Hexadecimal number system
Hexadecimal is the other main number system used in computing. It works in tandem with the binary number system and provides an easier and more accessible means of working with long sequences of binary numbers.
Hexadecimal place value
Unlike denary which uses base ten and binary which uses base two, hexadecimal uses base 16 as its place value.
Each place in a hexadecimal number represents a power of 16 and each place can be one of 16 symbols.
Hexadecimal values
The table below shows the symbols comprising hexadecimal alongside their denary and binary equivalents:
| Hexadecimal | Decimal | 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 |
This table shows the raw value of each hexadecimal place value:
| \(16^{3}\) | \(16^{2}\) | \(16^{1}\) | \(16^{0}\) |
|---|---|---|---|
| 4096 | 256 | 16 | 1 |
Hexadecimal prefix
The custom is to prefix a hexadecimal number with 0x to indicate that the number is hexadecimal.
Converting hexadecimal numbers
Using the previous table we can convert hexadecimal values to decimal.
For example we can convert 1A5 as follows, working from right to left:
\((5 \cdot 1 = 5) + (A \cdot 16 = 160) + (1 \cdot 256 = 256) = 421\)
The process is quite easy: we get the n from \(16^{n}\) based on the position of the digit and then multiply this by the value of the symbol (1,2,…F):
$$ 16^{n} \cdot 1,2,...F $$
As applied to 0x1A5:
| \(16^{3}\) | \(16^{2}\) | \(16^{1}\) | \(16^{0}\) |
|---|---|---|---|
| \(1\cdot 16^{3} = 4096\) | \(1\cdot 16^{2} = 256\) | \(A (10)\cdot 16^{1} = 160\) | \(5\cdot 16^{0} = 5\) |
Another example for 0xF00F:
\((15 \cdot 4096 = 61440) + (0 \cdot 256 = 0) + (0 \cdot 16 = 0) + (15 \cdot 1 = 15) = 61455\)
Using hexadecimal to simplify binary numbers
Whilst computers themselves do not use the hexadecimal number system (everything is binary), hexadecimal offers advantages for humans who must work with binary:
- It is much easier to read a hexadecimal number than long sequences of binary numbers
- It is easier to quickly convert binary numbers to hexadecimal than to convert binary numbers to decimal
Look at the following equivalences
| Number system | Example 1 | Example 2 |
|---|---|---|
| Binary | 1111 0000 0000 1111 | 1000 1000 1000 0001 |
| Hexadecimal | F00F | 8881 |
| Decimal | 61,455 | 34,945 |
It is obvious that a pattern is maintained between the hexadecimal and binary numbers and that this pattern is obscured by the decimal conversion. In the first example the binary half-byte 1111 is matched by the hexadecimal F00F.
Mathematically comparing hex F and binary 1111:
$$ \textsf{1111} = ((1 \cdot 2^{3}) + (1 \cdot 2^{2}) + (1 \cdot 2^{1}) + (1 \cdot 2^{0})) \\ = 8 + 4 + 2 + 1 \\ = 15 $$
$$ \textsf{F} = 15 \cdot 16^{0} \\ = 15 $$
Every four bits (or half byte) in binary corresponds to one symbol in hexadecimal. Therefore a byte can be easily represented with two hexadecimal symbols, a 16-bit number can be represented with four hex symbols, a 32-bit number can represented with eight hex symbols and so on.
