Hexadecimal: Conversions Between Binary, Denary, and Hex

Hexadecimal (base-16, often shortened to "hex") is the third number system you need to know for OCR J277 Section 2.6. Hexadecimal is widely used in computing because it provides a compact, human-readable representation of binary data.

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Why Hexadecimal?

Binary numbers are long and hard for humans to read. Hexadecimal solves this problem:

Binary (8 bits)	Denary	Hexadecimal
11111111	255	FF
10101010	170	AA
00000000	0	00

Hex is shorter and easier to read than binary, while still mapping directly to binary (each hex digit = exactly 4 bits). This is why hex is used for:

Use	Example
Colour codes	#FF5733 (red, green, blue values)
MAC addresses	4A:2B:7C:DE:01:FF
Memory addresses	0x7FFF0000
Error codes	0x80070005
HTML/CSS colours	color: #00FF00 (green)

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Hexadecimal Digits

Hex uses 16 digits. Since we only have 10 numeral symbols (0-9), the letters A-F represent values 10-15:

Hex digit	Denary value	Binary (4 bits)
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
A	10	1010
B	11	1011
C	12	1100
D	13	1101
E	14	1110
F	15	1111

OCR Exam Tip: You should memorise this table. In the exam, you can quickly write the hex-to-binary lookup in the margin to help with conversions.

flowchart LR
    subgraph Byte [8-bit byte split into two nibbles]
        N1[Upper nibble - 4 bits] --> H1[Hex digit 1]
        N2[Lower nibble - 4 bits] --> H2[Hex digit 2]
    end
    H1 --> Pair[Hex pair e.g. B7]
    H2 --> Pair
    Pair --> Use{Used as}
    Use --> C1[Colour code #B7...]
    Use --> C2[MAC address byte]
    Use --> C3[Memory address]
    Use --> C4[Error code]

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Converting Binary to Hexadecimal

1. Split the binary number into groups of 4 bits from the right
2. Convert each group to a hex digit using the table above

Example: Convert 10110111 to hex

Split: 1011 | 0111

• 1011 = B
• 0111 = 7

Answer: B7

Example: Convert 11111010 to hex

Split: 1111 | 1010

• 1111 = F
• 1010 = A

Answer: FA

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Converting Hexadecimal to Binary

1. Convert each hex digit to its 4-bit binary equivalent
2. Join the groups together

Example: Convert 3D to binary

• 3 = 0011
• D = 1101

Answer: 00111101

Example: Convert A5 to binary

• A = 1010
• 5 = 0101

Answer: 10100101

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Converting Hexadecimal to Denary

Multiply each hex digit by its place value (powers of 16):

Example: Convert 2F to denary

Hex digit	Place value	Calculation
2	16^1 = 16	2 x 16 = 32
F (15)	16^0 = 1	15 x 1 = 15

32 + 15 = 47

Example: Convert B4 to denary

• B (11) x 16 = 176
• 4 x 1 = 4

176 + 4 = 180

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Converting Denary to Hexadecimal

1. Divide the denary number by 16
2. The quotient is the first hex digit; the remainder is the second

Example: Convert 200 to hex

200 / 16 = 12 remainder 8

• 12 = C, 8 = 8

Answer: C8

Check: (12 x 16) + 8 = 192 + 8 = 200. Correct!

Alternative method: Convert to binary first, then to hex:

• 200 = 11001000
• Split: 1100 | 1000
• 1100 = C, 1000 = 8
• Answer: C8

OCR Exam Tip: If you find direct denary-to-hex conversion difficult, convert to binary first, then group into nibbles. This two-step method is perfectly acceptable in the exam.

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Practice

1. Convert hex A3 to binary: 10100011
2. Convert binary 01001110 to hex: 0100 = 4, 1110 = E, answer: 4E
3. Convert hex FF to denary: (15 x 16) + 15 = 255
4. Convert denary 127 to hex: 127 / 16 = 7 remainder 15 = 7F

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Worked Example: Decoding a MAC Address Byte-by-Byte

MAC addresses (media access control addresses) uniquely identify network devices and are written as six pairs of hex digits. Let us decode the first byte of the MAC address 4A:2B:7C:DE:01:FF.

Step 1: Identify the first byte.

The first hex pair is 4A.

Step 2: Convert 4A to binary.

• 4 = 0100
• A (10) = 1010

Concatenated: 01001010.

Step 3: Convert 4A to denary.

• 4 x 16 = 64
• A (10) x 1 = 10
• Total = 74

Step 4: Check via binary-to-denary.

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

64 + 8 + 2 = 74. Agrees with Step 3.

Step 5: Interpret the high-order bits of a MAC address.

Within the first byte, the two lowest bits of the first hex digit carry special meaning for networking. We see 4 = 0100 as the first nibble. Bit 1 (the "locally administered" bit) is 0 and bit 0 (the "multicast" bit) is 0, so this MAC address is a globally unique, unicast address — the standard case for a physical network card.

Step 6: Reconstruct the full binary representation of all six bytes.

Hex pair	Binary (8 bits)
4A	01001010
2B	00101011
7C	01111100
DE	11011110
01	00000001
FF	11111111

Full 48-bit MAC address: 01001010 00101011 01111100 11011110 00000001 11111111.