Talk:Binary numeral system

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Is this really the most basic numeral system? I can imagine base-1, where, for example, the number five would be 11111. Evercat 01:15 2 Jun 2003 (UTC)

Good point! So you can use a place to distinguish 1 and 2 like
  • 1 -> 1
  • 11 -> 2
  • 111 -> 3

and so on. I didn't realize that. But I have never heard of base-1. Anyway I remove the claim of most basic numeral system at a time. -- Taku 01:22 2 Jun 2003 (UTC)

Ah. I've just discovered its page. Unary. Evercat 01:24 2 Jun 2003 (UTC)

Heh, I totally missed that too. Having studied computer science for 6 years, when I see a claim that binary is the most basic numeral system, I don't even bat an eyelash... good call Evercat! -- Wapcaplet 02:08 2 Jun 2003 (UTC)
PS - Even better, it still follows the interpretation given for the others. 11111 is 1 times 14 plus 1 times 13... 1+1+1+1+1=5. Cool. -- Wapcaplet 02:09 2 Jun 2003 (UTC)
Yeah, it really is base-1, I think. Evercat 02:10 2 Jun 2003 (UTC)
But if base-1 were really the simplest, how would one represent 0?

Like this:

But seriously, the claim that binary is the "simplest" was removed from the article; even now it'd probably only make sense to call it the simplest "practical" system, but practicality is fairly dependent on purpose (for a 4-year-old learning to count on his fingers, the unary system is more practical, and he's not likely to need zero, since with zero things there's nothing to count). Unary is also limited in that it doesn't work well for real numbers (what would 111.111111 mean in unary?) These limitations are just a result of it being, in my current opinion, the simplest numeral system. Truly, though, it's hard to deny that unary is a numeral system, and that it is simpler than binary. -- Wapcaplet 22:38, 10 Sep 2004 (UTC)

Well, even binary doesn't work well for real numbers (what is 101011.0100110?) I assume you could use most of the same tricks used in binary floating point on unary...  :-) JesseW 09:00, 21 Oct 2004 (UTC)

Easy; 101001.0100110 binary is:

  • 1 × 25 +
  • 0 × 24 +
  • 1 × 23 +
  • 0 × 22 +
  • 0 × 21 +
  • 1 × 20 +
  • .
  • 0 × 2-1 +
  • 1 × 2-2 +
  • 0 × 2-3 +
  • 0 × 2-4 +
  • 1 × 2-5 +
  • 1 × 2-6 +
  • 0 × 2-7

or 41.296875. Assuming unary works like other numeral systems (position indicates power), and that the decimal point is only there to show where the power of 0 is, then every position in unary is multipled by 1 to some power; 111.111111 would be nine, then. Alternatively, the decimal could be used to separate whole numbers from fractional numbers, in which 111.111111 could mean 3.6. There's no doubt, at the very least, that unary would have to work differently than other systems. Adding in binary involves aligning two numbers and summing their individual digits; adding in unary is just a simple concatenation. Multiplication would probably have to work differently, too - make a table with width/height equal to the two numbers to be multiplied, and fill it with 1 to get the result of multiplication. Though, I guess there might be a way to contrive an arithmetic procedure that works similarly for all radixes... I really don't know :-) -- Wapcaplet 16:56, 21 Oct 2004 (UTC)

But the true point of binary is to have a simple system that does not ocuppy much RAM, and also the system of two digits is based on on/off. Base 1 is definetly more simple, but is not a system that would work for computers.--Matthew138s

Whee, converting is fun! I'm starting to wonder whether we should have some articles like Converting decimal to binary, Converting binary to octal, though a lot of times the conversion is simple and boring (like binary/octal, binary/hexadecimal). It may be going overboard to have all this conversion stuff in here, but I figure it warrants a brief mention, since binary is the lingua franca of the computer world. It's good to know how to convert it to other stuff, and outside of conversion there isn't a whole lot to say about it. Anyone else have ideas? What do people think of the organizational scheme I've started? -- Wapcaplet 21:45 2 Jun 2003 (UTC)

I think use of a table is good, that is, the table where each column is a numeral system from unary to hexadecimal and each row is a number probably from 0 to 16? -- Taku 01:59 3 Jun 2003 (UTC)
That is probably a good idea. Though, we'll have to be sure to make it clear that there is not always a direct substitution-type correspondence, as there is with octal and hex. Ex:
bin   dec  oct  hex
0000  0    0    0
0001  1    1    1
0010  2    2    2
0011  3    3    3
0100  4    4    4
0101  5    5    5
0110  6    6    6
0111  7    7    7
1000  8    10   8
1001  9    11   9
1010  10   12   A
1011  11   13   B
1100  12   14   C
1101  13   15   D
1110  14   16   E
1111  15   17   F
You can replace blocks of three bits with a corresponding octal digit, always, but not with decimal (since its radix is not a power of 2). So I'd be worried that a table of that sort might mislead people a bit unless we're careful.
We definitely need a good section on binary arithmetic, preferably towards the top. It's at the bottom now, but a nice overview of addition, subtraction (possibly two's complement), multiplication and division would be good. Some discussion of bitwise operations (AND/OR/XOR) would be good too. -- Wapcaplet 02:06 3 Jun 2003 (UTC)

I was looking for something else and noted that somebody "upstairs" thinks that this page needs further editing. I looks pretty good to me, has anybody any idea of what people might consider the rough spots to be?

