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Encyclopedia > Octal
Numeral systems by culture
Hindu-Arabic numerals
Western Arabic
Eastern Arabic
Khmer
Indian family
Brahmi
Thai
East Asian numerals
Chinese
Japanese
Korean
 
Alphabetic numerals
Abjad
Armenian
Cyrillic
Ge'ez
Hebrew
Ionian/Greek
Sanskrit
 
Other systems
Attic
Etruscan
Urnfield
Roman
Babylonian
Egyptian
Mayan
List of numeral system topics
Positional systems by base
Decimal (10)
2, 4, 8, 16, 32, 64
3, 9, 12, 24, 30, 36, 60, more…
v  d  e

The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. A numeral is a symbol or group of symbols, or a word in a natural language that represents a number. ... The Hindu-Arabic numeral system (also called Algorism) is a positional decimal numeral system documented from the 9th century. ... Numerals sans-serif Arabic numerals, known formally as Hindu-Arabic numerals, and also as Indian numerals, Hindu numerals, Western Arabic numerals, European numerals, or Western numerals, are the most common symbolic representation of numbers around the world. ... The Eastern Arabic numerals (also called Eastern Arabic numerals, Arabic-Indic numerals, Arabic Eastern Numerals) are the symbols (glyphs) used to represent the Hindu-Arabic numeral system in conjunction with the Arabic alphabet in Egypt, Iran, Pakistan and parts of India, and also in the no longer used Ottoman Turkish... Khmer numerals are the numerals used in the Khmer language of Cambodia. ... India has produced many numeral systems. ... The Brahmi numerals are an indigenous Indian numeral system attested from the 3rd century BCE (somewhat later in the case of most of the tens). ... The Abjad numerals are a decimal numeral system which was used in the Arabic-speaking world prior to the use of the Hindu-Arabic numerals from the 8th century, and in parallel with the latter until Modern times. ... Cyrillic numerals was a numbering system derived from the Cyrillic alphabet, used by South and East Slavic peoples. ... Note: This article contains special characters. ... The system of Hebrew numerals is a quasi-decimal alphabetic numeral system using the letters of the Hebrew alphabet. ... Greek numerals are a system of representing numbers using letters of the Greek alphabet. ... The Sanskrit alphabetic numerals were created in about A.D. 510 by Āryabhaa. ... Attic numerals were used by ancient Greeks, possibly from the 7th century BC. They were also known as Herodianic numerals because they were first described in a 2nd century manuscript by Herodianus. ... The Etruscan numerals were used by the ancient Etruscans. ... During the beginning of the Urnfield culture, around 1200 BC, a series of votive sickles of bronze with marks that have been interpreted as a numeral system, appeared in Central Europe. ... Roman numerals are a numeral system originating in ancient Rome, adapted from Etruscan numerals. ... Babylonian numerals by sarah nixon this number system was discovered by a 12 year old girl named sarah nixon were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a... Mayan numerals. ... This is a list of numeral system topics, by Wikipedia page. ... Positional notation is a system in which each position has a value represented by a unique symbol or character. ... The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ... The decimal (base ten or occasionally denary) numeral system has ten as its base. ... The binary numeral system (base 2 numerals) represents numeric values using two symbols, typically 0 and 1. ... Quaternary is the base four numeral system. ... In mathematics and computer science, base-16, hexadecimal, or simply hex, is a numeral system with a radix or base of 16, usually written using the symbols 0–9 and A–F or a–f. ... Base32 is a derivation of Base64 with the following additional properties: The resulting character set is all uppercase, which can often be beneficial when using a case-sensitive filesystem. ... Base 64 is a positional numeral system using a base of 64. ... Ternary or trinary is the base-3 numeral system. ... Nonary is a base 9 numeral system, typically using the digits 0-8, but not the digit 9. ... The duodecimal (also known as base-12 or dozenal) system is a numeral system using twelve as its base. ... As there are 24 hours in a day a numbering system based upon 24, and as the base 12 is convenient here some examples of the base 24 (quadrovigesimal) system. ... Base 30 or trigesimal is a positional numeral system using 30 as the radix. ... Base 36 refers to a positional numeral system using 36 as the radix. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... A numeral is a symbol or group of symbols, or a word in a natural language that represents a number. ... The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ...


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. The binary numeral system (base 2 numerals) represents numeric values using two symbols, typically 0 and 1. ...


