Talk:Binary number

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WikiProject Mathematics (Rated B-class, Mid-priority)
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Division is hard to understand[edit]

For binary arithmetic, I understood addition, subtraction and multiplication fine; but I felt division was poorly explained. It is only explained in one paragraph (which is too condensed) and it would be helpful if the provided example was displayed at different stages of the process. — Preceding unsigned comment added by (talk) 12:02, 11 October 2011 (UTC)

Ask here, not here. See also. (talk) 19:25, 23 November 2011 (UTC)

I agree with the comments above - the article doesn't really explain how you do binary division. For purposes of clarity, the quotient should be different from the divisor and it's also confusing that the dividend is shown as 110111 instead of 11011. It would be useful if the article could demonstrate how a computer would do it, as this would be different from doing it from the decimal way. There is a page on "Division (digital)", but that's not very clear either. Perhaps an external link would be useful. I've looked at the link that the poster 'should' have posted his comments to, but that takes you to a page entitled "Editing Wikipedia:Reference desk/Mathematics (new section)", which doesn't seem very appropriate. — Preceding unsigned comment added by (talk) 13:23, 2 January 2012 (UTC)

Further to the above, I've tried to add a wiki link for the word 'dividend' but it comes up with the wrong definition. I've read the editing tutorial and it doesn't really explain how to correct this, so I'll leave it to someone more knowledgeable.. hopefully they'll change the tutorial as well! — Preceding unsigned comment added by (talk) 13:56, 2 January 2012 (UTC)

See also[edit]

Talk:101010101010000100101#Not71.146.20.62 (talk) 19:26, 23 November 2011 (UTC)

Deleted per request. -R. S. Shaw (talk) 23:30, 24 June 2012 (UTC)


Any number can be represented by any sequence of bits (binary digits), which in turn may be represented by any mechanism capable of being in two mutually exclusive states. The following sequence of symbols could all be interpreted as the binary numeric value of 667:

1 0 1 0 0 1 1 0 1 1
| − | − − | | − | |
x o x o o x x o x x
y n y n n y y n y y

This does not equal 667. It equals 663. — Preceding unsigned comment added by (talk) 12:09, 23 June 2012 (UTC)

No, it does equal 667. 663 would be 1010010111. This is binary, not binary coded decimal. -R. S. Shaw (talk) 21:51, 24 June 2012 (UTC)

"computer-based devices such as mobile phones"[edit]

Is this recent addition to the lead topical? IMHO it is not enough. Incnis Mrsi (talk) 15:12, 28 September 2012 (UTC)

I don't think it's strictly necessary, but actually I think the lead reads nicely with it and I would support keeping it. --JBL (talk) 16:50, 28 September 2012 (UTC)
I think the addition is neutral-bordering-useful. In any case, the language and disposition of edit summaries by User:EEng are certainly unacceptable. Rschwieb (talk) 14:44, 30 September 2012 (UTC)

Why does the sentence say "almost" all modern computers? Are there some that don't use binary? What are they? - GroveGuy (talk) 23:49, 28 September 2012 (UTC)

analog computers? Are quantum computers binary? --JBL (talk) 00:00, 29 September 2012 (UTC)

Discussion of the name of this article[edit]

Because of a recent change in the title of this article, which I undid, I've started a discussion at Wikipedia_talk:WikiProject_Mathematics#Binary_numeral_system.2FBinary_number_system. Michael Hardy (talk) 19:21, 19 December 2012 (UTC)

Requested move[edit]

Consensus is that the article be moved, and two well-formed arguments are given for the new title being Binary number, with no objections. Drmies (talk) 04:16, 3 January 2013 (UTC)

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Binary numeral systemBinary number system – A numeral is a symbol for a number. A numbering system is the mathematics of using a particular number as a base. By over 40 to 1 books written choose "binary number system", not numbering, and not numeral. Apteva (talk) 19:54, 19 December 2012 (UTC)


