"Overflow just makes the number negative"	Overflow behaviour depends on the system. In simple systems, the extra bit is lost and the stored value wraps around, giving a completely incorrect result.
"Hex digits A–F are letters, not numbers"	A–F are just symbols representing the values 10–15. They function as numeric digits.

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Diagrams & Worked Examples

8-Bit Column Value Chart

┌─────┬────┬────┬────┬───┬───┬───┬───┐
│ 128 │ 64 │ 32 │ 16 │ 8 │ 4 │ 2 │ 1 │
├─────┼────┼────┼────┼───┼───┼───┼───┤
│  2⁷ │ 2⁶ │ 2⁵ │ 2⁴ │2³ │2² │2¹ │2⁰ │
└─────┴────┴────┴────┴───┴───┴───┴───┘

Binary Addition Carry Diagram

   Carry row:  → 1  1  1  1  0  0  0  ←
               0  1  1  0  1  1  0  1   (109)
             + 0  0  1  1  0  1  1  0   ( 54)
             ─────────────────────────
               1  0  1  0  0  0  1  1   (163)

Nibble-to-Hex Conversion Chart

Binary Nibble → Hex
──────────────────
0000 → 0      1000 → 8
0001 → 1      1001 → 9
0010 → 2      1010 → A
0011 → 3      1011 → B
0100 → 4      1100 → C
0101 → 5      1101 → D
0110 → 6      1110 → E
0111 → 7      1111 → F

Colour Code Example

HTML colour: #FF5733

FF → 1111 1111 → Red   = 255 (maximum)
57 → 0101 0111 → Green =  87
33 → 0011 0011 → Blue  =  51

Result: A warm orange-red colour

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Exam-Style Questions

Q1 [1 mark]

Convert the denary number 45 into 8-bit binary.

Q2 [2 marks]

Perform the following binary addition. Show your working including any carry values.

  01001110
+ 00101011
──────────

Q3 [3 marks]

Convert the binary number 11010110 into hexadecimal. Show all working, including the nibble split.

Q4 [2 marks]

A computer uses 8-bit binary numbers. Explain what overflow is and give an example of when it might occur, including the incorrect result stored.

Q5 [4 marks]

Explain why computers use binary to represent data. Include reference to transistors in your answer. Then explain why hexadecimal is commonly used by programmers instead of binary.

Multiple Choice Question

Which of the following correctly represents the hexadecimal digit D?

• A) 12 in denary
• B) 13 in denary and 1101 in binary
• C) 14 in denary and 1110 in binary
• D) 11 in denary

(Answer: B)

Fill in the Blank

Complete the sentence: "A __________ is a group of 4 bits, and it is directly equivalent to exactly one __________ digit."

(Answers: nibble; hexadecimal)

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Model Answers

Q1 Model Answer

Column values: | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

45 - 32 = 13  → 1 in 32 column
13 - 8  =  5  → 1 in  8 column
 5 - 4  =  1  → 1 in  4 column
 1 - 1  =  0  → 1 in  1 column

Answer: 0 0 1 0 1 1 0 1  →  00101101

Q2 Model Answer

  Carry:  0  1  0  0  0  0
          0  1  0  0  1  1  1  0   (78)
        + 0  0  1  0  1  0  1  1   (43)
        ─────────────────────────
          0  1  1  1  1  0  0  1   (121)

Answer: 01111001
Check: 78 + 43 = 121 ✓

Q3 Model Answer

11010110  → split into nibbles → 1101 | 0110

1101 = 8 + 4 + 1 = 13 = D
0110 = 4 + 2     =  6 = 6

Answer: D6

Q4 Model Answer

Overflow occurs when the result of a binary arithmetic operation is too large to be stored in the available number of bits. For example, if an 8-bit system tries to store 255 + 1:

  11111111  (255)
+ 00000001  (  1)
───────────
 100000000  (256) — 9 bits required

The 9th bit is discarded, leaving 00000000 stored, which incorrectly represents 0 instead of 256.

Q5 Model Answer

Computers use binary because they are built from transistors — electronic switches that have exactly two states: on or off. These two states are represented as 1 (on) and 0 (off). Because there are only two reliable states, all data must be encoded using just two digits, making binary the natural number system for computer hardware.

Programmers use hexadecimal because it is much more compact than binary — each hex digit represents exactly 4 binary bits (a nibble). This means an 8-bit binary number like 10110100 can be written as just B4 in hex, making it easier to read, write, and remember. Hex is used in colour codes, memory addresses, and error codes for this reason.

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Revision Checklist

Use this checklist to track your understanding before an assessment:

• I can explain why computers use binary (transistors, two states)
• I know the difference between a bit, nibble, and byte
• I can recall all 8 column values for an 8-bit binary number (1, 2, 4, 8, 16, 32, 64, 128)
• I can convert a denary number (0–255) to 8-bit binary using the subtraction method
• I can convert an 8-bit binary number to denary by summing column values
• I can apply all four binary addition rules including the carry
• I can perform 8-bit binary addition showing carry values
• I can explain what overflow is and give a numerical example
• I know all 16 hexadecimal digits (0–9, A–F) and their denary/binary equivalents
• I can explain at least two reasons why hexadecimal is used (compact, colour codes, addresses)
• I can convert binary to hex by splitting into nibbles
• I can convert hex to binary by expanding each digit to 4 bits
• I can convert a two-digit hex number to denary using place values (16¹, 16⁰)
• I can identify and correct common misconceptions about binary and hex