> For the complete documentation index, see [llms.txt](https://ret2basic.gitbook.io/ctfnote/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://ret2basic.gitbook.io/ctfnote/computer-science/computer-systems/hexadecimal.md).

# Hexadecimal

## Motivation: Easy Convertion

Computers work with binary numbers, but humans don't. We humans are used to decimal numbers, but the "binary-decimal" is not a natural idea. A better option is using hexadecimal numbers, or simply **hex**. Why hex? Take a look at the following convertions:

| Decimal | Binary    | Hex |
| ------- | --------- | --- |
| 0       | 0000      | 0   |
| 1       | 0001      | 1   |
| 2       | 0010      | 2   |
| 3       | 0011      | 3   |
| 4       | 0100      | 4   |
| 5       | 0101      | 5   |
| 6       | 0110      | 6   |
| 7       | 0111      | 7   |
| 8       | 1000      | 8   |
| 9       | 1001      | 9   |
| 10      | 1010      | A   |
| 11      | 1011      | B   |
| 12      | 1100      | C   |
| 13      | 1101      | D   |
| 14      | 1110      | E   |
| 15      | 1111      | F   |
| 16      | 0001 0000 | 10  |
| 17      | 0001 0001 | 11  |
| 55      | 0011 0111 | 37  |
| 195     | 1100 0011 | C3  |

**Key Ideas:**

* **1 hex digit <=> 4 bits.**
* For a long binary string, we can **pad 0's on the left** to some multiples of 4 and divide the binary string into chunks of length 4. For example, `10001` can be padded as `00010001` and divided into two chunks `0001 0001`.
* **1 byte <=> 2 hex digits.** The convertion is easy enough, even for humans.

## Binary => Decimal

For example, convert `0b10110001` to decimal. From right to left:

* 1 => 1 \* 2^0 = 1
* 0 => 0 \* 2^1 = 0 (skipped)
* 0 => 0 \* 2^2 = 0 (skipped)
* 0 => 0 \* 2^3 = 0 (skipped)
* 1 => 1 \* 2^4 = 16
* 1 => 1 \* 2^5 = 32
* 0 => 0 \* 2^6 = 0 (skipped)
* 1 => 1 \* 2^7 = 128

Therefore `0b10110001` = 1 + 16 + 32 + 128 = 177.

## Decimal => Binary

For example, convert 26 to binary:

* 26 / 2 = 13 ...... 0
* 13 / 2 = 6 ...... 1
* 6 / 2 = 3 ...... 0
* 3 / 2 = 1 ...... 1
* 1 / 2 = 0 ...... 1 (algorithm terminates)

Therefore 26 = `0b11010`.

## Hex => Decimal

For example, convert `0x125` to decimal. From right to left:

* 5 => 5 \* 16^0 = 5
* 2 => 2 \* 16^1 = 32
* 1 => 1 \* 16^2 = 256

Therefore `0x125` = 5 + 32 + 256 = 293.

## Decimal => Hex

For example, convert 293 to hex:

* 293 / 16 = 18 ...... 5
* 18 / 16 = 1 ...... 2
* 1 / 16 = 0 ...... 1

Therefore 293 = `0x125`.