Editing Octal (section)

{{Short description|Base-8 numeral system}}
{{Numeral systems for computation}}
{{Use dmy dates|date=August 2019|cs1-dates=y}}
{{Table Numeral Systems}}

'''Octal''' ('''base 8''') is a [[numeral system]] with [[8 (number)|eight]] as the [[radix|base]].

In the decimal system, each place is a [[power of ten]]. For example:
: <math>\mathbf{74}_{10} = \mathbf{7} \times 10^1 + \mathbf{4} \times 10^0</math>
In the octal system, each place is a power of eight. For example:
: <math>\mathbf{112}_8 = \mathbf{1} \times  8^2 + \mathbf{1} \times  8^1 + \mathbf{2} \times  8^0 </math>
By performing the calculation above in the familiar decimal system, we see why 112 in octal is equal to <math>64+8+2=74</math> in decimal.

Octal numerals can be easily converted from [[Binary numeral system|binary]] representations (similar to a [[quaternary numeral system]]) by grouping consecutive binary digits into groups of three (starting from the right, for integers). For example, the binary representation for decimal 74 is 1001010.  Two zeroes can be added at the left: {{nowrap|(00)1 001 010}}, corresponding to the octal digits {{nowrap|1 1 2}}, yielding the octal representation 112.

{| class="wikitable" style="float:right; text-align:center"
|+ The octal [[multiplication table]]
|-
| × || '''1''' || '''2''' || '''3''' || '''4''' || '''5''' || '''6''' || '''7''' || '''10''' 
|-
| '''1''' || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 10
|-
| '''2''' || 2 || 4 || 6 || 10 || 12 || 14 || 16 || 20
|-
| '''3''' || 3 || 6 || 11 || 14 || 17 || 22 || 25 || 30
|-
| '''4''' || 4 || 10 || 14 || 20 || 24 || 30 || 34 || 40
|-
| '''5''' || 5 || 12 || 17 || 24 || 31 || 36 || 43 || 50
|-
| '''6''' || 6 || 14 || 22 || 30 || 36 || 44 || 52 || 60
|-
| '''7''' || 7 || 16 || 25 || 34 || 43 || 52 || 61 || 70
|-
| '''10''' || 10 || 20 || 30 || 40 || 50 || 60 || 70 || 100
|}