Binary, Denary, and Hexadecimal Conversions

This lesson brings together all three number systems — binary, denary, and hexadecimal — and provides a comprehensive guide to converting between them. Fluency in these conversions is essential for the GCSE Computer Science exam.

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Conversion Summary Table

From	To	Method
Binary → Denary	Add up the place values of all 1-bits	Use the column headings: 128, 64, 32, 16, 8, 4, 2, 1
Denary → Binary	Subtract place values or divide by 2	Use place value method or successive division
Binary → Hex	Split into nibbles (4-bit groups)	Convert each nibble to its hex digit
Hex → Binary	Replace each hex digit with 4 bits	Use the hex-to-binary lookup table
Hex → Denary	Multiply digits by powers of 16	First digit × 16 + second digit × 1
Denary → Hex	Divide by 16	Quotient = first digit, remainder = second digit

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Conversion Routes Between Bases

flowchart LR
    D[Denary] -->|place value or div by 2| B[Binary]
    B -->|sum place values| D
    B -->|nibble groups| H[Hexadecimal]
    H -->|each digit -> 4 bits| B
    D -->|divide by 16| H
    H -->|digit x 16 + digit| D

Worked Examples: Full Conversion Practice

Example 1: Convert denary 156 to binary and hex

Denary to Binary (place value method):

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

• 156 ≥ 128? Yes → 1, remainder 28
• 28 ≥ 64? No → 0
• 28 ≥ 32? No → 0
• 28 ≥ 16? Yes → 1, remainder 12
• 12 ≥ 8? Yes → 1, remainder 4
• 4 ≥ 4? Yes → 1, remainder 0
• 0 ≥ 2? No → 0
• 0 ≥ 1? No → 0

156 in binary = 10011100

Binary to Hex:

• Split: 1001 | 1100
• 1001 = 9
• 1100 = C

156 in hex = 9C

Check (Hex to Denary): 9 × 16 + 12 × 1 = 144 + 12 = 156. Correct!

Example 2: Convert hex E7 to binary and denary

Hex to Binary:

• E = 1110
• 7 = 0111

E7 in binary = 11100111

Hex to Denary:

• E (14) × 16 = 224
• 7 × 1 = 7
• 224 + 7 = 231

Check (Binary to Denary): 128 + 64 + 32 + 4 + 2 + 1 = 231. Correct!

Example 3: Convert binary 01011010 to denary and hex

Binary to Denary:

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

• 64 + 16 + 8 + 2 = 90

Binary to Hex:

• Split: 0101 | 1010
• 0101 = 5
• 1010 = A

01011010 in binary = 5A in hex

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Common Mistakes to Avoid

1. Misaligning place values: Always start from the right when assigning binary place values (1, 2, 4, 8, 16...). A common error is writing them left to right.

2. Forgetting to pad with leading zeros: When converting binary to hex, if the leftmost group has fewer than 4 bits, add leading zeros. For example, 110101 should be split as 0011 | 0101, not 1101 | 01.

3. Confusing hex letters with denary values: Remember A=10, B=11, C=12, D=13, E=14, F=15. Students sometimes write A=1 by mistake.

4. Losing the carry in binary addition: When you have 1+1+1 in a column, the result is 1 with a carry of 1. Do not forget to add the carry to the next column.

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Practice Conversions

Try these conversions yourself, then check your answers:

Denary	Binary	Hexadecimal
45	00101101	2D
100	01100100	64
170	10101010	AA
199	11000111	C7
240	11110000	F0
255	11111111	FF
128	10000000	80
63	00111111	3F

Exam Tip: In the exam, always show your working. Even if you get the final answer wrong, you can pick up method marks by showing clear working. Write out the place values, show each step, and circle or underline your final answer.

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Maximum Values

The maximum value that can be represented depends on the number of bits:

Bits	Maximum Unsigned Value	In Hex
4 bits (nibble)	15	F
8 bits (byte)	255	FF
16 bits	65,535	FFFF

The formula for the maximum value of n bits is: 2ⁿ – 1

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Summary

• You must be confident converting between binary, denary, and hexadecimal in any direction.
• Use the place value method or successive division for denary-to-binary conversions.
• Group binary into 4-bit nibbles for binary-to-hex conversions.
• Multiply hex digits by powers of 16 for hex-to-denary conversions.
• Always show your working in exam answers and double-check by converting back.
• The maximum value of an n-bit unsigned integer is 2ⁿ – 1.

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Comprehensive Triple Conversion Walkthroughs

Walkthrough 1: Start with hex 7B

Hex → Binary: 7 = 0111, B = 1011, so 7B = 01111011.

Binary → Denary:

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

64 + 32 + 16 + 8 + 2 + 1 = 123.

Cross-check Hex → Denary directly: 7 × 16 + 11 = 112 + 11 = 123. The two routes agree.

Walkthrough 2: Start with denary 199