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The octal
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner. The same s ...
, or oct for short, is the
base Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications operator *Base CRM Base CRM (originally Future Simple or PipeJump) is an enterprise software company based in Mountain Vie ...

base
-8 number system, and uses the digits 0 to 7, that is to say 10 represents 8 in decimal and 100 represents 64 in decimal. However, English uses a
base-10 The decimal numeral system (also called the base-ten positional numeral system, and occasionally called denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hi ...

base-10
number language system, hence a true octal system might use different language to avoid confusion with the decimal system. In the decimal system, each place is a power of ten. For example: : \mathbf_ = \mathbf \times 10^1 + \mathbf \times 10^0 In the octal system, each place is a power of eight. For example: : \mathbf_8 = \mathbf \times 8^2 + \mathbf \times 8^1 + \mathbf \times 8^0 By performing the calculation above in the familiar decimal system we see why 112 in octal is equal to 64+8+2 = 74 in decimal. Octal numerals can be easily converted from
binary Binary may refer to: Science and technology Mathematics * Binary number In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: ty ...
representations (similar to a
quaternary numeral system A quaternary numeral system is Base (radix), base-. It uses the numerical digit, digits 0, 1, 2 and 3 to represent any real number. Conversion from Binary number, binary is straightforward. Four is the largest number within the subitizing ran ...

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: , corresponding to the octal digits , yielding the octal representation 112.


Usage


By Native Americans

* The
Yuki language Yuki, also known as Ukomno'm, is an extinct language of California California is a U.S. state, state in the Western United States. With over 39.3million residents across a total area of approximately , it is the List of states and territor ...
in
California California is a state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * The State (newspaper), ''The State'' (newspaper), a daily newspaper i ...

California
has an octal system because the speakers count using the spaces between their fingers rather than the fingers themselves. * The Pamean languages in
Mexico Mexico, officially the United Mexican States, is a country A country is a distinct territorial body or political entity A polity is an identifiable political entity—any group of people who have a collective identity, who are organi ...

Mexico
also have an octal system, because their speakers count on the knuckles of a closed fist.


By Europeans

* It has been suggested that the reconstructed Proto-Indo-European (PIE) word for "nine" might be related to the PIE word for "new". Based on this, some have speculated that proto-Indo-Europeans used an octal number system, though the evidence supporting this is slim. * In 1668,
John Wilkins John Wilkins, (14 February 161419 November 1672) was an Anglican ministry, Anglican clergyman, natural philosophy, natural philosopher and author, and was one of the founders of the Royal Society. He was Bishop of Chester from 1668 until his d ...

John Wilkins
in ''
An Essay towards a Real Character, and a Philosophical Language ''An Essay Towards a Real Character, and a Philosophical Language'' (London, 1668) is the best-remembered of the numerous works of John Wilkins, in which he expounds a new universal language, meant primarily to facilitate international communic ...
'' proposed use of base 8 instead of 10 "because the way of Dichotomy or Bipartition being the most natural and easie kind of Division, that Number is capable of this down to an Unite". * In 1716, King
Charles XII of Sweden Charles XII, sometimes Carl XII ( sv, Karl XII) or Carolus Rex (17 June 1682 – 30 November 1718 Adoption of the Gregorian calendar, O.S.), was the King of Sweden from 1697 to 1718. He belonged to the House of Palatinate-Zweibrücken, a branc ...
asked
Emanuel Swedenborg Emanuel Swedenborg (, ; born Emanuel Swedberg; 29 March 1772) was a -Christian , , and . He became best known for his book on the , (1758). Swedenborg had a prolific career as an and . In 1741, at 53, he entered into a phase in which he be ...