Feel free to state your position on the renaming proposal by beginning a new line in this section with *'''Support''' or *'''Oppose''', then sign your comment with ~~~~. Since polling is not a substitute for discussion, please explain your reasons, taking into account Wikipedia's policy on article titles.
  • Weak oppose. A number system is a system of numbers (real, rational, integer, hyperreal, etc). A numeral is a means of representing a number. So the title with the word "numeral" as opposed to "number" is more precise and correct, since the actual numbers that are described by binary numerals are precisely the same numbers that are described by any other system of numerals. I'm open to debate about whether the word "system" is needed. Sławomir Biały (talk) 21:27, 19 December 2012 (UTC)
    • That would be correct if the article was about the two numerals used, and did not go on to add system, and include the mathematics of the binary numbering system. There is a difference between the symbol used to describe a number, the numeral 1, and the number one, which is unary. Apteva (talk) 22:07, 19 December 2012 (UTC)
  • Support - Consistent with the policy on common names which says that the title should agree with the most common use in reliable sources. A Google Scholar search turns up 4150 results for "binary number system" and 391 for "binary numeral system". No less an authority than Donald Knuth has a chapter entitled "Positional number systems" in The Art of Computer Programming. Volume 2. RockMagnetist (talk) 21:43, 19 December 2012 (UTC)
  • Support. But I would prefer "binary number" (i.e. for the title to refer to the individual pieces of notation described by the system, rather than to the system as a whole) for the same reason that we have articles titled Leopardus pardalis and not Leopardus pardalis species, addition and not addition operation, real number and not real number field, etc — the excess verbiage just makes the title look more technical and intimidating without adding any useful meaning. Additionally the Google scholar hit count argument applies even more strongly for this as it does for the proposed move. —David Eppstein (talk) 22:17, 19 December 2012 (UTC)
  • Support per WP:COMMONNAME which is a policy I fully agree with. I also agree with David Eppstein that the 'system' part should just be dropped to leave 'Binary number'. System doesn't add anything useful and doesn't disambiguuate with some other common term. Dmcq (talk) 22:34, 19 December 2012 (UTC)
    The problem with that is the scope of the article would be restricted to numbers like 1010 and 1000010101011, and would not properly include how they are used or what they are used for. Addition is an operation. Leopardus pardalis is a species. The field of real numbers is covered in the article real number. We do not need to add explanatory words like that. Apteva (talk) 02:09, 20 December 2012 (UTC)
    WP:TITLE says 'Precision – Titles usually use names and terms that are precise enough to unambiguously identify the topical scope of the article, but not overly precise.' WP:SCOPE says 'The lead, ideally the introductory sentence or at least introductory paragraph, of an article, should make clear what the scope of the article is.' The scope of an article is not narrowly determined by the title. Dmcq (talk) 02:37, 20 December 2012 (UTC)


Any additional comments:
Why are people using numeral for things like that? We're not here to correct the world, we're here to say what's said in reliable sources. Just check out google ngrams for positional number system compared to positional numeral systems. [2] There's no comparison and numeral just sounds awkward there. Dmcq (talk) 23:36, 20 December 2012 (UTC)
Using singular rather than plural makes a difference to your comparison [3] but I think it still makes the same point. —David Eppstein (talk) 01:29, 21 December 2012 (UTC)
I tried numbering and number wins. Apteva (talk) 08:35, 23 December 2012 (UTC)
The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

merge from base 32[edit]

Propose a merge from the base 32 stub to binary number as it is essentially a notation for five binary bits. It would best be covered as a section of the binary number article. Bcharles (talk) 03:11, 12 July 2013 (UTC)

