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-8 number system, and uses the
digits 0 to 7. This is to say that 10
octal represents eight and 100
octal represents sixty-four. However, English, like most languages, uses a
base-10
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
number system, hence a true octal system might use different vocabulary.
In the decimal system, each place is a
power of ten
A power of 10 is any of the integer exponentiation, powers of the number 10 (number), ten; in other words, ten multiplication, multiplied by itself a certain number of times (when the power is a positive integer). By definition, the number one is ...
. For example:
:
In the octal system, each place is a power of eight. For example:
:
By performing the calculation above in the familiar decimal system, we see why 112 in octal is equal to
in decimal.
Octal numerals can be easily converted from
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that t ...
representations (similar to a
quaternary numeral system
A quaternary numeral system is base-. It uses the digits 0, 1, 2 and 3 to represent any real number. Conversion from binary is straightforward.
Four is the largest number within the subitizing range and one of two numbers that is both a sq ...
) 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
In China
![Bagua-name-earlier](https://upload.wikimedia.org/wikipedia/commons/4/4d/Bagua-name-earlier.svg)
The eight
bagua
The bagua or pakua (八卦) are a set of eight symbols that originated in China, used in Taoist cosmology to represent the fundamental principles of reality, seen as a range of eight interrelated concepts. Each consists of three lines, each li ...
or trigrams of the
I Ching
The ''I Ching'' or ''Yi Jing'' (, ), usually translated ''Book of Changes'' or ''Classic of Changes'', is an ancient Chinese divination text that is among the oldest of the Chinese classics. Originally a divination manual in the Western Zho ...
correspond to octal digits:
* 0 = ☷, 1 = ☳, 2 = ☵, 3 = ☱,
* 4 = ☶, 5 = ☲, 6 = ☴, 7 = ☰.
Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
made the connection between trigrams, hexagrams and binary numbers in 1703.
By Native Americans
* The
Yuki language
Yuki, also known as Ukomno'm, is an extinct language of California, formerly spoken by the Yuki people. The Yuki are the original inhabitants of the Eel River area and the Round Valley Reservation of northern California. Yuki ceased to be used ...
in
California
California is a U.S. state, state in the Western United States, located along the West Coast of the United States, Pacific Coast. With nearly 39.2million residents across a total area of approximately , it is the List of states and territori ...
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 (Spanish: México), officially the United Mexican States, is a country in the southern portion of North America. It is bordered to the north by the United States; to the south and west by the Pacific Ocean; to the southeast by Guatema ...
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 1614 – 19 November 1672) was an Anglican clergyman, natural philosopher, and author, and was one of the founders of the Royal Society. He was Bishop of Chester from 1668 until his death.
Wilkins is one of the fe ...
in ''
An Essay towards a Real Character, and a Philosophical Language'' 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 O.S.), was King of Sweden (including current Finland) from 1697 to 1718. He belonged to the House of Palatinate-Zweibrücken, a branch line of t ...
asked
Emanuel Swedenborg
Emanuel Swedenborg (, ; born Emanuel Swedberg; 29 March 1772) was a Swedish pluralistic-Christian theologian, scientist, philosopher and mystic. He became best known for his book on the afterlife, ''Heaven and Hell'' (1758).
Swedenborg had ...
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, England, by Edward Cave in January 1731. It ran uninterrupted for almost 200 years, until 1922. It was the first to use the term ''magazine'' (from the French ''magazine'' ...
'', (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 (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 James Anderson may refer to:
Arts
*James Anderson (American actor) (1921–1969), American actor
*James Anderson (author) (1936–2007), British mystery writer
*James Anderson (English actor) (born 1980), British actor
* James Anderson (filmmaker) ...
criticized the French for basing the
metric system
The metric system is a system of measurement that succeeded the Decimal, decimalised system based on the metre that had been introduced in French Revolution, France in the 1790s. The historical development of these systems culminated in the d ...
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-Saxon and Roman systems of units. Various standards have applied to English units at ...
was already, to a remarkable extent, an octal system.
* In the mid-19th century, Alfred B. Taylor concluded that "Our octonary
ase 8radix
In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is t ...
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 1050 The UNIVAC 1050 was a variable word-length (one to 16 characters) decimal and binary computer.
Instructions were fixed length (30 bits – five characters), consisting of a five-bit " op code", a three-bit index register specifier, one reser ...
,
PDP-8
The PDP-8 is a 12-bit computing, 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 ...
,
ICL 1900
ICT 1900 was a family of mainframe computers released by International Computers and Tabulators (ICT) and later International Computers Limited (ICL) during the 1960s and 1970s. The 1900 series was notable for being one of the few non-American ...
and
IBM mainframe
IBM mainframes are large computer systems produced by IBM since 1952. During the 1960s and 1970s, IBM dominated the large computer market. Current mainframe computers in IBM's line of business computers are developments of the basic design of th ...
s employed
6-bit,
12-bit,
24-bit or
36-bit
36-bit computers were popular in the early mainframe computer era from the 1950s through the early 1970s.