Emanuel Swedenborg
to elaborate a number system based on 64 instead of 10. Swedenborg argued, however, that for people with less intelligence than the king such a big base would be too difficult and instead proposed 8 as the base. In 1718 Swedenborg wrote (but did not publish) a manuscript: "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. *Writing under the pseudonym "Hirossa Ap-Iccim" in ''
The Gentleman's Magazine ''The Gentleman's Magazine'' was a monthly magazine founded in London London is the capital Capital most commonly refers to: * Capital letter Letter case (or just case) is the distinction between the letters that are in larg ...
'', (London) July 1745, Hugh Jones proposed an octal system for British coins, weights and measures. "Whereas reason and convenience indicate to us an uniform standard for all quantities; which I shall call the ''Georgian standard''; and that is only to divide every integer in each ''species'' into eight equal parts, and every part again into 8 real or imaginary particles, as far as is necessary. For tho' all nations count universally by ''tens'' (originally occasioned by the number of digits on both hands) yet 8 is a far more complete and commodious number; since it is divisible into halves, quarters, and half quarters (or units) without a fraction, of which subdivision ''ten'' is uncapable...." In a later treatise on
Octave computation
Octave computation
(1753) Jones concluded: "Arithmetic by ''Octaves'' seems most agreeable to the Nature of Things, and therefore may be called Natural Arithmetic in Opposition to that now in Use, by Decades; which may be esteemed Artificial Arithmetic." * In 1801, James Anderson criticized the French for basing the
metric system The metric system is a system of measurement A system of measurement is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been important, regulated and defined for the purpose ...

metric system
on decimal arithmetic. He suggested base 8, for which he coined the term ''octal''. His work was intended as recreational mathematics, but he suggested a purely octal system of weights and measures and observed that the existing system of
English units English units are the units of measurement used in England up to 1826 (when they were replaced by Imperial units), which evolved as a combination of the Anglo-Saxons, Anglo-Saxon and Ancient Roman units of measurement, Roman systems of units. Va ...
was already, to a remarkable extent, an octal system. * In the mid-19th century, Alfred B. Taylor concluded that "Our octonary
ase 8 Ase may refer to: * , a town in Delta State, Nigeria * , a West African philosophical concept * (ISO 639-3 code: ase) See also

* * ASE (disambiguation) {{disambiguation ...
radix In a positional numeral system Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any of the (or ). More generally, a positional system is a numeral system in which the contribution ...

radix
is, therefore, beyond all comparison the "''best possible one''" for an arithmetical system." The proposal included a graphical notation for the digits and new names for the numbers, suggesting that we should count "''un'', ''du'', ''the'', ''fo'', ''pa'', ''se'', ''ki'', ''unty'', ''unty-un'', ''unty-du''" and so on, with successive multiples of eight named "''unty'', ''duty'', ''thety'', ''foty'', ''paty'', ''sety'', ''kity'' and ''under''." So, for example, the number 65 (101 in octal) would be spoken in octonary as ''under-un''. Taylor also republished some of Swedenborg's work on octal as an appendix to the above-cited publications.


In computers

Octal became widely used in computing when systems such as the
UNIVAC 1050The UNIVAC 1050 was a variable word-length (one to 16 characters) decimal and binary computer. Instruction set, Instructions were fixed length (30 bits – five characters), consisting of a five-bit "opcode, op code", a three-bit index register ...
,
PDP-8 The PDP-8 is a 12-bit minicomputer that was produced by Digital Equipment Corporation, Digital Equipment Corporation (DEC). It was the first commercially successful minicomputer, with over 50,000 units being sold over the model's lifetime. Its bas ...

