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 Field: Basics



For example, the binary representation for decimal 74 is 1001010, which groups into 1 001 010 — so the octal representation is 112.


Can anyone explain this "representation?" -- Anonymous

It may help to first understand the binary numeral system. Put simply, 1001010 binary is equivalent to 1 × 26 + 1 × 23 + 1 × 21 = 64 + 8 + 2 = 74. Since each octal digit directly corresponds to a group of three binary digits, any binary representation can be condensed into an octal form. The octal numeral 112 above can be expanded to mean 1 × 82 + 1 × 81 + 2 × 80 = 64 + 8 + 2 = 74. -- Wapcaplet 01:40, 10 Dec 2004 (UTC)

I changed the above quoted paragraph to be slightly clearer, I hope.

I'm not sure about a couple of things in the article though. First, Knuth/TAOCP says Karl ("Charles") XII "hit on the idea of radix-8 arithmetic about 1717" and that this "was probably his own invention, although he had met Leibniz briefly in 1707". The article makes it sound like Knuth is certain about this, and forgets the .. well, qualified statement. Perhaps it should not even mention it at all?

Second, novem/novus. Any specific reason to believe this is true? Most of these kind of similarities are entirely random.

Third, fractions. Why should octal be better for fractions? (I'm assuming the author really meant numbers like 0.25, not fractions like 1/4, but then English is not my first language.) It has all the advantages and disadvantages of binary, but if anything it is worse than most bases: it has fewer divisors, so more numbers would tend to "expand into infinity". (Compare with radix 12, that would be nice considering how common 1/3 is.)

Fourth, does the "8" in "base 8 number system" really need to be an article link?

magetoo 16:36, 12 Dec 2004 (UTC)

  • I can't answer for the Knuth reference or novem/novus, but I agree on the matter of fractions; I don't think fractions should even be brought into the picture. When octal is the preferred numeral system, fractions aren't likely to be of much interest. I raised a similar question about the hexadecimal article a while back (the author in that case had claimed that hexadecimal was "quite good" for fractions, which is only true if one considers infinitely repeating expansions desirable). If it were up to me, discussion of fractions would be removed from both articles. In fact, I think I will go ahead and do so... -- Wapcaplet 17:48, 12 Dec 2004 (UTC)

Think I made it a bit more clear. Pseudoanonymous 20:44, 9 August 2007 (UTC)

Number prefixes[edit]

Please add description of 0x = hexadecimal, 0b = binary, 0 = octal. - Omegatron 16:48, Jun 23, 2005 (UTC)

This is just a C rule. Why then not use the '$' prefix, as in Pascal? ;) — Preceding unsigned comment added by (talk) 2005-10-11T19:02:42

move Knuth?[edit]

Seems like that reference to Knuth (that King Charles XII of Sweden was the inventor of octal in Europe) should be removed from the "By Native Americans" section and placed in its own section. Perhaps "Historically" or "In History"? Meonkeys 01:00, 14 November 2006 (UTC)

Benefits of octal[edit]

Here's some original research of mine (though easily verifiable by anyone) that I wisely refrained from adding to the article. I'm placing it here for purposes of discussion.

Octal has three specific and distinct properties that decimal does not:

  • Easy powers of two: The powers of two, which in decimal are written 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, etc., would be written as 1, 2, 4, 10, 20, 40, 100, 200, 400, 1000, 2000, 4000, etc. Also, inverse powers of two, decimal .5, .25, .125, .625 (equivalent to 1/2, 1/4, 1/8, 1/16) would be just as simply written .4, .2, .1, .04, .02, .01, etc., or 1/2, 1/4, 1/10, 1/20, 1/40, etc.
  • Rational appearance of nonprimes: The first nonprime that is not divisible by two is nine (3x3). In octal, it would be written 11. Thus, single-digit numbers' primality is easily found just by checking evenness. 3, 5, and 7 would be easily seen as prime, while 4 and 6 would be obviously nonprime.
  • Easy divisibility by 7 and 9: Take a multiple-digit number in octal, such as 527 (decimal 343). Take the sum of the digits (16 octal, 14 decimal), and repeat the process if the result also has multiple digits. If the resulting single digit is 7, the original number is divisible by seven. Also, since nine is written 11, it has the same property of eleven in decimal, namely that multiples are visually obvious: 22, 55, 121, etc. In octal, 374 is thus obviously a product of nine, seven, and two, since it is even, its digits eventually add up to 7, and it is obviously the sum of 340 and 34.

