Hexadecimal Representation

Hexadecimal (hex) is a base-16 number system. Where binary uses two digits (0–1) and denary uses ten (0–9), hexadecimal uses sixteen: 0–9 followed by A–F to represent the values ten through fifteen in a single character.

Computers store everything in binary, but long binary strings — for example, a 32-bit memory address — are difficult to read and prone to transcription error. Hexadecimal solves this by providing a compact notation: every 4 binary bits map exactly to a single hex digit, making the conversion lossless.

Where hex appears	Example
Web colour codes	#FF5733
Memory addresses	0x1A2F
MAC addresses	A8:5E:45:6B:3C:71
Error codes	0x000000ED

Hex is used in computing not because it is intuitive, but because 4 bits = 1 hex digit. Binary-to-hex conversion is exact, with no rounding or information loss.

The sixteen hex digits and their denary equivalents must be memorised. Digits 0–9 behave identically to denary; A–F replace the two-digit denary values 10–15 with a single character.

Hex	Denary	4-bit 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
A	10	1010
B	11	1011
C	12	1100
D	13	1101
E	14	1110
F	15	1111

The most common error here is treating A as 11 rather than 10. A is the first letter after the digits, and 10 is the first value after 9. Each subsequent letter adds one: B = 11, C = 12, D = 13, E = 14, F = 15.

Each hex digit represents a power of 16, starting from $16^0=1$ at the rightmost position. Multiply each digit by its place value, substituting the denary equivalent for any letter, then sum the results.

For a 2-digit hex number:

Position	Place value
Left digit	$16^1=16$
Right digit	$16^0=1$

Worked example 1 — convert 2A to denary:

• 2 × 16 = 32
• A (= 10) × 1 = 10
• Total: 32 + 10 = 42

Worked example 2 — convert B7 to denary:

• B (= 11) × 16 = 176
• 7 × 1 = 7
• Total: 176 + 7 = 183

Substitute the denary value for each letter before multiplying. Multiplying "A" directly is a common source of arithmetic errors.

Use the division-by-16 method: divide the number by 16 repeatedly, recording the remainder after each step. Read the remainders bottom to top, replacing any remainder of 10–15 with its letter.

Worked example 1 — convert 174 to hex:

Division	Quotient	Remainder	Hex digit
174 ÷ 16	10	14	E
10 ÷ 16	0	10	A

Reading bottom to top: AE

Check: (10 × 16) + 14 = 160 + 14 = 174 ✓

Worked example 2 — convert 255 to hex:

Division	Quotient	Remainder	Hex digit
255 ÷ 16	15	15	F
15 ÷ 16	0	15	F

Result: FF — the maximum value of an 8-bit byte.

Check: (15 × 16) + 15 = 240 + 15 = 255 ✓

Four binary bits correspond exactly to one hex digit — this is the reason hex is used in computing. The nibble-split method exploits this relationship directly.

Method: starting from the right, divide the binary number into groups of 4 bits (nibbles). Pad the leftmost group with leading zeros if it has fewer than 4 bits. Convert each nibble to its hex digit using the place values 8, 4, 2, 1.

Worked example 1 — convert 10110101 to hex:

Split from right: 1011 | 0101

Nibble	Bit values	Denary	Hex
Left	1×8 + 0×4 + 1×2 + 1×1	11	B
Right	0×8 + 1×4 + 0×2 + 1×1	5	5

Result: B5

Check: (11 × 16) + 5 = 176 + 5 = 181; 10110101 = 128 + 32 + 16 + 4 + 1 = 181 ✓

Worked example 2 — convert 00001111 to hex:

Split: 0000 | 1111 → 0 and 15 → 0F

Hex-to-binary is the reverse of the nibble-split method: replace each hex digit with its 4-bit nibble. Always write all 4 bits, even when the value is small.

Worked example 1 — convert 3C to binary:

Hex digit	Denary	4-bit nibble
3	3	0011
C	12	1100

Result: 00111100

Check: 32 + 16 + 8 + 4 = 60; (3 × 16) + 12 = 48 + 12 = 60 ✓

Worked example 2 — convert 7F to binary:

• 7 → 0111
• F (= 15) → 1111
• Result: 01111111 = 64 + 32 + 16 + 8 + 4 + 2 + 1 = 127; (7 × 16) + 15 = 112 + 15 = 127 ✓

Never write 3 bits for a nibble. The digit 3 in hex is 0011, not 011. Dropping the leading zero shifts every nibble left and produces a completely wrong result.

1. Off-by-one on A–F

A = 10, not 11. The letters start where the single digits end: after 9 comes A, and A = 10. Each subsequent letter adds one — B = 11, C = 12, D = 13, E = 14, F = 15.

2. Not padding nibbles to 4 bits

When converting binary to hex, 101 is not a valid nibble — write it as 0101. Unpadded nibbles cause all subsequent hex digits to be wrong.

3. Reading division remainders top to bottom

In the division-by-16 method, the first remainder you write becomes the rightmost hex digit. Remainders are read bottom to top, not top to bottom.

4. Writing two denary digits instead of a letter

174 in hex is AE, not 1014. Once a remainder reaches 10 or above, convert it to a letter before writing the result.

5. Using denary place values for hex

Hex place values are 1, 16, 256 — not 1, 10, 100. Mixing these up on multi-digit conversions produces a subtly wrong answer that is hard to spot without a check step.

Always verify your answer by converting back: if hex AE gives denary 174, confirm (10 × 16) + 14 = 174 before moving on.