PDP-8
, ICL 1900 and
IBM mainframe IBM mainframes are large computer systems produced by IBM International Business Machines Corporation (IBM) is an American multinational technology company headquartered in Armonk, New York, with operations in over 170 countries. The com ...
s employed 6-bit,
12-bit Possibly the best-known 12-bit CPU is the PDP-8 The PDP-8 is a 12-bit Possibly the best-known 12-bit CPU is the PDP-8 and its relatives, such as the Intersil 6100 microprocessor produced in various incarnations from August 1963 to mid-1990. Man ...
,
24-bit Notable 24-bit machines include the CDC 924 The CDC 1604 was a 48-bit computer designed and manufactured by Seymour Cray and his team at the Control Data Corporation (CDC). The 1604 is known as one of the first commercially successful transis ...
or
36-bit 36-bit computers were popular in the early mainframe computer A mainframe computer, informally called a mainframe or big iron, is a computer used primarily by large organizations for critical applications like bulk data processing for tasks ...
words. Octal was an ideal abbreviation of binary for these machines because their word size is divisible by three (each octal digit represents three binary digits). So two, four, eight or twelve digits could concisely display an entire
machine word In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and softwa ...
. It also cut costs by allowing
Nixie tube A Nixie tube ( ), or cold cathode display, is an electronic device used for displaying numerals or other information using glow discharge. The glass tube contains a wire-mesh anode An anode is an electrode through which the conventional ...
s,
seven-segment display A seven-segment display is a form of electronic display device s, LED A light-emitting diode (LED) is a semiconductor A semiconductor material has an Electrical resistivity and conductivity, electrical conductivity value falling be ...
s, and
calculator An electronic calculator is typically a portable device used to perform s, ranging from basic to complex . The first calculator was created in the early 1960s. Pocket-sized devices became available in the 1970s, especially after the , the f ...

calculator
s to be used for the operator consoles, where binary displays were too complex to use, decimal displays needed complex hardware to convert radices, and
hexadecimal In mathematics and computing, the hexadecimal (also base 16 or hex) numeral system is a Numeral system#Positional systems in detail, positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system repres ...
displays needed to display more numerals. All modern computing platforms, however, use 16-, 32-, or 64-bit words, further divided into eight-bit bytes. On such systems three octal digits per byte would be required, with the most significant octal digit representing two binary digits (plus one bit of the next significant byte, if any). Octal representation of a 16-bit word requires 6 digits, but the most significant octal digit represents (quite inelegantly) only one bit (0 or 1). This representation offers no way to easily read the most significant byte, because it's smeared over four octal digits. Therefore, hexadecimal is more commonly used in programming languages today, since two hexadecimal digits exactly specify one byte. Some platforms with a power-of-two word size still have instruction subwords that are more easily understood if displayed in octal; this includes the
PDP-11 The PDP-11 is a series of 16-bit 16-bit microcomputer A microcomputer is a small, relatively inexpensive computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically ...
and
Motorola 68000 family The Motorola 68000 series (also known as 680x0, m68000, m68k, or 68k) is a family of 32-bit 32-bit microcomputers are computers in which 32-bit microprocessors are the norm. Range for storing integers A 32-bit register can store 232 differe ...
. The modern-day ubiquitous
x86 architecture x86 is a family of instruction set architecture In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for ...
belongs to this category as well, but octal is rarely used on this platform, although certain properties of the binary encoding of opcodes become more readily apparent when displayed in octal, e.g. the ModRM byte, which is divided into fields of 2, 3, and 3 bits, so octal can be useful in describing these encodings. Before the availability of
assembler Assembler may refer to: Arts and media * Nobukazu Takemura Nobukazu Takemura (竹村延和 ''Takemura Nobukazu'') is a Japanese musician and artist. Born in Hirakata, Osaka in August 1968, he became interested in music at a young age by listenin ...
s, some programmers would handcode programs in octal; for instance, Dick Whipple and John Arnold wrote Tiny BASIC Extended directly in machine code, using octal. Octal is sometimes used in computing instead of hexadecimal, perhaps most often in modern times in conjunction with
file permissionsMost file system In computing, a file system or filesystem (often abbreviated to fs) controls how data is Computer data storage, stored and retrieved. Without a file system, data placed in a storage medium would be one large body of data with no wa ...
under
Unix Unix (; trademarked as UNIX) is a family of multitasking, multiuser Multi-user software is computer software Software is a collection of Instruction (computer science), instructions that tell a computer how to work. This is in contrast t ...