Patrick0Moran 01:29, 1 Aug 2003 (UTC)

The section on arithmetic, especially, is in need of work. Multiplication is only briefly discussed, and division not at all. I don't know if there perhaps needs to be a separate article for binary arithmetic, similar to what we have for Bitwise operation. Integral data type covers some of the complement stuff; Computer numbering formats is probably the most comprehensive overview of the subject. There's probably a lot of refactoring to be done of many articles, so they don't overlap too much, but I am not sure the best way to do it. Perhaps we need a WikiProject for computer numbering formats. It's certainly a huge topic! -- Wapcaplet 20:02, 7 Aug 2003 (UTC)

It's interesting,but I don't think the "binary image" really improved the article, so I removed it. It's not exactly an intuitive representation and I don't think it will help anyone understand binary any better. Computer novices might find it interesting-looking, but it could also be confusing, etc. Daniel Quinlan 04:22, Sep 20, 2003 (UTC)

I put them in as a "visual overview" to counting in the numeration systems. Counting in these systems can be modelled via an odometer, and the action of this can be visualised. The reason I put them in was to show the growth in the length of the representation (logarithmic) and also the repetitive fractal-like pattern occurring on all scales. The base-phi one was the original, and has pleasing irregularity. Maybe I should have also added comments to describe their intent. If anyone feels that such images belong, or just wants to look at them to decide what they are, look at http://www.wikipedia.org/wiki/Image:Binary.png http://www.wikipedia.org/wiki/Image:Ternary.png and http://www.wikipedia.org/wiki/Image:Phinary.png --AndrewKepert 05:08, 20 Sep 2003 (UTC)

I don't want to be harsh, but speaking as a person with a degree in Computer Science, it's just about least intuitive visualization of any mathematical concept that I've ever seen. It might be worth an interesting web page somewhere, but it really doesn't help explain the base concepts at all. And while comments would help explain what they represent, that's really missing my point. Daniel Quinlan 06:37, Sep 20, 2003 (UTC)

Ordinarily I'm against "summary executions", but having had a look at these several images I have to agree that most people would not even have the visual acuity to see how they are built up. I didn't see any image of an odometer, but it sounds to me like a useful idea. Most people do not have a clear idea of what they are doing when they write 4959 or 4a4f9 or 100101000, and they can really only operate in decimal -- and only then on the basis of habit. So the image of a wheel the moves one notch for each tick and that kicks the next wheel once as it rotates back to its own zero position will give such a student a clear idea of what is going on. We had one of the first digital clocks in our kitchen in the 1950s. It worked that way, and the wheels were large enough for me to see clearly what was going on.

We should not assume that we are writing for people who already know what we are talking about. Unless people on the "no child left behind" train are moving both faster and in better cars than I think they are, most high school students who view the article on binary numbers will be trying to read 10110 as ten thousand one hundred and ten even as they try to get their minds around the "fact" that it refers to some other quantity of objects.

I'm 100% in favor of improving the article and making sure anyone can understand it. I'm the last person who needs an explanation of the binary numeral system -- I want the article to be simpler and more intuitive, not less. When explaining a potentially difficult subject, such as binary math, I think the last thing someone needs is someone trying to explain this complicated image. I gave a "summary execution" because I'm quite confident that the images did not help bring the article towards that goal, quite the opposite. If an image helps explain a concept, especially in a way that text cannot, then by all means, we should keep it. Daniel Quinlan 08:14, Sep 20, 2003 (UTC)

By the way, in the 50s there was a series of "Tutor Texts" (I think that was a trade name), and the first of them was excellent. It was called Computer Mathematics and it was extremely well done. It covered binary in detail and then showed how things were done in base 8 and maybe base 12 or base 16. I wish I still had my copy, as it was an exemplary "teaching machine" device.

