OCTAL

The 'octal' numeral system, or 'oct' for short, is the base-8 number system, and uses the digits 0 to 7.
Octal numerals can be made from binary numerals by grouping consecutive digits into groups of three (starting from the right). For example, the binary representation for decimal 74 is 1001010, which groups into 001 001 010 — so the octal representation is 112. In decimal systems each decimal place is a base of 10 for example, '74' = '7' x 10^1 + '4' x 10^0. since '7' x 10^1 = '70' + '4' x 10^0= 4, 70+4 = '74'. In octal numerals each place is a power with base 8 for example '112'= '1' x 8^2 + '1' x 8^1 + '2' x 8^0 since '1' x 8^2 = 64 + '1' x 8^1 = 8 + '2' x 8^0 = 2,
112 in octal is equal to 64+8+2 = 74. This is why 112 in octal is equal to 74 in decimal.
Octal is sometimes used in computing instead of hexadecimal.

Contents

Usage

By Native Americans

In Europe

In fiction

In computers

Conversion between bases

Decimal – Octal Conversion

Method of consecutive divisions by 8

Method of consecutive multiplications by 8

Octal – Decimal Conversion

Octal – Binary Conversion

Binary – Octal Conversion

Octal – Hexadecimal Conversion

Hexadecimal – Octal Conversion

See also

External links

References

Usage

By Native Americans

The Yuki language in California and the Pamean languages in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves.
^[1]

In Europe

In 1716 King Charles XII of Sweden asked Emanuel Swedenborg to elaborate a number system based on 64 instead of 10. Swedenborg however argued that for people with less intelligence than the king such a big base would be too difficult and instead proposed 8 as base. In 1718 Swedenborg wrote a manuscript, which has not been published: "En ny rekenkonst som om vexlas wid Thalet 8 i stelle then wanliga wid Thalet 10" ("A new arithmetic (or art of counting) which changes at the Number 8 instead of the usual at the Number 10"). The numbers 1-7 are there denoted by the consonants l, s, n, m, t, f, u (v) and zero by the vowel o. Thus 8 = "lo", 16 = "so", 24 = "no", 64 = "loo", 512 = "looo" etc. Numbers with consecutive consonants are pronounced with vowel sounds between in accordance with a special rule.^[2]

In fiction

The fictional alien felinoid species Kilrathi of the ''Wing Commander'' universe, count in octal, since their paws have 4 toes instead of 5.
The Octospider species of ''Rendezvous with Rama'' and the computer game ''RAMA'' use a colour code based on octal system, and its comprehension is a puzzle of the game scenario.
The Alterans from Stargate SG-1 used octal, even though they had ten fingers. It's possible that they counted the gaps between each finger, ignored the thumb on each hand, or used the thumb as a base-2 counter (as on an abacus) allowing them to count up to 30 (24 in decimal) on their hands.
The satirist Tom Lehrer famously remarked in his song parodying new math that "base 8 is just like base 10... if you're missing two fingers."
In the first-person shooter ''Prey'', numerical codes to open doors are entered in octal.
The Tau race in the Warhammer 40,000 universe use the octal system.
In The Beekeeper's Apprentice, Laurie R. King's first Sherlock Holmes pastiche featuring Mary Russell, base eight math played a key role in solving the mystery.

In computers

Octal is sometimes used in computing instead of hexadecimal, perhaps most often in modern times in conjunction with file permissions under Unix systems (see chmod). It has the advantage of not requiring any extra symbols as digits (the hexadecimal system is base-16 and therefore needs six additional symbols beyond 0–9). It is also used for digital displays.
At the time when octal originally became widely used in computing, systems such as the IBM mainframes employed 24-bit (or 36-bit) words. Octal was an ideal abbreviation of binary for these machines because eight (or twelve) digits could concisely display an entire machine word (each octal digit covering three binary digits).
All modern computing platforms, however, use 16-, 32-, or 64-bit words, with eight bits making up a byte. On such systems three octal digits would be required, with the most significant octal digit inelegantly representing only two binary digits (and in a series the same octal digit would represent one binary digit from the next byte). Hence hexadecimal is more commonly used in programming languages today, since a hexadecimal digit covers four binary digits and all modern computing platforms have machine words that are evenly divisible by four.
The prefix customarily used to represent an octal number is zero (e.g. '0'73).

Conversion between bases

''For more information and other bases, see Conversion between bases.''

Decimal – Octal Conversion

Method of consecutive divisions by 8

Is used to convert integer decimals to octals and consists in dividing the original number by the largest possible factor of 8 and successively dividing the remainders by successively smaller factors of 8 until the factor is 0. The octal number is formed by the quotients, written in the order of its obtention.
for example
convert the following to octal
125
solution:-
125/8^2='1'
125-((8^2)
★ 1)=61
61/8^1='7'
61-((8^1)
★ 7)='5'
125(base10)=175(base8)
900/8^3='1'
900-((8^3)
★ 1)=388
388/8^2='6'
388-((8^2)
★ 6)=4
4/8^1='0'
4-((8^1)
★ 0)=4
4/8^0='4'
900(base10)=1604(base8)

Method of consecutive multiplications by 8

Is used to convert a decimal fraction to octal. The decimal fraction is multiplied by 8, and the integer part of the result is the first digit of the octal fraction. The process is repeated with the fractionary part of the result, until it is null or within acceptable error parameter.
Example:
Convert 0.140625 to octal:
0.140625 x 8 = 1.125

0.125 x 8 = 1.0
Previous methods can be combined to convert decimal numbers with integer and fractionary parts.

Octal – Decimal Conversion

Use the formula:
:$sum\_\{i=0\}^n\; left(\; a\_i\; imes\; B^i$
ight).
Example:
Convert octal 764 to decimal system.
764 (base 8) = 7 x 8² + 6 x 8¹ + 4 x 8° = 448 + 48 + 4 = 500 (base 10)
A simpler way for double digit numbers is to take the first number, multiply it by 8 and then add the second to get the total.
Example: 65 in octal would be 53 in decimal (6
★ 8 + 5 = 53)

Octal – Binary Conversion

To convert octals to binaries, replace each digit of octal number by its binary correspondent.
Example:
Convert octal 1572 to binary.

1 5 7 2 = 001 101 111 010

Binary – Octal Conversion

The process is the reverse of previous algorithm. The binary digits are grouped 3 by 3, from the decimal point to the left and to the right. Then each trio is substituted by the equivalent octal digit.
For instance, conversion of binary 1010111100 to octal:

001	010	111	100
1	2	7	4

Thus 1010111100₂ = 1274₈

Octal – Hexadecimal Conversion

The conversion is made in two steps using binary as an auxiliary base. Octal is converted to binary and then to hexadecimal, grouping digits 4 by 4, which correspond each to an hexadecimal digit.
For instance, convert octal 1057 to hexadecimal:
:To binary:
:

1	0	5	7
001	000	101	111

:To hexadecimal:
:

0010	0010	1111
2	2	F

Thus 1057₈ = 22F₁₆

Hexadecimal – Octal Conversion

Reverse the previous algorithm.

See also

★ Computer numbering formats

External links

★ Online Converter for Decimal/Octal Numerals (JavaScript, GPL)

★ The typology of Pame number systems and the limits of Mesoamerica as a linguistic area

★ octomatics is a project trying to invent a numeral system in octal.

References

1. Ethnomathematics: A Multicultural View of Mathematical Ideas Marcia Ascher
2. Donald Knuth, ''The Art of Computer Programming''