Unix
systems (see
chmod In Unix Unix (; trademarked as UNIX) is a family of multitasking, multiuser Multi-user software is computer software Software is a collection of Instruction (computer science), instructions that tell a computer how to work. This is in ...

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 programming languages, octal
literal Literal may refer to: * Interpretation of legal concepts: ** Strict constructionism ** The plain meaning rule (a.k.a. "literal rule") * Literal (mathematical logic), certain logical roles taken by propositions * Literal (computer programming ...
s are typically identified with a variety of
prefix A prefix is an affix In linguistics Linguistics is the scientific study of language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) ...
es, including the digit 0, the letters o or q, the digit–letter combination 0o, or the symbol & or $. In ''Motorola convention'', octal numbers are prefixed with @, whereas a small (or capital) letter o or q is added as a postfix following the ''Intel convention''. In
Concurrent DOS Multiuser DOS is a real-time multi-user multi-tasking operating system An operating system (OS) is system software System software is software designed to provide a platform for other software. Examples of system software include operati ...
,
Multiuser DOS Multiuser DOS is a real-time multi-user multi-tasking operating system An operating system (OS) is system software System software is software designed to provide a platform for other software. Examples of system software include operatin ...
and
REAL/32 Multiuser DOS is a Real-time operating system, real-time multi-user multi-tasking operating system for IBM PC-compatible microcomputers. An evolution of the older Concurrent CP/M-86, Concurrent DOS and Concurrent DOS 386 operating systems, it wa ...
as well as in
DOS Plus DOS Plus (erroneously also known as DOS+) was the first operating system An operating system (OS) is system software that manages computer hardware, computer software, software resources, and provides common daemon (computing), services for co ...
and
DR-DOS DR-DOS (written as DR DOS, without a hyphen, in versions up to and including 6.0) is an operating system of the DOS family, written for IBM PC-PC compatible, compatible personal computers. It was originally developed by Gary A. Kildall's ...
various
environment variable An environment variable is a dynamic-named value Value or values may refer to: * Value (ethics) In ethics Ethics or moral philosophy is a branch of philosophy Philosophy (from , ) is the study of general and fundamental questions, ...
s like $CLS, $ON, $OFF, $HEADER or $FOOTER support an \nnn octal number notation, and DR-DOS
DEBUG In computer programming Computer programming is the process of designing and building an executable In computing, executable code, an executable file, or an executable program, sometimes simply referred to as an executable or binary, caus ...

DEBUG
utilizes \ to prefix octal numbers as well. For example, the literal 73 (base 8) might be represented as 073, o73, q73, 0o73, \73, @73, &73, $73 or 73o in various languages. Newer languages have been abandoning the prefix 0, as decimal numbers are often represented with leading zeroes. The prefix q was introduced to avoid the prefix o being mistaken for a zero, while the prefix 0o was introduced to avoid starting a numerical literal with an alphabetic character (like o or q), since these might cause the literal to be confused with a variable name. The prefix 0o also follows the model set by the prefix 0x used for hexadecimal literals in the
C language C (, as in the C, letter ''c'') is a General-purpose language, general-purpose, procedural programming, procedural computer programming language supporting structured programming, lexical variable scope, and Recursion (computer science), recurs ...
; it is supported by Haskell,
OCaml OCaml ( , formerly Objective Caml) is a general-purpose, multi-paradigm programming language Programming paradigms are a way to classify programming languages based on their features. Languages can be classified into multiple paradigms. S ...
,
Python Python may refer to: * Pythonidae The Pythonidae, commonly known as pythons, are a family of nonvenomous snakes found in Africa, Asia, and Australia. Among its members are some of the largest snakes in the world. Ten genera and 42 species ...
as of version 3.0, Raku,
Ruby A ruby is a pink-ish red to blood-red colored gemstone A gemstone (also called a fine gem, jewel, precious stone, or semi-precious stone) is a piece of mineral In geology and mineralogy, a mineral or mineral species is, broadly spea ...
,
Tcl Tcl (pronounced "tickle" or as an initialism An acronym is a word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, prac ...