Patrick0Moran 07:16, 20 Sep 2003 (UTC)

Fair enough. I generated the base-phi image for my own benefit (I was trying to visualise how "carry" worked in a system where it "spills" in both directions). As a mathematician (also a CS major from a while back) I like to have several ways to view a concept -- algebraic, algorithmic, visual. I put it up because I thought it was interesting. My initial response to Daniel was to see what others thought. If the consensus is "no" due to the reasons stated, then that is fine. I will think about whether I write something more coherent on odometers. (the image is sort of like ticker tape from an odometer... 8-) --AndrewKepert 12:46, 20 Sep 2003 (UTC)~

Is there already a sort of schematic image of an odometer and/or digital clock somewhere that we could just borrow (legally)? I'm fairly good at mechanical drawing, but doing it right could take a fair amount of time. I think that a hundreds disk, a tens disk, and a ones disk arranged horizontally and showing the appropriate "kicker knobs" could be very helpful when compared with a similar counter of a few disks for representing binary numbers. If we could animate the images it would be even neater.

Sorry about my mention of "summary execution" above, but that's the way it felt to me at first. After I actually saw the graphics I decided that the little vertical strips didn't say anything to me, although they were attractive in some way, and that they probably needed to go.

Patrick0Moran 01:06, 21 Sep 2003 (UTC)

Here's what I would suggest (note, I'm talking about ideas, which are not copyrightable, not text, which we can't copy willy-nilly):
  • Read web pages on binary numbers and see how they explain binary math. Don't set out on a quest to find an image or an odometer image. Quest to find good explanations.
  • Read introductory computer science and mathematics textbooks and see how they introduce the concept. Also try encyclopedias.
Which seem to be the best explanations? Enlist the help of a non-technical person you know to evaluate the best few. Just be careful, if you set out after images, merely because I removed an image or because someone added an image not to everyone's liking, your focus may not result in the best possible article. Odometer images may not help explain this concept better, or maybe they will. I think it's better to figure out what works well (not necessarily limiting ourselves to paper formats, of course) and do that. Daniel Quinlan 01:53, Sep 21, 2003 (UTC)
One more thing: bear in mind that not every web browser can display images and there are also sight-impaired readers. Daniel Quinlan 01:55, Sep 21, 2003 (UTC)

Sounds reasonable to me. I'm not stuck on images. Speaking of sight-impaired users, when I use my IBM to view these pages I get huge characters as a default, and I have to hit control-[ four times to get the page to manageable size. Switching to the Wiktionary at that point gives characters that appear to be about 2 point, and the same thing happens if you send somebody e-mail from within the Wikipedia environment. But on my Mac I don't see this problem. Maybe it's just because my IBM is running an old copy of Netscape (but then so is my Mac), but it looks like something funny is going on.

If a web browser can't handle images the user will by now be used to the situation, and also probably will experience difficulties visiting most WWW sites. I guess it would not be a good idea to put crucial info in an image that was not covered elsewhere. As for the sight-impaired user, it is always possible to have a clickable link to a larger image. And sight-impaired users are probably adept at grabbing an image and blowing it up to a size they can view. That can't be any fun, however.

There is much to be said for presenting the same information in different formats. Some people (Einstein included) think in images first and then laboriously do the math. Other people (like me) can't visualize much more than a circle, and need to get the info in words. Some people may even need spoken words.

The Tutor Text I mentioned is excellent. I wonder if any libraries have kept it. Maybe they have regarded it as an ephemeral publication and purged it from their stacks. The two other books in that series were dogs and so the whole series may often have been purged.

Patrick0Moran 02:15, 21 Sep 2003 (UTC)

Don't forget that there are some sight-impaired users who cannot see. Having larger images or "clickable" anything is not going to help them :-) I believe what Daniel is talking about is finding a good way to explain things in text, rather than relying too much on anything visual.
I agree that some work could be done on making this article more accessible to the layperson. Perhaps all we need are a few adventurous non-mathematicians who would be willing to critique the article's weaknesses. List it on Wikipedia:Peer review, perhaps? -- Wapcaplet 17:00, 21 Sep 2003 (UTC)

Some thoughts about refactoring:

We have an article Computer numbering formats, which was essentially lifted from another public-domain source. It's a good introduction to the binary system in general, but duplicates a lot of what we have here, and is long, ugly, and unwikified. Probably one of our first goals should be to merge that article with Binary numeral system; the term "computer numbering formats" is pretty vague, and that article should probably redirect to Binary numeral system.

A whole article could probably be written simply on how to count in binary. One could simply say "counting in binary is exactly the same as counting in decimal", which is true, but the concept is so new to many people that a thorough kindergarten-level explanation would be quite useful. I propose Counting in binary be split off from this article for that reason.

An entire article, possibly several articles, could be written on binary arithmetic. Addition, subtraction, multiplication and division should of course be covered; other techniques such as left/right shifting to perform multiplication/division by powers of two could be covered here as well. I propose Binary arithmetic be split off for this purpose.