Starting in the 1960s, but especially the 1970s, the introduction of 7-bit ASCII and 8-bit EBCDIC led to the move to machines using 8-bit ...
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, a word is the natural unit of data used by a particular processor design. A word is a fixed-sized datum handled as a unit by the instruction set or the hardware of the processor. The number of bits or digits in a word (the ''word s ...
. 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 and multiple cathodes, shaped like numerals or other symbo ...
s,
seven-segment display
A seven-segment display is a form of electronic display device for displaying decimal numerals that is an alternative to the more complex dot matrix displays.
Seven-segment displays are widely used in digital clocks, electronic meters, basic ...
s, and
calculator
An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics.
The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...
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 simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, hexa ...
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 minicomputers sold by Digital Equipment Corporation (DEC) from 1970 into the 1990s, one of a set of products in the Programmed Data Processor (PDP) series. In total, around 600,000 PDP-11s of all models were sold, ...
and
Motorola 68000 family
The Motorola 68000 series (also known as 680x0, m68000, m68k, or 68k) is a family of 32-bit complex instruction set computer (CISC) microprocessors. During the 1980s and early 1990s, they were popular in personal computers and workstations and w ...
. The modern-day ubiquitous
x86 architecture
x86 (also known as 80x86 or the 8086 family) is a family of complex instruction set computer (CISC) instruction set architectures initially developed by Intel based on the Intel 8086 microprocessor and its 8088 variant. The 8086 was introd ...
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, avant-garde electronic musician, stage name Assembler
* Assemblers, a fictional race in the ''Star Wars'' universe
* Assemblers, an alternative name of the superhero group Champions of ...
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 permissions
Most file systems include attributes of files and directories that control the ability of users to read, change, navigate, and execute the contents of the file system. In some cases, menu options or functions may be made visible or hidden dependin ...
under
Unix
Unix (; trademarked as UNIX) is a family of multitasking, multiuser computer operating systems that derive from the original AT&T Unix, whose development started in 1969 at the Bell Labs research center by Ken Thompson, Dennis Ritchie, and ot ...
systems (see
chmod
In Unix and Unix-like operating systems, is the command (computing), command and system call used to change the File-system permissions, access permissions and the #Special modes, special mode flags (the setuid, ''setuid'', ''setgid'', and stick ...
). 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
literals are typically identified with a variety of
prefix
A prefix is an affix which is placed before the Word stem, stem of a word. Adding it to the beginning of one word changes it into another word. For example, when the prefix ''un-'' is added to the word ''happy'', it creates the word ''unhappy'' ...
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 operating system, real-time multi-user multi-tasking operating system for IBM Personal Computer, IBM PC-compatible microcomputers.
An evolution of the older Concurrent CP/M-86, Concurrent DOS and Concurrent DOS 386 ...
,
Multiuser DOS
Multiuser DOS is a 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 was originally developed by ...
and
REAL/32
Multiuser DOS is a 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 was originally developed by ...
as well as in
DOS Plus
DOS Plus (erroneously also known as DOS+) was the first operating system developed by Digital Research's OEM Support Group in Newbury, Berkshire, UK, first released in 1985. DOS Plus 1.0 was based on CP/M-86 Plus combined with the PCMODE ...
and
DR-DOS
DR-DOS (written as DR DOS, without a hyphen, in versions up to and including 6.0) is a disk operating system for IBM PC compatibles. Upon its introduction in 1988, it was the first DOS attempting to be compatible with IBM PC DOS and MS-D ...
various
environment variable
An environment variable is a dynamic-named value that can affect the way running processes will behave on a computer. They are part of the environment in which a process runs. For example, a running process can query the value of the TEMP env ...
s like
$CLS,
$ON,
$OFF,
$HEADER or
$FOOTER support an
\nnn
octal number notation,
and DR-DOS
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 (''pronounced like the letter c'') is a general-purpose computer programming language. It was created in the 1970s by Dennis Ritchie, and remains very widely used and influential. By design, C's features cleanly reflect the capabilities o ...
; it is supported by
Haskell
Haskell () is a general-purpose, statically-typed, purely functional programming language with type inference and lazy evaluation. Designed for teaching, research and industrial applications, Haskell has pioneered a number of programming lan ...
,
OCaml
OCaml ( , formerly Objective Caml) is a general-purpose programming language, general-purpose, multi-paradigm programming language which extends the Caml dialect of ML (programming language), ML with object-oriented programming, object-oriented ...
,
Python
Python may refer to:
Snakes
* Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia
** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia
* Python (mythology), a mythical serpent
Computing
* Python (pro ...
as of version 3.0,
Raku,
Ruby
A ruby is a pinkish red to blood-red colored gemstone, a variety of the mineral corundum ( aluminium oxide). Ruby is one of the most popular traditional jewelry gems and is very durable. Other varieties of gem-quality corundum are called sa ...