why merge base 32 into binary ? jjbernardiscool (talk) 15:52, 27 July 2013 (UTC)
I think the best resolution would be to redirect "base 32" to "base32" encoding scheme, add an "about (positional numerical system)" template which links to "binary number" at the top of the "base32" article, and include a section on "base 32 numerical system" in the binary article. This is similar to how "base 64" is handled.
It is helpful to distinguish numerical/mathematical systems from specific applications such as a data encoding scheme. Base 32 is not notable as a numerical system, it amounts to a compact notation for 5 binary bits, a section of the binary article could fully cover the content of this stub and link to its application for data encoding. Bcharles (talk) 17:19, 28 July 2013 (UTC)
  • oppose I see no reason why this would be an improvement. I would also oppose merging to Base32 - its use as a real, if obscure, counting system has no overlap with the encoding scheme. Andy Dingley (talk) 11:03, 19 September 2013 (UTC)
  • oppose I agree - Base-32 is a real counting system which is in use. However, why not mention it in this article (if it hasn't already been?). Digitalhamster (talk) 08:14, 14 April 2016 (UTC)

Square root section[edit]

Could someone who knows about it work on Section 5.5, Square root? The opening sentence and equation are very unclear to me, and the subsequent algorithm needs some commentary. Thanks. (talk) 13:35, 28 June 2014 (UTC)

I'm going to tweak it myself. Further edits would be appreciated. (talk) 18:19, 29 June 2014 (UTC)

In music[edit]

I think there should be a section titled In music explaining how music relates to an amount of time expressed in binary based on the way its sections are divided up. I know music also sometimes uses a mixed base where the last digit is expressed in base 3 and the other digits are expressed in base 2. Blackbombchu (talk) 00:43, 13 November 2014 (UTC)

It should also discuss theories of how human brain evolved that made it use binary to make music or even why it didn't instead have make disorderly music that plays only on whole number beats but there's no distinguishing feature between an odd number beat and an even number beat. Blackbombchu (talk) 00:57, 13 November 2014 (UTC)

You mean you never heard anything in 3/4? Or 6/8? Or something like Musorgsky's Promenade from Pictures at an Exhibition? (・・? I guess in a sense traditional Western music notation is binary (at least, for note values below a half; from half notes upward the notation is more ad hoc), because the way non-binary rhythms like triplets, quintuplets, and septuplets are represented (by writing "3", "5", or "7") is really an extension, akin to writing vulgar fractions instead of non-terminating decimals. But the groupings are not always binary, and by the time you get to the phrasing-level, there's nothing that forces you to have eight-bar units instead of, say, seven-bar units, save an already long-eroded tradition, that one does not need to follow anymore. Double sharp (talk) 14:13, 3 January 2016 (UTC)

BIASED History of Binary code[edit]

There is no mention of Pingala in this section. His work is an important ancient contribution to the concept. Here's a summary of the information missing from this article:

An Indian scholar Pingala (2nd century BC) developed the binary system for describing prosody in his Chandaḥśāstra. The Chandaḥśāstra presents the first known description of a binary numeral system in connection with the systematic enumeration of meters with fixed patterns of short and long syllables. The procedure of Pingala's system is as follows:

  • Divide the number by 2. If divisible register 1, otherwise register 0.
  • If first division yields 1 as remainder, add 1 and divide again by 2. If fully divisible, register 1, otherwise write 0 to the right of first 1.
  • If first division yields 0 as remainder that is, it is fully divisible, add 1 to the remaining number and divide by 2. If divisible, register 1, otherwise register 0 to the right of first 0.
  • This procedure is continued until 0 as final remainder is obtained.
Pingala is mentioned. His contributions are described in great detail in the fourth paragraph. As per David Eppstein's edit summary, there's just no reason for him to appear at the very beginning of the article when other binary systems clearly predate Pingala's work. For example, one of the binary-like systems described in the article dates back to the 9th century BC. Why should it appear after Pingala? It makes most sense to structure the history section chronologically.--Rurik the Varangian (talk) 17:27, 25 April 2015 (UTC)
where would you suggest we put the section noted above?-JG (talk) 18:43, 25 April 2015 (UTC)
As the comment you responded to clearly states, Pingala's contribution are already there, in the history section. They do not need to be added anywhere, because they are already there. But, the history section is also missing any mention of binary systems in Ancient Egypt, both Ancient Egyptian multiplication and the Horus-eye system that was a predecessor to Egyptian fractions. Those should be added, too. —David Eppstein (talk) 18:47, 25 April 2015 (UTC)
Are you volunteering or is this something that we should work on together?