We've already split off Bitwise operation to discuss AND, OR, XOR, etc. That article currently discusses shifting, which is often classified as a bitwise operation even though it is not. That article is probably good as it is, although more could certainly be added by way of explaining some of the really cool (to geeks, at least) things you can do with bitwise operations, such as AND for checking flags, OR for setting multiple flags, XOR for clearing a register, etc. I'm not entirely sure that bitwise operations should be discussed at all in this article, since they have very little to do with the binary numeral system.

Comparing binary with other numbering systems could take volumes. Obviously, some comparison to decimal is good to have; hex and octal are also useful, since those are also commonly seen in computing environments. Any more than that may be excessive. Since the splitting off of the other topics mentioned above would result in an article dominated by the "Binary compared with..." sections, I'd probably vote to split that section off also, unless we can fill the void with something else useful. It may also make sense to merge Binary coded decimal with this article (though it is not really part of the binary numeral system), as well as Binary coding, which seems to be an aborted and rather poor attempt at summarizing binary in general. Truncated binary encoding may also need refactoring. Perhaps these can be brought under a different heading, along with things like Gray code, which talk about using binary as a form of symbolic code, rather than as a numeral system.

A good thing to keep in mind is that this article is about the binary numeral system. That is, it's about how to represent numbers in a symbolic format with a radix of 2. It should definitely not talk much, if at all, about binary versus ASCII files, hardware implementations of binary numerals, binary trees, etc.

Any thoughts? -- Wapcaplet 17:31, 21 Sep 2003 (UTC)

The important question is the depth of the article; do we really want to cover those topic throughtly? Articles in wikipedia should not be something like you found in textbooks. The purpose of them should not teach people how to handle binary numeral systems. I think it would be nice to focus on more about topics interesting for the general audience, things like 1.1 cannot be expressed in binary system, history. I'd rather create an article that is compact but well-written than spliting off subtopics. Subprogram is a good example. Many subtle discussions are completely omitted, instead the article discusses why you want to use subprograms and will discuss history. Perhaps detailed discussions can be immigrated to wikibook. -- Taku 18:24, 21 Sep 2003 (UTC)

Yes, I think this topic should be covered thoroughly. Not to make the already-present geek-bias in Wikipedia even stronger, but to present the topic(s) in a way that non-geeks can understand. It's unfortunate that, in a world so centered around computers and digital devices, the majority of the population still has barely any notion of how the underlying numeral system works. Granted, most people don't care, but for those that do, it'd be nice to provide an accessible introduction for them. We can also focus on the interesting things that arise, such as 1.1 (decimal) being an irrational number in the binary system, but I hardly think that's a matter of interest to a general audience! At any rate, in order to even begin explaining that fact, it's necessary to explain in depth how the binary numeral system works. What good is an article about the binary numeral system if it doesn't even tell the reader how to use it or why it's useful? -- Wapcaplet 20:36, 21 Sep 2003 (UTC)

correction: 1.1 is not irrational in binary numeral system. It is a rational number (11/10) and consequently its representation in any radix-based numeration system is a numeral that either terminates (no) or recurs (yes). In this case 1.1ten = 1.0 0011 0011 0011 ...two. See decimal or my comment below.

I like the above observations. I have seen some entries where so much technical jargon is used that only the person who is already an expert in the field can read the discussion. It would be hard to find anything incorrect in such an article, but it would also be hard to find anything useful to the general reader who wants to find an entryway to a new subject.

I am not a mathematician. I did not have a good background in mathematics, and proved that to myself when I got into a physics major at a major univeristy -- and back out again. Nevertheless, I am interested in the subject. I can remember having 3 calculus teachers. The first-semester calculus teacher worked calculus problems at blinding speed, asked, "Does everybody understand?" Saw the engineering students in the front row all nod "yes", and went on to the next problem. I never learned a thing from her lectures. The next trimester I had a teacher from South America who never wrote more than a single blackboard full of stuff per class session, but he let students understand why he was performing the operations that got him the answers. Frequently Dr. Restrepo would finish with a problem that looked like it would take all night to calculate and then he would prod us into seeing how we could do it in our heads. The third trimester I was stuck with the chairman of the math department who surely could do all the math but could not teach much better than the first teacher.

Eventually I remediated a lot of my math education on my own, and what I learned is that it is not mastering the operations or remembering the formulas that is crucial, but understanding why the operations and the formulas are as they are.

The Computer Mathematics book I mentioned earlier is an exemplary textbook. It was followed by many "teaching machine" texts that were poor to middling, unfortunately.

Anyway, if memory serves me at all after 50 years, the key to my own understanding of a lot of this stuff was understanding what we are actually doing when we count. That's why I agree with the idea that this article should be supplemented by one on "counting in binary." And, is there an article on "counting"?