Octal is sometimes used in computing instead of hexadecimal. Originally, the word computing was synonymous with counting and calculating, and a science and technology that deals with the original sense of computing mathematical calculations. ... In mathematics and computer science, base-16, hexadecimal, or simply hex, is a numeral system with a radix or base of 16, usually written using the symbols 0–9 and A–F or a–f. ...

Contents

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. [[Image:YukiTribe. ... Official language(s) English Capital Sacramento Largest city Los Angeles Area  Ranked 3rd  - Total 158,302 sq mi (410,000 km²)  - Width 250 miles (400 km)  - Length 770 miles (1,240 km)  - % water 4. ...


In Europe

Donald Knuth wrote in his book The Art of Computer Programming that King Charles XII of Sweden was the inventor of octal in Europe. In 1716 the king 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 1-10 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. Donald Knuth at a reception for the Open Content Alliance. ... Cover of books The Art of Computer Programming[1] is a comprehensive monograph written by Donald Knuth which covers many kinds of programming algorithms and their analysis. ... Carl XII, Karl XII or Carolus Rex, (June 17, 1682 – November 30, 1718), the Alexander of the North, nicknamed in Turkish as Demirbaş Şarl (Charles the Habitué), was a King of Sweden from 1697 until his death in 1718. ... Emanuel Swedenborg, 75, holding the manuscript of Apocalypsis Revelata (1766). ...


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 Kilrathi are a race of warlike, felinoid aliens in the popular computer game series Wing Commander by Origin Systems. ... It has been suggested that this article or section be merged with Wing Commander game series. ...


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. Rendezvous with Rama is a novel by Arthur C. Clarke first published in 1972. ... RAMA is a first-person adventure game developed and published by Sierra Entertainment in 1996. ...


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. To meet Wikipedias quality standards, this article or section may require cleanup. ... Stargate SG-1 (often abbreviated as SG-1) is a science fiction television series part of the Stargate franchise. ... It has been suggested that Abacus system be merged into this article or section. ...


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." Tom Lehrer in 1960. ... New math is a term referring to a brief dramatic change in the way mathematics was taught in American grade schools during the 1960s. ...


In the first-person shooter Prey, numerical codes to open doors are entered in octal. This article or section does not cite its references or sources. ...


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. In mathematics and computer science, base-16, hexadecimal, or simply hex, is a numeral system with a radix or base of 16, usually written using the symbols 0–9 and A–F or a–f. ... Filiation of Unix and Unix-like systems Unix (officially trademarked as UNIX®) is a computer operating system originally developed in the 1960s and 1970s by a group of AT&T employees at Bell Labs including Ken Thompson, Dennis Ritchie and Douglas McIlroy. ... The chmod command (abbreviated from change mode) is a shell command in Unix and Unix-like environments. ...


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). SAS 8 on an IBM mainframe under 3270 emulation An IBM mainframe is a mainframe computer made by IBM. // From 1952 into the late 1960s, IBM manufactured and marketed several large computer models, known as the IBM 700/7000 series. ...


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 0 (i.e. 073).


Conversion between bases

For more information and other bases, see Conversion between bases. In mathematics, the base or radix is the number of various unique symbols (digits), including zero, that a positional numeral system uses to represent numbers in a given counting system. ...


Decimal – Octal Conversion

Method of consecutive divisions by 8

Is used to convert integer decimals to octals and consists in dividing successively the original number and resulting quotients by 8, until quotient is 0. The octal number is formed by the remainders, written in the reverse order of its obtention.


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:

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)


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

The term computer numbering formats refers to the schemes implemented in digital computer and calculator hardware and software to represent numbers. ...

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.

  Results from FactBites:
 
What is octal? - A Word Definition From the Webopedia Computer Dictionary (154 words)
Programs often display data in octal format because it is relatively easy for humans to read and can easily be translated into binary format, which is the most important format for computers.
By contrast, decimal format is the easiest format for humans to read because it is the one we use in everyday life, but translating between decimal and binary formats is relatively difficult.
For example, the octal number 3456 is 011 100 101 110 in binary.
Octal Number System (533 words)
The typical method to convert from decimal to octal is repeated division by 8.
This process is continued by dividing he quotient by 8 and writing the remainder until the quotient is 0.
When performing the division, the remainders which will represent the octal equivalent of the decimal number are written beginning at the least significant digit (right) and each new digit is written to the next more significant digit (the left) of the previous digit.
  More results at FactBites »

 
 

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