Eye of Horus in History section[edit]

The article regarding the eye of horus states that the idea that it was used for mathematical purposes is false. If that is the case the picture regarding it should removed and the relevant section edited. Personally I have no knowledge in this area I was merely jumping from link to link, I just thought I ought to mention it. Sonzaisuru (talk) 02:34, 4 June 2015 (UTC)

Just to clarify: the source clearly states that these symbols were used for binary fractions. The disputed part is whether they also depict pieces of an eye or are just abstract symbols. And it also says that the idea that these symbols form an eye has "become standard among historians of mathematics". So my impression (with no more knowledge than you beyond what we have both read from the same source) is that there is a disagreement between scholars, or at least between that source's author and other scholars. I think this article is not the right place for describing this controversy (Eye of Horus seems most likely) but that we should describe both sides there rather than trying to take sides ourselves. As for what we put in this article, although we should mention the Egyptian numbering system (that part seems to be uncontroversial and highly relevant), I don't think the question of whether or not these symbols can be put together to form an eye is particularly important for binary numbers. So we should probably avoid mentioning this question. That may mean removing the image as you suggest, since even if we don't say that the image was used historically in that way, just having the image there implies it. —David Eppstein (talk) 02:51, 4 June 2015 (UTC)
Rather than removing referense to eyes alltogether, I'd recommend (in general terms) to (1) cut it down, (2) qualify it ("allegedly", "possibly", "according to some scholars"), and (3) link it to a discussion elsewhere on Wikipedia. One reason for this is that it is likely to creep back into this article anyway as other editors come by, so why not do it right in a way that is likely to be stable.-- (talk) 06:23, 4 June 2015 (UTC)
Good idea. I've taken a stab at it; feel free to reword. —David Eppstein (talk) 07:00, 4 June 2015 (UTC)

Fantastic example of an analogue binary clock[edit]

I have found a fantastic example of a binary analogue clock in a programme on BBC iPlayer. However, I am unsure of the copyright implications of posting a still image of it. I mean, there's fair use and all that, but I don't know where to start. It is such a brilliant example of such a rare thing, that it would be sad if we could not find a way to allow a still image of it.

Those of you who have access to iPlayer, please check out "How They Dug The Victoria Line" (It's in the Archive programmes section, so won't vanish any time soon). An analogue binary clock is featured at 36:56, measuring in hours on the inner dial, and half-minutes on the outer dial. Having been in trouble for just going ahead and doing my own thing on Wikipedia before (well, we all try to learn from our mistakes...) I thought I should bring it up on the Talk page for discussion first. Meaning of Fife (talk) 22:31, 16 June 2015 (UTC)

Date of binary system creation[edit]

Can someone cite a reference for 1679 as a date of release of Explication de l'Arithmétique Binaire... the document is dated on 1703.... and one more thing... I came across a book ISBN: 978-84-323-1667-8 (p. 30-32) that suggest that Leibniz created the binary system before 1673 marked by the date of the dead of the Maguncia Bishop, employer or main source of income for Leibniz, event that urge him to create a device (or first calculator) to sell it, obtaining a steady source of income .Ayala.canela (talk) 00:39, 22 December 2015 (UTC)

ternary-coded decimal numeral[edit]

I'd like to know how to create a ternary clock using LEDs to express ternary values, so that I can add it to the ternary wikipedia page. Backinstadiums (talk) 01:50, 3 December 2016 (UTC)

You should only add that sort of thing to Wikipedia articles when it can be reliably sourced — otherwise it is original research, something that is fine on other sites but not here. I doubt you will find much sourcing for ternary-coded-decimal digital clocks. —David Eppstein (talk) 04:58, 3 December 2016 (UTC)

Title seems a little misguided[edit]