As an item of possibly related interest: There was a great deal of discussion a few years ago on why Chinese students seemed to have so much less trouble with math than did American kids. One of the advantages they have is that from the very beginning they understand the format, J(1000)+K(1000)+L(100)+M(10)+N -- because that is the way you say a number in Chinese (and Japanese too) (except that you don't actually say the "plus" signs): J wan K qian L bai M shi N. To me, growing up, it was a major discovery to realize that "fourteen" meant "four plus ten" and that "ten" and "teen" were really the same word. Maybe some people define "fourteen" as "the number that comes after thirteen" all their lives and never get beyond that truism.

Patrick0Moran 18:55, 21 Sep 2003 (UTC)

Just checked. There is an article on counting. I do not believe that it can be very helpful. It doesn't disclose the reasons that counting was such a neat discovery.

Patrick0Moran 19:01, 21 Sep 2003 (UTC)

My little images have provoked much discussion, and now the structure of the whole page is in question. 8-) As I see it, the page starts out okay, but then wallows. I would go:

  • intro (currently good)
  • representation of whole numbers/integers (currently ok)
    • Uniqueness of representation for whole numbers (This is really a key point - there is a one-to-one correspondence between bit strings and whole numbers. This could be mentioned, then explained in the conversion section)
  • counting (needs work, and maybe a discussion of the odometer)
  • arithmetic (ok, better examples would help, or maybe a separate page)
  • conversions from other representations such as decimal (condense, and maybe shift octal and hexadecimal to a page where appliations to computing are being discussed)
  • Representation of real numbers -- rewrite "fractional" section to include recurring expansions, irrationals, non-uniqueness of repn: 1=0.111111..., somewhat akin to what I did in decimal and golden mean base.
  • applications to computing, such as bitwise ops, one-complement vs two-complement for negs, octal and hexadecimal representation, binary encoded decimal, etc -- list these with links to external pages

HTH --AndrewKepert 00:25, 22 Sep 2003 (UTC)

Now that you focus attention on that introduction, I don't think it is as effective as it could be. The average reader will be snowed by that first paragraph and will be looking for something to latch on to. I think it might start with things that readers might have experienced and might find familiar. Most people have seen binary written out in 1s and 0s. That might be compared to a picture of lights on an antique computer showing the same binary number, and by a sound recording of a corresponding series of prominent beeps and mild clicks. Patrick0Moran 01:05, 22 Sep 2003 (UTC)

Binary can be considered the most basic practical numeral system, especially for mechanical and electronic devices (the unary system is simpler, but is impractical for computation).

I think we have to be clear on what practical and simple are supposed to mean; otherwise, the above sounds like a contributor's own POV. Did the ancient pre-literate peoples of the world use unary to count herds of sheep? Young American children sometimes give their age by holding up "this many" fingers; that's certainly practical for kids too young to understand numerals. On the other hand, when I teach math I always introduce addition by using unary. You can see 2 + 3 visually by combining a pile of 2 tokens with a pile of 3 tokens, and then counting the new pile. --Uncle Ed 16:26, 24 Sep 2003 (UTC)

Little rewording just would help well. See my changes. -- Taku 04:54, 25 Sep 2003 (UTC)

I rephrased the intro a bit and removed some of the questionable statements (the bits regarding unary or binary being the simplest, and the vague stuff about early 20th century computers (were there digital computers in the early 20th century?) being based on on/off Boolean algebra). A clear, concise, and correct explanation of why computers use binary would be wonderful, but I am not the person to write it. Such an explanation likely belongs in the "History" section, anyway. -- Wapcaplet 20:15, 25 Sep 2003 (UTC)

p.s. - Also, I think it's fairly important to avoid using literal numerals in the prose sections of this article, especially numerals with a 1 or 0 in them. One of the examples I rephrased was about how 999 (decimal) has 10 digits in binary. The mathematically precise approach would be to subscript every literal, but to me it seems more natural to say "ten", "two", etc. to avoid any ambiguity, particularly before the notation of subscripting has been explained. It should always be totally clear what number or system we're talking about. -- Wapcaplet 20:22, 25 Sep 2003 (UTC)

Temporarily removed the following text:

In binary, multiplying by two amounts to moving all digits one position to the left and inserting a leftmost zero. This observation leads to the following faster conversion algorithm, a variant of the Horner scheme which does not require the computation of the powers of two: start with a result of zero and scan over the binary string from left to right. If the current digit in the string is a zero, multiply the result by two; if the current digit is a one, multiply the result by two and add one. Continue until you hit the right end of the binary string. Example:

To convert the binary number 101101001 into decimal: 0 × 2 + 1 = 1, 1 × 2 = 2; 2 × 2 + 1 = 5; 5 × 2 + 1 = 11; 11 × 2 = 22; 22 × 2 + 1 = 45; 45 × 2 = 90; 90 × 2 = 180; 180 × 2 + 1 = 361.