Tcl
as of version 9,
PHP PHP is a general-purpose scripting language A scripting language or script language is a programming language A programming language is a formal language comprising a Instruction set architecture, set of instructions that produce various k ...

PHP
as of version 8.1 and it is intended to be supported by
ECMAScript ECMAScript () (or ES) is a JavaScript JavaScript (), often abbreviated JS, is a programming language A programming language is a formal language In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of s ...
6 (the prefix 0 originally stood for base 8 in
JavaScript JavaScript (), often abbreviated JS, is a programming language A programming language is a formal language In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), ma ...

JavaScript
but could cause confusion, therefore it has been discouraged in ECMAScript 3 and dropped in ECMAScript 5). Octal numbers that are used in some programming languages (C,
Perl Perl is a family of two high-level High-level and low-level, as technical terms, are used to classify, describe and point to specific Objective (goal), goals of a systematic operation; and are applied in a wide range of contexts, such as, for ...
,
PostScript PostScript (PS) is a page description language In digital printing, a page description language (PDL) is a computer language that describes the appearance of a printed page in a higher level than an actual output bitmap (or generally raster gr ...

PostScript
…) for textual/graphical representations of byte strings when some byte values (unrepresented in a code page, non-graphical, having special meaning in current context or otherwise undesired) have to be to escaped as \nnn. Octal representation may be particularly handy with non-ASCII bytes of
UTF-8 UTF-8 is a variable-width character encoding Character encoding is the process of assigning numbers to Graphics, graphical character (computing), characters, especially the written characters of Language, human language, allowing them to be ...
, which encodes groups of 6 bits, and where any start byte has octal value \3nn and any continuation byte has octal value \2nn. Octal was also used for
floating point In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and soft ...
in the
Ferranti Atlas The Atlas Computer was one of the world's first supercomputer by the largest supercomputer over time A supercomputer is a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical oper ...
(1962), Burroughs B5500 (1964), Burroughs B5700 (1971), Burroughs B6700 (1971) and Burroughs B7700 (1972) computers.


In aviation

Transponders 120px, A Ontario Highway 407, Highway 407 toll transponder In telecommunication, a transponder is a device that, upon receiving a signal, emits a different signal in response. The term is a portmanteau of ''transmitter'' and ''responder''. It is va ...
in aircraft transmit a "squawk"
code In communication Communication (from Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', mean ...
, expressed as a four-octal-digit number, when interrogated by ground radar. This code is used to distinguish different aircraft on the radar screen.


Conversion between bases


Decimal to octal conversion


Method of successive Euclidean division by 8

To convert integer decimals to octal, divide the original number by the largest possible power of 8 and divide the remainders by successively smaller powers of 8 until the power is 1. The octal representation is formed by the quotients, written in the order generated by the algorithm. For example, to convert 12510 to octal: :125 = 82 × 1 + 61 :61 = 81 × 7 + 5 :5 = 80 × 5 + 0 Therefore, 12510 = 1758. Another example: :900 = 83 × 1 + 388 :388 = 82 × 6 + 4 :4 = 81 × 0 + 4 :4 = 80 × 4 + 0 Therefore, 90010 = 16048.


Method of successive multiplication by 8

To convert a decimal fraction to octal, multiply by 8; the integer part of the result is the first digit of the octal fraction. Repeat the process with the fractional part of the result, until it is null or within acceptable error bounds. Example: Convert 0.1640625 to octal: :0.1640625 × 8 = 1.3125 = 1 + 0.3125 :0.3125 × 8 = 2.5 = 2 + 0.5 :0.5 × 8 = 4.0 = 4 + 0 Therefore, 0.164062510 = 0.1248. These two methods can be combined to handle decimal numbers with both integer and fractional parts, using the first on the integer part and the second on the fractional part.


