Hexadecimal, or simply "hex", is a base-16 numbering system that uses 16 digits, 0-9 and A-F. The purpose of using hex is to represent large numbers in a compact and easy-to-read format. Each digit in a hex number represents a power of 16, starting with 16^0, which is equal to 1. For example, the hex number "1F" represents the decimal number 31, which is calculated as (1 x 16^1) + (15 x 16^0) = 16 + 15 = 31.

Octal, on the other hand, is a base-8 numbering system that uses 8 digits, 0-7. It was commonly used in early computing systems because it is easy to convert between octal and binary, which is the native language of computers. Each digit in an octal number represents a power of 8, starting with 8^0, which is equal to 1. For example, the octal number "34" represents the decimal number 28, which is calculated as (3 x 8^1) + (4 x 8^0) = 24 + 4 = 28.

In programming, hexadecimal is often used to represent colors, memory addresses, and other low-level data. It is also commonly used to encode and decode binary data. Octal is less commonly used in modern programming, but it still appears in some legacy systems and older programming languages.

To convert a number from one base to another, you can use built-in functions in most programming languages. For example, in PHP, you can use the `hexdec()`

function to convert a hex number to decimal, or the `decoct()`

function to convert a decimal number to octal. Conversely, you can use the `decbin()`

function to convert a decimal number to binary, and the `bindec()`

function to convert binary to decimal.

Sure! Here are some examples of how to use hexadecimal and octal numbers in PHP:

```
<?php
// Hexadecimal example
$hex = '1F'; // Hex number representing decimal 31
$dec = hexdec($hex); // Convert hex to decimal
echo $dec; // Outputs: 31
// Octal example
$oct = '34'; // Octal number representing decimal 28
$dec = octdec($oct); // Convert octal to decimal
echo $dec; // Outputs: 28
?>
```

In this example, the `hexdec()`

function is used to convert the hex number "1F" to decimal, which is stored in the `$dec`

variable. The resulting decimal number is then printed to the screen using the `echo`

statement.

Similarly, the `octdec()`

function is used to convert the octal number "34" to decimal, which is also stored in the `$dec`

variable. The resulting decimal number is then printed to the screen using the `echo`

statement.

Note that both `hexdec()`

and `octdec()`

functions take a string as an argument, and return the equivalent decimal number.