End of removed text

I removed the above because of its predominantly "how-to" nature, which is out of place in the current article. I hope it may be re-inserted into the upcoming Binary arithmetic article, since it is quite interesting. Additional information in the arithmetic section may also need to be removed to that location. -- Wapcaplet 20:50, 3 Oct 2003 (UTC)

Right now, some closely related information is scattered across a maze of articles. There is Binary numeral system, there is two's complement, there is a near-stub called Binary arithmetic, there is a near-book called Computer numbering formats. Not to mention octal and hexadecimal. Frankly, I think that as they stand, these articles do a terrible job of serving anyone who doesn't already know their way around the material. Would anyone be interested in forming up a "project" team to devise a proposed multi-article structure for this material, then start moving it around to where it belongs, then see what's still missing and add it? Maybe revive the moribund Wikipedia:WikiProject Computing and start with this? It's never going to get fixed by a series of small edits without an overarching plan. This topic is way off of my usual interests as a wikipedian (although well within my area of actual expertise), but I'd be willing to lend a hand. I'm not interested enough to go it alone, though. This looks like person-weeks of work. -- Jmabel 07:49, 29 Feb 2004 (UTC)

I completely agree. As you can see, I proposed some changes (above) a while back that never really went anywhere. A WikiProject might be a good idea, since there is so much material to cover. WikiProject Computing is probably a good place for it. The first major task is deciding what to do with Computer numbering formats; Taku suggested merging that article with Integer (computer science), but that would be silly since Computer numbering formats covers floating point, ASCII encoding, BCD, and other stuff mostly unrelated to integers. I still think that article should be merged with this one, since they both cover much of the same information. Anyhow, I think I will start a list of articles we need to consider for merging, deletion, redirection, and other refactoring, at Wikipedia talk:WikiProject Computing. -- Wapcaplet 16:52, 29 Feb 2004 (UTC)

[edit] What about Fu Xi?

I think the Fu Xi sequence of the Yi Jing probably deserves a mention here. It predates Pingala by 2-3000 years, although it does not define any rules for arithmetic, it does define a binary representation of the numbers 0-64. As I understand it, it is where Leibniz got the idea from. Perhaps an addition to the Yi Jing entry on the Fu Xi sequence, and a one-sentence reference to it in the binary numeral system page, is the way to go.

I have never edited anything here before so I am a bit reticent to just go ahead do it. I would value encouragement or discouragement from, e.g., the other people who have contributed to this entry. Redwards2 15:59, 15 Apr 2004 (UTC)

It may deserve a brief mention, but from what I can tell (please correct me if I'm wrong), it's not really a numeral system. There are certainly some principles in common with binary representation---six two-state entities in each hexagram, giving a total of 26=64 unique symbols---but I don't see how the symbols in the I Ching comprise a numeral system, unless there are numerical relationships between the symbols. Do such relationships exist? -- Wapcaplet 16:10, 15 Apr 2004 (UTC)

[P0M:] Leibniz was startled when he saw a Song dynasty chart that arranges the hexagrams in what appears to be binary order, but two tbings have decided people like Joseph Needham (Science and Civilization in China) against the idea that they functioned as numbers. (1) They were never used to count anything. (1) In the series that shocked Leibniz, the numbers go from 00 to the "binary" equivalent of 31 and then from 64 back to the "binary" equivalent of 32. The reason is that when they are arranged around a circle each "1" is matched with a "0". A better way to look at it would be to say that each +1 is matched with a -1 so that the sum along a diameter is 0. The whole idea is of male parts fitting into and filling female parts. So the connection between hexagrams and binary numbers is only a strange footnote on the apparent similarity of symbols. P0M 19:02, 15 Apr 2004 (UTC)

Interesting, though I don't really see the particular ordering in a circular arrangement that results in a complementary relationship between opposite symbols; symbol 01 (||||||) would be opposite symbol 02 (::::::); symbol 03 (|:::|:) opposite symbol 50 (:|||:|); 04 opposite 49, 5 opposite 35. Also, a linear arrangement that would result in what we today call "binary order" (positional numbering with significance ascending to the left) would arrange the symbols as one of:

  • 02, 23, 08, 20, 16...
  • 02, 24, 07, 19, 15...
  • 01, 43, 14, 34, 09...
  • 01, 44, 13, 33, 10...

