122 lines
4 KiB
Markdown
122 lines
4 KiB
Markdown
---
|
|
tags:
|
|
- number-systems
|
|
- computer-architecture
|
|
---
|
|
|
|
# Hexadecimal number system
|
|
|
|
Hexadecimal is the other main number system used in computing. It works in
|
|
tandem with the [binary number system](Binary_number_system.md) 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:
|
|
|
|
1. It is much easier to read a hexadecimal number than long sequences of binary
|
|
numbers
|
|
2. 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.**
|