,
Tcl
TCL or Tcl or TCLs may refer to:
Business
* TCL Technology, a Chinese consumer electronics and appliance company
**TCL Electronics, a subsidiary of TCL Technology
* Texas Collegiate League, a collegiate baseball league
* Trade Centre Limited ...
as of version 9,
PHP
PHP is a general-purpose scripting language geared toward web development. It was originally created by Danish-Canadian programmer Rasmus Lerdorf in 1993 and released in 1995. The PHP reference implementation is now produced by The PHP Group ...
as of version 8.1,
Rust
Rust is an iron oxide, a usually reddish-brown oxide formed by the reaction of iron and oxygen in the catalytic presence of water or air moisture. Rust consists of hydrous iron(III) oxides (Fe2O3·nH2O) and iron(III) oxide-hydroxide (FeO(OH ...
and it is intended to be supported by
ECMAScript
ECMAScript (; ES) is a JavaScript standard intended to ensure the interoperability of web pages across different browsers. It is standardized by Ecma International in the documenECMA-262
ECMAScript is commonly used for client-side scripting o ...
6 (the prefix
0
originally stood for base 8 in
JavaScript
JavaScript (), often abbreviated as JS, is a programming language that is one of the core technologies of the World Wide Web, alongside HTML and CSS. As of 2022, 98% of Website, websites use JavaScript on the Client (computing), client side ...
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, general-purpose, interpreted, dynamic programming languages. "Perl" refers to Perl 5, but from 2000 to 2019 it also referred to its redesigned "sister language", Perl 6, before the latter's name was offici ...
,
PostScript
PostScript (PS) is a page description language in the electronic publishing and desktop publishing realm. It is a dynamically typed, concatenative programming language. It was created at Adobe Systems by John Warnock, Charles Geschke, Doug Br ...
...) 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 encoding, variable-length character encoding used for electronic communication. Defined by the Unicode Standard, the name is derived from ''Unicode'' (or ''Universal Coded Character Set'') ''Transformation Format 8-bit'' ...
, 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, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be ...
in the
Ferranti Atlas (1962),
Burroughs B5500 (1964),
Burroughs B5700
The Burroughs Large Systems Group produced a family of large 48-bit mainframes using stack machine instruction sets with dense syllables.E.g., 12-bit syllables for B5000, 8-bit syllables for B6500 The first machine in the family was the B5000 in ...
(1971),
Burroughs B6700
The Burroughs Large Systems Group produced a family of large 48-bit mainframes using stack machine instruction sets with dense syllables.E.g., 12-bit syllables for B5000, 8-bit syllables for B6500 The first machine in the family was the B5000 i ...
(1971) and
Burroughs B7700 (1972) computers.
In aviation
Transponders
In telecommunications, a transponder is a device that, upon receiving a signal, emits a different signal in response. The term is a blend of ''transmitter'' and ''responder''.
In air navigation or radio frequency identification, a flight trans ...
in aircraft transmit a "squawk"
code
In communications and information processing, code is a system of rules to convert information—such as a letter, word, sound, image, or gesture—into another form, sometimes shortened or secret, for communication through a communication ...
, 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 125
10 to octal:
:125 = 8
2 × 1 + 61
:61 = 8
1 × 7 + 5
:5 = 8
0 × 5 + 0
Therefore, 125
10 = 175
8.
Another example:
:900 = 8
3 × 1 + 388
:388 = 8
2 × 6 + 4
:4 = 8
1 × 0 + 4
:4 = 8
0 × 4 + 0
Therefore, 900
10 = 1604
8.
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, 0.1640625
10 = 0.124
8.
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:
:
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 764
8 to decimal:
:764
8 = 7 × 8
2 + 6 × 8
1 + 4 × 8
0 = 448 + 48 + 4 = 500
10
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: 65
8 = 6 × 8 + 5 = 53
10
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 51
8 to binary:
:5
8 = 101
2
:1
8 = 001
2
Therefore, 51
8 = 101 001
2.
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, 1010111100
2 = 1274
8.
Convert binary 11100.01001 to octal:
:
Therefore, 11100.01001
2 = 34.22
8.
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, 1057
8 = 22F
16.
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 3FA5
16:
:To binary:
:
:then to octal:
:
Therefore, 3FA5
16 = 37645
8.
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, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
s in decimal and octal.
See also
*
*
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 as a generalization of Nim, Kayles, and similar games. Revised and repr ...
, a game numbering system used in
combinatorial game theory
Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a ''position'' that the players ...
*
Split 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 Ele ...
, a 16-bit octal notation used by the Heath Company, DEC and others
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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 transponders to assist ...
, 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, the Gillham interface, that is used to transmit uncorrected barometric altitude between an encoding altimeter or analog air data computer and a d ...
*
Syllabic octal, 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 a disk operating system for IBM PC compatibles. Upon its introduction in 1988, it was the first DOS attempting to be compatible with IBM PC DOS and MS-D ...
and OpenDOS 7.01
DR-DOS (written as DR DOS, without a hyphen, in versions up to and including 6.0) is a disk operating system for IBM PC compatibles. Upon its introduction in 1988, it was the first DOS attempting to be compatible with IBM PC DOS and MS-D ...
, 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 ...
, 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
Octomaticsis 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 converterperforms bidirectional conversions between the octal and decimal system.
Binary arithmetic
Power-of-two numeral systems