Last Sync: 2022-10-04 11:30:21

Hardware/Binary/Hexadecimal_number_system.md

@@ -3,5 +3,70 @@ title: Hexadecimal number system
 categories:
   - Computer Architecture
   - Mathematics
-tags: []
+tags: [number-systems]
 ---
+
+# Hexadecimal number system
+
+Hexadecimal is the other main number system used in computing. It works in tandem with the [binary number system](/Hardware/Binary/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        |
+
+## 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 `1A5`:
+
+| $16^{2}$              | $16^{1}$                   | $16^{0}$            |
+| --------------------- | -------------------------- | ------------------- |
+| $1\cdot 16^{2} = 256$ | $A (10)\cdot 16^{1} = 160$ | $5\cdot 16^{0} = 5$ |
+
+Another example for `F00F`:
+
+$(15 \cdot 4096 = 61440) + (0 \cdot 256 = 0) + (0 \cdot 16 = 0) + (15 \cdot 1 = 15) = 61455$
+
+// TODO: Relation to binary and bytes