Octal

The octal numeral system is a base 8 numeral system. It uses the numerals 0 through 7. The system is similar to binary (base 2) and hexadecimal (base 16). Octal numerals are written using the letter o before the numeral, for example, o04 or o1242. Octal numbers are also sometimes written with a small 8 to the lower right, as in 1242₈.

At one time, the octal system was used mainly for work with computers. It provided an easier way to work with binary numbers. As computers changed from using 24-bit systems to 32- and 64-bit systems, hexadecimal replaced octal for most uses. Certain groups, for example, Native Americans using the Yuki language in California and the Pamean languages^[1] in Mexico, also use an octal numbering system. They do this because when they count, they use the spaces between their fingers instead of counting the actual fingers.

Octal and binary

The octal numeral system uses a "three-bit" binary coding. Each digit in an octal numeral is the same as three digits in a binary numeral. The grouping of the binary digits is done from right to left. The first three binary digits from the right are grouped into the last part of the octal numeral, then the next three digits form the next to the last part of the numeral.

Octal	Binary
1	001
2	010
3	011
4	100
5	101
6	110
7	111
10	001 000

Octal	Binary
11	001 001
12	001 010
36	011 110
45	100 101
53	101 011
64	110 100
100	001 000 000
357	011 101 111

Binary	Groupings				Octal
11				011	3
010111			010	111	27
101000110		101	000	110	506
01011010101	001	011	010	101	1325

Octal and decimal

In the decimal system (base 10), each digit in octal is equal to that digit multiplied by the exponent of 8 that is equal to its location minus one.

	Location
	6	5	4	3	2	1
Value	32768 (8⁵)	4096 (8⁴)	512 (8³)	64 (8²)	8(8¹)	1 (8⁰)

Example: o3425 to decimal

Octal		Decimal
o3425	=	( 5 × 1 )	+	( 2 × 8)	+	( 4 × 64 )	+	( 3 × 512)
o3425	=	5	+	16	+	256	+	1536
o3425	=	1813

Octal and hexadecimal

Octal is similar to hexadecimal because they are both easily converted to binary. Where octal is equal to three-digit binary, hexadecimal is equal to four-digit binary. Where octal numerals start with the letter "o", hexadecimal numerals end with the letter "h". The easiest way to convert from one to the other is to convert to binary and then to the other system.

Octal	Binary									Hexadecimal
Octal	three digit					four digit				Hexadecimal
o4					100				0100	04h
o15				001	101				1101	0Dh
o306			011	000	110			1100	0110	C6h
o54253	101	100	010	101	011	0101	1000	1010	1011	58ABh

Octal Media

Fuxi's "Earlier Heaven" Arrangement of the Eight Trigrams

Related pages

References

↑ Avelino, Heriberto (2006), "The typology of Pame number systems and the limits of Mesoamerica as a linguistic area" (PDF), Linguistic Typology, 10: 41–60, doi:10.1515/LINGTY.2006.002, S2CID 20412558

Other websites

Online Converter Archived 2007-01-04 at the Wayback Machine for Decimal/Octal Numerals (JavaScript, GPL)

[1] Avelino, Heriberto (2006), "The typology of Pame number systems and the limits of Mesoamerica as a linguistic area" (PDF), Linguistic Typology, 10: 41–60, doi:10.1515/LINGTY.2006.002, S2CID 20412558

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