(depending on which symbol is equivalent to 0, and which direction you read the vertically-oriented symbols from). In any case, it doesn't seem very numerical to me, although there are patterns. In some google searching, I've turned up conflicting reports ([1] [2] [3] [4] [5]) about whether Leibniz actually got the idea of binary from the I Ching, though most indicate that he published De progressione dyadica in 1679, twenty years before being introduced to the I Ching hexagrams by Joachim Bouvet. So, binary it is, but numeral system it is not. Perhaps discussion of the I Ching's relevance should go into Binary Code, Binary encoding, or one of the many other binary articles in need of refactoring. -- Wapcaplet 20:18, 15 Apr 2004 (UTC)

I am from germany, and my english isn't that good. But isn't it so, that a numerical system is any code, coding numerical values with symbols, but a positional notation or a positional numeral system (dict.leo.org is also refering "radix notation") is the thing you are talking about? In a positional notation, every position in a symbol string has a place value wich is multiplied with the symbols value. The place value is calculated with the same radix for every position with the number of the position as exponent. The Fu Xi sequence is like the roman numeral system a numeral system: A system for noting numeral values. --Supaari 15:23, July 28, 2005 (UTC)

You're right; any system coding numerical values is (by definition) a numeral system. But as near as I can tell, the I Ching symbols are not numerical at all (unless someone can correct me...) -- Wapcaplet 06:43, 27 November 2005 (UTC)

I think this is a great article, in fact pretty close to feature quality. With my addition of an explanation of multiplication, what remains to be done except explaining division? - Fredrik | talk 23:50, 27 Jun 2004 (UTC)

I agree, it's getting there. Before I'd be comfortable with a nomination for featured article, I'd like to see the "history" section expanded, at least into the 3-4 paragraph range. Division, as you note, needs to get done. And it'd be nice to have a better and more in-depth explanation of representing real numbers. Other than that, I can't think of anything... -- Wapcaplet 09:14, 28 Jun 2004 (UTC)

Is there a name for the set of numbers, where all the 1s in the binary representation are in one block?

Such a number is the product of a power of two and a number one less than a power of two. All known perfect numbers belong to this set. The first few are

1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 24, 28, 30, 31, 32, 48, ...

I recall shortly before 1984, discovering that 1984 is such a number.

--Karl Palmen 9 September 2004

[edit] Headers

I removed the fourth-level headers splitting up conversion to and from various other numeral systems. I don't think there's enough material in each of those sections to necessitate having a separate heading, and they break up the flow somewhat. -- Wapcaplet 19:35, 15 Sep 2004 (UTC)

[edit] Conversion table

I think we should be putting a conversion table in for hex, decimal, octal, binary, hindu-arabic etc. in, to help random people that can't do it themselves. Brokenvcr 07:14, 21 March 2006 (UTC)Brokenvcr

[edit] Math codes

Shouldn't we replace the "dummy" math formulas that are used in this article (simply indenting each line with a space so that spaces inside the formula are retained) with TeX_markup? The current solution does look a little ugly. That is, the addition and subtraction looks okay like this, but I don't like the division. --Spug 13:17, 3 Nov 2004 (UTC)

Maybe we should... You are free to do that if you know enough Tex. --ZeroOne 15:56, 3 Nov 2004 (UTC)
Sure, feel free. I did most of them the way they are now because it was easy to write, and easy to edit. Doing it the TeX way, I imagine, would be gruesomely complex, but I would have no problem with it, so long as I'm not the one that has to do it :-) -- Wapcaplet 18:34, 3 Nov 2004 (UTC)

[edit] Fractions

I added some conversions of binary fractions, because I need them for another article. But then the "Decimal" section got too long. So I also removed the alternate decimal to binary conversion algorithms. They aren't really used in practice anyway. -- Scottcraig 00:21, 16 Nov 2004 (UTC)

[edit] Number prefixes

The Representation section only mentions suffixed representations.

Prefixes: Binary

Assemblers for Motorola microprocessors commonly use % as a prefix for base 2 (binary).

Prefixes: Hexadecimal

See: http://en.wikipedia.org/wiki/Hexadecimal

--Aremond 20:33, 9 December 2005 (UTC)

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

[edit] Division Remainders

In the division example of 11011/101 the answer is reported to be 1012. Should not the answer actually be 10111 so all digits are in binary? --Zippanova 21:59, 27 September 2005 (UTC)

No! Reading all digits as binary, the base subscript 11 would mean three, which is not the base we want. Then, one might write 10110 instead, which, reading all as binary, means 101-read-in-base-two, i.e. five, as desired. However, reading it as decimal, it means 101-read-in-base-ten, i.e. hundred-and-one, which is incorrect. In fact, adopting a convention about always writing the base subscript in the base in question will cause no end of confusion, as the base would always appear as 10. Instead, the arbitrary convention is that the base subscript is always written in decimal.--Niels Ø 12:15, 30 September 2005 (UTC)

[edit] Binary 'humor'

Removed this:

  • "01 01 was a race horse, 10 10 was 01 10, 01 01 01 01 race, 10 10 01 01 10."