However, octal loses the easy divisibility by five, three, and eleven. Anyone else have other such? -- 22:58, 22 May 2007 (UTC)

You mean easy divisibility checking, not divisibility itself. If you prefer powers of two, you must give up divisibility by any prime but two. - TAKASUGI Shinji (talk) 03:53, 25 April 2008 (UTC)
The third item is nothing special: divisibility by n–1 and n+1 is easy to check in base n for any n. Elphion (talk) 17:41, 2 March 2009 (UTC)
You don't lose the test for 3: it follows from the test for 9. Double sharp (talk) 04:09, 20 June 2014 (UTC)

Wiki page for "significant digit"[edit]

Is there any page we can link to for the line "most significant octal digit"? I gather that this refers to the leftmost digit out of the three, but I've never seen a definition of "significant digit" in this sense - it certainly does not mean the same thing as significant figure. --Keflavich 21:50, 24 August 2007 (UTC)

The Simpson's should use Base 8[edit]

As the animated cartoon series, The Simpon's, have only four fingers on each hand their numeral system should be octal. It is not. — Preceding unsigned comment added by (talk) 2008-04-24T19:25:16

It's not just the Simpsons. By convention, most cartoon figures have only four fingers; evidently five looks too fussy. Elphion (talk) 19:01, 21 July 2009 (UTC)

Corrected Rama inaccuracy[edit]

I have updated the In Fiction section. It stated that octal was used by the octospiders in 'Rendezvous with Rama'. However, the octospiders are not from that book, but from the book 'Rama Revealed'. To my shame, I was not logged in. I am the anon-IP that did it, and I assume blame for any further inaccuracy in this update! Dfrauzel (talk) 09:12, 20 July 2008 (UTC) 1 2 3 4 5 6 7 10

Base 8 in the Indus Valley[edit]

I have read claims by archaeologists (e.g., Walter Fairservice) than the numbers found in inscriptions in the early Indus Valley civilization indicate a base-8 system; and that there are vestiges of this in modern Dravidian languages (where often nine is "eight and one" and ten "eight and two"), suggesting an ancestral relationship. Any truth in this? Elphion (talk) 17:50, 2 March 2009 (UTC)


The following was added recently:

Finnish language and Estonian language (both belonging to Finno-Ugric languages[1]) represent certain cardinal numbers in a way which suggests the former use of an octal system. Number nine (9) - yhdeksän - [2] in Finnish could be decomposed as "one-from-ten" while number eight (8) - kahdeksan - [3] could be decomposed as "two-from-ten". In Estonian language the relevant cardinals sound very similar: 8=kaheksa and 9=üheksa.
  1. ^ Finno-Ugric languages#Numbers,
  2. ^ yhdeksän -
  3. ^ kahdeksan -

So Finnish uses "overcounting" - subtraction from a larger number - as does vigesimal Danish "halvtreds" (i.e. half third score = 2.5 scores = 2.5 x 20 = 50) and Roman numeral IX = 10-1. But what's that got to do with octal?--Noe (talk) 20:15, 17 December 2009 (UTC)

I've now removed the paragraph, pending a reply or source.--Noe (talk) 08:48, 15 January 2010 (UTC)

(I hope this is the right place for comments - I am a newbie here).

