72 lines
2.4 KiB
Markdown
72 lines
2.4 KiB
Markdown
---
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title: Hexadecimal number system
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categories:
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- Computer Architecture
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- Mathematics
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tags: [number-systems]
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---
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# Hexadecimal number system
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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.
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## Hexadecimal place value
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Unlike denary which uses base ten and binary which uses base two, hexadecimal uses base 16 as its place value.
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> Each place in a hexadecimal number represents a power of 16 and each place can be one of 16 symbols.
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## Hexadecimal values
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The table below shows the symbols comprising hexadecimal alongside their denary and binary equivalents:
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| Hexadecimal | Decimal | Binary |
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| ----------- | ------- | ------ |
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| 0 | 0 | 0000 |
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| 1 | 1 | 0001 |
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| 2 | 2 | 0010 |
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| 3 | 3 | 0011 |
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| 4 | 4 | 0100 |
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| 5 | 5 | 0101 |
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| 6 | 6 | 0110 |
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| 7 | 7 | 0111 |
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| 8 | 8 | 1000 |
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| 9 | 9 | 1001 |
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| A | 10 | 1010 |
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| B | 11 | 1011 |
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| C | 12 | 1100 |
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| D | 13 | 1101 |
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| E | 14 | 1110 |
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| F | 15 | 1111 |
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This table shows the raw value of each hexadecimal place value:
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| $16^{3}$ | $16^{2}$ | $16^{1}$ | $16^{0}$ |
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| -------- | -------- | -------- | -------- |
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| 4096 | 256 | 16 | 1 |
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## Converting hexadecimal numbers
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Using the previous table we can convert hexadecimal values to decimal.
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For example we can convert `1A5` as follows, working from right to left:
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$(5 \cdot 1 = 5) + (A \cdot 16 = 160) + (1 \cdot 256 = 256) = 421$
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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):
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$$
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16^{n} \cdot 1,2,...F
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$$
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As applied to `1A5`:
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| $16^{2}$ | $16^{1}$ | $16^{0}$ |
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| --------------------- | -------------------------- | ------------------- |
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| $1\cdot 16^{2} = 256$ | $A (10)\cdot 16^{1} = 160$ | $5\cdot 16^{0} = 5$ |
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Another example for `F00F`:
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$(15 \cdot 4096 = 61440) + (0 \cdot 256 = 0) + (0 \cdot 16 = 0) + (15 \cdot 1 = 15) = 61455$
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// TODO: Relation to binary and bytes
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