Information about Octal

Numeral systems by culture
Hindu-Arabic numerals
Western Arabic Eastern Arabic Khmer	Indian family Brahmi Thai
East Asian numerals
Chinese Chinese counting rods	Korean Japanese
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,
•  • [ e]

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.

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 (i.e. 073).

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:

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
• Octet, a grouping of eight bits

References

1. ^ Marcia Ascher. Ethnomathematics: A Multicultural View of Mathematical Ideas. The College Mathematics Journal. Retrieved on 2007-04-13.
2. ^ Donald Knuth, The Art of Computer Programming

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 numeral system enabling simple visual calculation in octal.

numeral system (or system of numeration) is a framework where a set of numbers are represented by numerals in a consistent manner. It can be seen as the context that allows the numeral "11" to be interpreted as the binary numeral for three
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Hindu-Arabic numeral system (also called Algorism) is a positional decimal numeral system documented from the 9th century.

The symbols (glyphs) used to represent the system are in principle independent of the system itself.
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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.
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The Eastern Arabic numerals (also called 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, Afghanistan, Pakistan and parts of India, and also in
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Khmer numerals are the numerals used in the Khmer language of Cambodia. In informal spoken language one can ignore the last "sep" (30 to 90) and it is still understood.
e.g.
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symbols used in various modern Indian scripts for the numbers from zero to nine:

Variant 0 1 2 3 4 5 6 7 8 9 Used in
Eastern Nagari numerals ০ ১ ২ ৩ ৪ ৫ ৬ ৭ ৮ ? Bengali language
Assamese language

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Brahmi numerals are an indigenous Indian numeral system attested from the 3^rd century BCE (somewhat later in the case of most of the tens). They are the direct graphic ancestors of the modern Indic and Hindu-Arabic numerals.
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Thai numerals (ตัวเลขไทย) are traditionally used in Thailand, although the Arabic numerals (also known as Western numerals) are more common.
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This page contains Chinese text.
Without proper rendering support, you may see question marks, boxes, or other symbols instead of Chinese characters.

Numeral systems by culture
Hindu-Arabic numerals
Western Arabic
Eastern Arabic
Khmer Indian family
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Counting rods (Traditional Chinese: 籌; Simplified Chinese: 筹; Pinyin: chou2
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sset
• 여덟 권 yeodeolgwon (eight (books)) is pronounced like [여덜꿘] yeodeolkkwon

Several numerals have long vowels, namely 둘 (two), 셋 (three) and 넷 (four), but these become short when
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Japanese numerals is the system of number names used in the Japanese language. The Japanese numerals in writing are entirely based on the Chinese numerals and the grouping of large numbers follow the Chinese tradition of grouping by 10,000.
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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.
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Armenian numerals is a historic numeral system created using the majuscules (uppercase letters) of the Armenian alphabet.

There was no notation for zero in the old system, and the numeric values for individual letters were added together.
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Cyrillic numerals was a numbering system derived from the Cyrillic alphabet, used by South and East Slavic peoples. The system was used in Russia as late as the 1700s when Peter the Great replaced it with the Hindu-Arabic numeral system.
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Hebrew numerals is a quasi-decimal alphabetic numeral system using the letters of the Hebrew alphabet.

In this system, there is no notation for zero, and the numeric values for individual letters are added together. Each unit (1, 2, ...
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Greek numerals are a system of representing numbers using letters of the Greek alphabet. They are also known by the names Milesian numerals, Alexandrian numerals, or alphabetic numerals.
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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 Herodian.
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Etruscan numerals were used by the ancient Etruscans. The system was adapted from the Greek Attic numerals and formed the inspiration for the later Roman numerals.

Etruscan Decimal Symbol *
θu 1 I
ma? 5 ?
śar 10 X
muval? 50
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/» and the fifths place with a stroke from the top-left to the bottom-right «\». The numbers from 1 = / to 29 = ////\\\\\ have been found.

Interpretation

These embossed marks, unique in objects from the Bronze Age, were introduced in cast-iron molds and were not
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Roman numerals is a numeral system originating in ancient Rome, adapted from Etruscan numerals. The system used in classical antiquity was slightly modified in the Middle Ages to produce the system we use today. It is based on certain letters which are given values as numerals.
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Babylonian numerals 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 permanent record.
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Egyptian numerals was a numeral system used in ancient Egypt. It was a decimal system, often rounded off to the higher power, written in hieroglyphs. The hieratic form of numerals stressed an exact finite series notation, being ciphered one:one onto the Egyptian alphabet.
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Maya numerals is very simple. [1]
Addition is performed by combining the numeric symbols at each level:

If five or more dots result from the combination, five dots are removed and replaced by a bar.
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This is a list of numeral system topics (and "numeric representations"), by Wikipedia page. It does not systematically list computer formats for storing numbers (computer numbering formats). See also number names.
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A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system.
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base or radix is usually the number of various unique digits, including zero, that a positional numeral system uses to represent numbers. For example, the decimal system, the most common system in use today, uses base ten, hence the maximum number a single digit will ever
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decimal (base ten or occasionally denary) numeral system has ten as its base. It is the most widely used numeral system, perhaps because humans have four fingers and a thumb on each hand, giving a total of ten digits over both hands.
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binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2.
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Quaternary is the base-4 numeral system. It uses the digits 0, 1, 2 and 3 to represent any real number.

It shares with all fixed-radix numeral systems many properties, such as the ability to represent any real number with a canonical representation (almost unique) and the
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