IMHO the title should really be binary numeral system (as I think it used to be) or just binary or binary (mathematics), because you don't so much have "binary numbers" as real numbers that happen to be expressed in base 2. You could just as well express the reals in base 3 or base 4 or base 10 or base 720720 if you invented enough symbols, but that doesn't mean that we somehow have "base-3 numbers" or "base-720720 numbers", as those would be exactly the same set; only "numbers expressed in base 3" and "numbers expressed in base 720720". Double sharp (talk) 14:37, 21 February 2017 (UTC)

Have you looked above at #Requested move? JBL (talk) 14:46, 21 February 2017 (UTC)
Ah, OK, I just noticed that. Still, since that one was conducted just over four years ago, it may be reasonable to open another one now. Double sharp (talk) 14:53, 21 February 2017 (UTC)
Yeah, I didn't mean that the question was settled always and forever, just that it has been discussed before. (I do not personally have a strong feeling one way or the other about what the title should be.) --JBL (talk) 21:35, 22 February 2017 (UTC)

Computer Binary[edit]

I think that the binary system used here is different from the binary of computer science. May I request to add a section in this article or computer binary is already included in another page? Redpo888 (talk) 12:49, 24 May 2017 (UTC)

I don't think there's anything different about binary in CS. In what way do you think it's different? --Deacon Vorbis (talk) 12:58, 24 May 2017 (UTC)
Computer implementations treat negative values in various ways - is that what Redpo888 has in mind?-- (talk) 17:14, 24 May 2017 (UTC)
That's handled at Two's complement. —David Eppstein (talk) 18:24, 24 May 2017 (UTC)

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About the lead of history part[edit]

This is the my previous edition [4] and it was reverted by others. Most books divide the history of binary to before and after Leibniz. Every development before Leibniz was claimed as the predecessors. For example, See "History of Binary and Other Nondecimal Numeration"[5], "Development of the Binary Number System and the Foundations of Computer Science" [6] and "Gottfried Wilhelm Leibniz invents the Binary System" [7]. Hence, the common theory considered Leibniz as the start of modern binary. If we defined all European scholar who studied this area before Leibniz as the modern binary, why don't we define the development in Egypt, China and India as the real start of modern binary number so that we can become more neutral for each region. —Miracle dream (talk) 15:50, 28 December 2017 (UTC)

"Most books" are ignorant of modern scholarship, which points to the work of Harriot and Caramel as providing complete descriptions of modern binary notation before Leibniz. Leibniz was certainly important, as his contributions are the ones that stuck, but it is inaccurate and wrong to write the summary paragraph of the history section in a way that assigns him all the credit and completely snubs the other pioneers of the subject. To put it another way: that paragraph is supposed to be a summary of the whole history section. Your change rewrites it to completely omit any mention of one of the main subsections in the section. That is not an improvement. —David Eppstein (talk) 21:10, 28 December 2017 (UTC)

Leading with the history?[edit]

Are you kidding? Most people are here looking for a mathematical explanation -- not a silly history lesson. (talk) 16:17, 4 April 2019 (UTC)

I know where you coming from. As a C/S teacher it is hard to forget the past and explain current affairs moving history lessens to where they belong. But, however binary numbers where at the cradle of electronic computing this was not always the case and binary was invented outside the realm of electronic computing. Having said that, I do agree that currently binary numbers exclusively bear a meaning in electronic computing and the history is a fringe show. It should be move to a section history of binary theory and not be at the core of the article. Theking2 (talk) 14:31, 21 August 2021 (UTC)

colums order: Fraction, Binary, Decimal, Fractional approximation[edit]

The current order Decimal - Binary misgui is misleading, leading to believe that 1/1 is 0.9999... is the norm. --Backinstadiums (talk) 20:32, 13 July 2019 (UTC)

binary fractions in wikibooks. --Adam majewski (talk) 23:14, 13 July 2019 (UTC)

Negativ numbers[edit]

Two complements is mentioned but I do believe it should have at least a section explaining the basics and eventually referring to Two's_complement Theking2 (talk) 14:15, 21 August 2021 (UTC)