What the addition has to do with octal? The usage of indo-europaean root words for these numbers suggests that the previous (lost) system for ugris was octal. I have to investigate yet the issue but considering that virtually all ugri languages have this feature, it should be rather old - 3-10 thousands years. While "overcounting" probably is more related to a culturally and economically developed nation. Ugris probably were not having high civilization at that period, thus they only needed some very basic counting system, not more.

You can suggest another (even non-existing) article for including this information. My primary interest is that some rare features of my language should be described and accessible for further investigation. The proper place hasn't to be "octal". I am open to ideas. Overcounting? No-no, a too crowded termin.

About sources - a no-no again. A nation or 1.4 mln probably has not too many linguists to deep issues. I haven't yet met this issue in written.

Considering the "half third" example above, in Estonian we say (as German Dreiviertel adaptation): "the clock is 5 minutes before 3/4 of five - kell on viie pärast kolmveerand viis - actually it is 20:40" and - and I have noticed this amuses indo-europeans much. You are right these shortcuts have nothing to do with the counting system, and, I think they rather reflect some difficulties grasping a foreign value system (like the clock was for ugris). For cardinal numbers, it could be very weird to think that numbers up to 7 were "own" and the larger these became, the more too difficult it got to grasp these ... ;) . Well, that is my personal understanding, not more. Tunguuz (talk) 15:54, 22 January 2010 (UTC)

Split Octal[edit]

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... Therefore, hexadecimal is more commonly used...

Surely this calls for a discussion of Split Octal as used by Digital Equipment Corporation as well as microcomputers such as the Heathkit H8? Wlindley (talk) 12:43, 14 April 2010 (UTC)

This is an especially egregious error and should be corrected as an octal representation of the maximum of the range of a 16 bit word requires only five octal digits: 17777 - not six. (2^16)-1 == 0d65,535 == 0q17777 == 0hFFFF —Preceding unsigned comment added by (talk) 03:17, 2 February 2011 (UTC)

You've dropped a digit: 65535 = 0177777, not 017777; it does in fact require 6 octal digits. -- Elphion (talk) 03:31, 2 February 2011 (UTC)

External links[edit]

Is any of the external links worth keeping? (I get "Forbidden" on one them anyway.) -- Elphion (talk) 23:59, 4 December 2011 (UTC)

Pamean number system doesn't appear to be base 8[edit]

Going through the article linked as a citation, if you scroll down to the actual words, you'll notice a few things. First, the number 8 does have significance. The word for 9 (nda ntsaw? nda) is made from putting together the words for 8 (nda ntsaw?) and 1 (nda). But that's not enough to say that they have a base 8 number system.

The word for 10 isn't made from putting together the words for 8 and 2. 10 gets its own word: seska?ai. The word for 11 isn't made from putting together 8 and 3. It's made from putting together 10 and 1: seska?ai nda

Similarly for all the numbers between 10 and 20. 16 is 10 and 6, 19 is 10 and 9, etc. There's no entry on the list for a word meaning 64. A language that uses base 8 obviously would have a separate word for 64.

There is a word for 100: nda ntswa?. And 200: nui ntswa? And 300: renhu? ntswa?

Unsurprisingly, nda, nui, and renhu? are similar to the words for 1, 2, and 3, respectively.

It's not a base 8 language. It's a base 10 language with a special word for 8. (talk) 20:20, 4 January 2013 (UTC)

Use of mod operator in examples[edit]

I've reverted the example using the mod operator, in favor of the basic well-known arithmetic operators. The mod operator obscures what is going on, and its use in the examples does not match the surrounding text. -- Elphion (talk) 01:25, 14 October 2014 (UTC)

I agree. But the real idea that is behind is the Euclidean division, where the quotient and the remainder are determined at the same time. This wasn't clear, and I've rewritten the examples to use the usual a=bq+r form, and added a link to the Euclidean division article on this subject. — Vincent Lefèvre (talk) 22:54, 14 October 2014 (UTC)
Yes, that's a big improvement. -- Elphion (talk) 05:32, 15 October 2014 (UTC)