Method of successive duplication

To convert integer decimals to octal, prefix the number with "0.". Perform the following steps for as long as digits remain on the right side of the radix: Double the value to the left side of the radix, using ''octal'' rules, move the radix point one digit rightward, and then place the doubled value underneath the current value so that the radix points align. If the moved radix point crosses over a digit that is 8 or 9, convert it to 0 or 1 and add the carry to the next leftward digit of the current value. ''Add'' ''octally'' those digits to the left of the radix and simply drop down those digits to the right, without modification. Example:
 0.4 9 1 8 decimal value
  +0
 ---------
   4.9 1 8
  +1 0
  --------
   6 1.1 8
  +1 4 2
  --------
   7 5 3.8
  +1 7 2 6
  --------
 1 1 4 6 6. octal value


Octal to decimal conversion

To convert a number to decimal, use the formula that defines its base-8 representation: :k = \sum_^n \left( a_i\times 8^i \right) In this formula, is an individual octal digit being converted, where is the position of the digit (counting from 0 for the right-most digit). Example: Convert 7648 to decimal: :7648 = 7 × 82 + 6 × 81 + 4 × 80 = 448 + 48 + 4 = 50010 For double-digit octal numbers this method amounts to multiplying the lead digit by 8 and adding the second digit to get the total. Example: 658 = 6 × 8 + 5 = 5310


Method of successive duplication

To convert octals to decimals, prefix the number with "0.". Perform the following steps for as long as digits remain on the right side of the radix: Double the value to the left side of the radix, using ''decimal'' rules, move the radix point one digit rightward, and then place the doubled value underneath the current value so that the radix points align. ''Subtract'' ''decimally'' those digits to the left of the radix and simply drop down those digits to the right, without modification. Example:
 0.1 1 4 6 6  octal value
  -0
 -----------
   1.1 4 6 6
  -  2
  ----------
     9.4 6 6
  -  1 8
  ----------
     7 6.6 6
  -  1 5 2
  ----------
     6 1 4.6
  -  1 2 2 8
  ----------
     4 9 1 8. decimal value


Octal to binary conversion

To convert octal to binary, replace each octal digit by its binary representation. Example: Convert 518 to binary: :58 = 1012 :18 = 0012 Therefore, 518 = 101 0012.


Binary to octal conversion

The process is the reverse of the previous algorithm. The binary digits are grouped by threes, starting from the least significant bit and proceeding to the left and to the right. Add leading zeroes (or trailing zeroes to the right of decimal point) to fill out the last group of three if necessary. Then replace each trio with the equivalent octal digit. For instance, convert binary 1010111100 to octal: : Therefore, 10101111002 = 12748. Convert binary 11100.01001 to octal: : Therefore, 11100.010012 = 34.228.


Octal to hexadecimal conversion

The conversion is made in two steps using binary as an intermediate base. Octal is converted to binary and then binary to hexadecimal, grouping digits by fours, which correspond each to a hexadecimal digit. For instance, convert octal 1057 to hexadecimal: :To binary: : :then to hexadecimal: : Therefore, 10578 = 22F16.


Hexadecimal to octal conversion

Hexadecimal to octal conversion proceeds by first converting the hexadecimal digits to 4-bit binary values, then regrouping the binary bits into 3-bit octal digits. For example, to convert 3FA516: :To binary: : :then to octal: : Therefore, 3FA516 = 376458.


Real numbers


Fractions

Due to having only factors of two, many octal fractions have repeating digits, although these tend to be fairly simple:


Irrational numbers

The table below gives the expansions of some common
irrational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s in decimal and octal.


