Hexadecimal

Hexadecimal (often shortened to hex) is a base-16 number system. It is widely used in computing as a more human-friendly way of representing binary data. This lesson explains how hexadecimal works and why it is useful.

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What Is Hexadecimal?

Hexadecimal uses 16 symbols to represent values:

Denary	Hex	Binary (4-bit)
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

Because hex needs 16 symbols but we only have 10 numerical digits (0–9), the letters A to F are used to represent the values 10 to 15.

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Why Is Hexadecimal Used in Computing?

Hexadecimal is used because:

1. It is a shorthand for binary. Each hex digit maps exactly to 4 binary bits (a nibble). This makes conversion between hex and binary very quick and straightforward.

2. It is more compact. An 8-bit binary number like 11111111 can be written as just FF in hex. This reduces the chance of making errors when reading or writing long binary strings.

3. It is easier for humans to read and remember. Hex values are much shorter than their binary equivalents, making them more practical for programmers and engineers.

Common Uses of Hexadecimal

• Colour codes in web design: Colours are represented as 6-digit hex values, e.g. #FF5733 (red = FF, green = 57, blue = 33).
• MAC addresses: Network hardware addresses use hex, e.g. 00:1A:2B:3C:4D:5E.
• Memory addresses: Locations in computer memory are typically displayed in hex.
• Error codes: Many system error messages use hexadecimal values.
• Assembly language and machine code: Instructions and data are often shown in hex.

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Hex Conversion Process

flowchart TD
    A[Hex digit] --> B[Lookup 4-bit nibble]
    B --> C[Concatenate nibbles]
    C --> D[8-bit binary number]
    A --> E[digit x 16 + next digit]
    E --> F[Denary value]
    D --> F

Converting Between Hexadecimal and Binary

Hex to Binary

Each hex digit converts directly to a 4-bit binary nibble. Simply replace each hex digit with its 4-bit binary equivalent.

Worked Example: Convert 3F to binary.

• 3 = 0011
• F = 1111

3F in hex = 00111111 in binary

Worked Example: Convert A7 to binary.

• A = 1010
• 7 = 0111

A7 in hex = 10100111 in binary

Binary to Hex

Split the binary number into groups of 4 bits from the right, then convert each group to its hex equivalent.

Worked Example: Convert 11010110 to hex.

• Split into nibbles: 1101 | 0110
• 1101 = D
• 0110 = 6

11010110 in binary = D6 in hex

Worked Example: Convert 101111001 to hex (9 bits).

• Pad with leading zeros to make groups of 4: 0001 | 0111 | 1001
• 0001 = 1
• 0111 = 7
• 1001 = 9

101111001 in binary = 179 in hex

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Converting Between Hexadecimal and Denary

Hex to Denary

Each hex digit has a place value based on powers of 16:

16¹ (16s)	16⁰ (1s)
First digit	Second digit

Multiply each hex digit by its place value and add the results.

Worked Example: Convert 2E to denary.

• 2 × 16 = 32
• E (14) × 1 = 14
• 32 + 14 = 46

Worked Example: Convert B4 to denary.

• B (11) × 16 = 176
• 4 × 1 = 4
• 176 + 4 = 180

Denary to Hex

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

Worked Example: Convert 200 to hex.

• 200 ÷ 16 = 12 remainder 8
• 12 = C, 8 = 8

200 in denary = C8 in hex

Worked Example: Convert 255 to hex.

• 255 ÷ 16 = 15 remainder 15
• 15 = F, 15 = F

255 in denary = FF in hex

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Summary

• Hexadecimal is a base-16 number system using digits 0–9 and letters A–F.
• Each hex digit corresponds exactly to 4 binary bits (a nibble).
• Hex is used because it is a compact, human-readable shorthand for binary.
• Common uses include colour codes, MAC addresses, memory addresses, and error codes.
• To convert hex to binary, replace each hex digit with its 4-bit binary equivalent.
• To convert binary to hex, group the bits into nibbles (from the right) and convert each group.
• To convert hex to denary, multiply each digit by its power of 16 and sum the results.
• To convert denary to hex, divide by 16 — the quotient and remainder give the hex digits.

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Extended Worked Examples

Worked Example: Convert hex 4D to binary, then to denary

Hex to Binary

• 4 = 0100
• D = 1101

So 4D in hex = 01001101 in binary.

Binary to Denary

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

64 + 8 + 4 + 1 = 77.

Cross-check via direct hex-to-denary: 4 × 16 + 13 × 1 = 64 + 13 = 77. The two routes agree.

Worked Example: Convert denary 220 to hex

220 ÷ 16 = 13 remainder 12.

• Quotient 13 = D
• Remainder 12 = C

So 220 in denary = DC in hex.

Verification via binary: 220 = 128 + 64 + 16 + 8 + 4 = 11011100. Splitting into nibbles gives 1101 | 1100 = D | C = DC. The two methods agree.