Octal

 ( Binary Arithmetic ) 

Octal ( base 8) is a numeral system with eight as the radix.

In the decimal system, each place is a power of ten. For example:

$\backslash mathbf\{74\}\_\{10\}\; =\; \backslash mathbf\{7\}\; \backslash times\; 10^1\; +\; \backslash mathbf\{4\}\; \backslash times\; 10^0$

$\backslash mathbf\{112\}\_8\; =\; \backslash mathbf\{1\}\; \backslash times\; 8^2\; +\; \backslash mathbf\{1\}\; \backslash times\; 8^1\; +\; \backslash mathbf\{2\}\; \backslash times\; 8^0$

Octal numerals can be easily converted from binary representations (similar to a 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.

• 0 = ☷, 1 = ☳, 2 = ☵, 3 = ☱,
• 4 = ☶, 5 = ☲, 6 = ☴, 7 = ☰.

Gottfried Wilhelm Leibniz made the connection between trigrams, hexagrams and binary numbers in 1703.

• The Yuki language in California has an octal system because the speakers count using the spaces between their fingers rather than the fingers themselves.
• The Pame language in Mexico also have an octal system, because their speakers count on the knuckles of a closed fist.

• 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.
  Werner Winter (1991). 9783110113228, Mouton de Gruyter. ISBN 9783110113228
• In 1668, John Wilkins 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 asked 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.Donald Knuth, The Art of Computer Programming
• Writing under the pseudonym "Hirossa Ap-Iccim" in The Gentleman's 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."See H. R. Phalen, "Hugh Jones and Octave Computation," The American Mathematical Monthly 56 (August–September 1949): 461-465.
• In 1801, James Anderson criticized the French for basing the 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 was already, to a remarkable extent, an octal system.James Anderson, On Octal Arithmetic title, Recreations in Agriculture, Natural-History, Arts, and Miscellaneous Literature , Vol. IV, No. 6 (February 1801), T. Bensley, London; pages 437-448.
• In the mid-19th century, Alfred B. Taylor concluded that "Our octonary base 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.Alfred B. Taylor, Report on Weights and Measures, Pharmaceutical Association, 8th Annual Session, Boston, 1859-09-15. See pages 48 and 53.Alfred B. Taylor, Octonary numeration and its application to a system of weights and measures, Proc. Amer. Phil. Soc. Vol XXIV , Philadelphia, 1887; pages 296-366. See pages 327 and 330. Taylor also republished some of Swedenborg's work on octal as an appendix to the above-cited publications.

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 and Motorola 68000 family. The modern-day ubiquitous x86 architecture 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 assemblers, some programmers would handcode programs in octal; for instance, Dick Whipple and John Arnold wrote Tiny BASIC 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 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).

In programming languages, octal literals are typically identified with a variety of , 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 suffix following the Intel convention.

     
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
     
     

     
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