See also

*
Computer numbering formats A computer number format is the internal representation of numeric values in digital device hardware and software, such as in programmable computers and calculators. Numerical values are stored as groupings of bits, such as bytes and words. The en ...
*
Octal games The octal games are a class of two-player games that involve removing tokens (game pieces or stones) from heaps of tokens. They have been studied in combinatorial game theory at a combinatorial game theory workshop Combinatorial game theory (CGT) ...
, a game numbering system used in
combinatorial game theory at a combinatorial game theory workshop Combinatorial game theory (CGT) is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (al ...
* Split octal, a 16-bit octal notation used by the Heath Company, DEC and others *
Squawk code A transponder (short for ''trans''mitter-res''ponder'' and sometimes abbreviated to XPDR, XPNDR, TPDR or TP) is an electronic device that produces a response when it receives a radio-frequency interrogation. Aircraft have transponder Image:407 Tr ...
, a 12-bit octal representation of
Gillham code Gillham code is a zero-padded 12-bit binary code using a parallel nine- to eleven-wire interface (computing), interface, the Gillham interface, that is used to transmit uncorrected Barometer, barometric altitude between an encoding altimeter or a ...
*
Syllabic octal Syllabic octal and split octal are two similar notations for 8-bit and 16-bit octal numbers, respectively, used in some historical contexts. Syllabic octal ''Syllabic octal'' is an 8-bit octal number representation that was used by English Electric ...
, an octal representation of 8-bit syllables used by English Electric


References

{{reflist, refs= {{cite book , title=NWDOS-TIPs — Tips & Tricks rund um Novell DOS 7, mit Blick auf undokumentierte Details, Bugs und Workarounds , work=MPDOSTIP , author-first=Matthias R. , author-last=Paul , date=1997-07-30 , edition=3 , version=Release 157 , language=de , url=http://www.antonis.de/dos/dos-tuts/mpdostip/html/nwdostip.htm , access-date=2014-08-06 , url-status=live , archive-url=https://web.archive.org/web/20161104235829/http://www.antonis.de/dos/dos-tuts/mpdostip/html/nwdostip.htm , archive-date=2016-11-04 (NB. NWDOSTIP.TXT is a comprehensive work on
Novell DOS 7 DR-DOS (written as DR DOS, without a hyphen, in versions up to and including 6.0) is an operating system An operating system (OS) is system software that manages computer hardware, computer software, software resources, and provides com ...
and OpenDOS 7.01, including the description of many undocumented features and internals. It is part of the author's yet larger MPDOSTIP.ZIP collection maintained up to 2001 and distributed on many sites at the time. The provided link points to a HTML-converted older version of the NWDOSTIP.TXT file.)
{{cite web , title=Updated CLS posted , author-first=Matthias R. , author-last=Paul , date=2002-03-26 , url=http://marc.info/?l=freedos-dev&m=101717593306186&w=2 , publisher=freedos-dev mailing list , access-date=2014-08-06 , url-status=live , archive-url=https://archive.today/20190427173821/https://marc.info/?l=freedos-dev&m=101717593306186&w=2 , archive-date=27 April 2019 {{cite book , title=CCI Multiuser DOS 7.22 GOLD Online Documentation , id=HELP.HLP , date=1997-02-10 , publisher= Concurrent Controls, Inc. (CCI) {{cite book , title=CP/M-86 - Operating System - Programmer's Guide , chapter=2.4.1 Numeric Constants , date=January 1983 , orig-year=1981 , edition=3 , publisher=
Digital Research Digital Research, Inc. (DR or DRI) was a company created by Gary Kildall to market and develop his CP/M operating system and related 8-bit, 16-bit and 32-bit systems like MP/M, Concurrent DOS, FlexOS, Multiuser DOS, DOS Plus, DR DOS and Gr ...
, location=Pacific Grove, California, USA , page=9 , url=http://www.bitsavers.org/pdf/digitalResearch/cpm-86/CPM-86_Programmers_Guide_Jan83.pdf , access-date=2020-02-27 , url-status=live , archive-url=https://web.archive.org/web/20200227225328/http://www.bitsavers.org/pdf/digitalResearch/cpm-86/CPM-86_Programmers_Guide_Jan83.pdf , archive-date=2020-02-27}

(1+viii+122+2 pages)


External links


Octomatics
is a
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner. The same s ...
enabling simple visual calculation in octal.
Octal converter
performs bidirectional conversions between the octal and decimal system. Binary arithmetic Power-of-two numeral systems