Because this is only binary humor because the numbers happen to be notated in binary. I know this pun as "1 1 was a racing horse, 2 2 was 1 2..." In contrast to the "10 kinds of people" joke, this is not specific to binary. It relies on homophones and would work in any base. Those who want to argue it's funnier in binary because of the wacky wacky notation are left on their own, I'm afraid. 82.92.119.11 15:28, 28 September 2005 (UTC)

I remember a quote we could put in the article, but not its source. "Binary is how people with two fingers count." I think it's funny.DL 22:58, 15 June 2006 (UTC)

There are 10 types of people in this world, those who understand binary and those who don't.

Translated into decimal: One-one was a race horse, Two-two was one too, One-one won one race, Two-two won one too. Scott Gall 01:03, 26 February 2007 (UTC)

[edit] Bit article trivia needs?

There's a Reqest for Comment over at Talk:Bit#RfC on trivial bits about whether the bit article needs to have some "examples" which might be called "trivia". If you have thoughts on it, please add them there. -R. S. Shaw 23:27, 5 November 2005 (UTC)

[edit] "Binary Variable" rolls over

The topic "binary variable" rolls over to this topic, though they differ. I recommend a divorce but am unsure of how to accomplish this.

[edit] Repeating Binary Decimals

Floating point rounding errors [6] are not mentioned. I think it should be in the article; but i don't know enough about binary to put it in myself.(83.118.38.37 03:28, 21 January 2006 (UTC))

I see that's with Computer numbering formats. Shouldn't there be a link on this page to Computer numbering formats then.(83.118.38.37 03:41, 21 January 2006 (UTC))

[edit] what is the binary equivalent of 14

1110 = (1 x 8) + (1 x 4) + (1 x 2) + (0 x 1) = 14

hope this helps. It's worth learning the binary numbers 0-15 (0000-1111) by heart, as you can then do hexadecimal conversions in your head and decimal ones much more quickly - but if you understand that the 1s and 0s fall into columns ...64,32,16,8,4,2,1 then conversion is really straightforward. Graham 21:38, 24 January 2006 (UTC)

[edit] It's Magical

there is a card trick where you have 6 cards. the numerals are written in base 10 but arranged in base 2. for instance the card that starts with 4 has 4-7(100-111), 8-15(1000-1111), 20-27(10100-10111)... in others words if the 4s place is one the card has that number(shown in base 10 of course) the cards start with 6 different numbers 1,2,4,8,16,and 32. anyway you have someone think of a number and pick all the cards that have that number and hand you the cards, then you just add up the first number on the cards mentally and name the number. example: they think of 42, 42 is on the card that starts with 4,8, and 32. 32+8+4=42. don't know if you find this interesting. 24.237.198.91 01:27, 3 February 2006 (UTC)

For the sake of non-mathematicians like me, wouldn't the use of more familiar terms like "base" be preferable to terms like "radix"? (Since the link to "radix" leads to an article called "Base", I assume they are interchangeable in this context?)

[edit] What do you think of my contribution on binary

I recently posted a article (binary simplified in laymans terms) which I aimed to simplify the works and working out of binary to a more understandable level. This has to be my first contribution to the site.So could you please post what you think of the article and how simple it really is. Weather you're young old, a computer noob (which it is aimed out)or vetern please post your views on it.

Thnks

Gav

At present, that thing needs a at least a rewrite. First, we do not sign contributions in the articles. Second, it looks like original research at the moment. Sjakkalle (Check!) 14:21, 25 April 2006 (UTC)


[edit] Counting in Binary

Someone just blanked this section, but didn't give any reason for the deletion. At this point I can only assume it's vandalism, so I have reverted the article. capitalist 03:00, 31 May 2006 (UTC)

Should we add a table to this section to show say the first 50 decimal numbers and their binary equivalent. As someone who is learning binary for the first time I think the article is a bit hard to understand and to show this type of table will help. 11 Sept. 2007.

[edit] About external links

Wikipedia is not a web directory and the External links section should be limited. Unfortunately the list on this article tends to grow, and tends to collect repetitive and inappropriate links.

In particular, a link to an irrelevant page called "Binary Blocks" or "Binary Tutorial" (on aegis-bearing dot com) is being repeatedly reinserted by an editor. This page appears to be original research, but in any case is not about binary numeral systems. It isn't very coherent, either. To the extent interpretable, it is about stacking blocks, some of which are powers of two numbers of units, and others of which are not. To most eyes, I'm sure, this is not a valuable resource about Binary numeral systems, and hence should not be linked to from this article.

If editors would like to help keep the External links section tidy, the help would be appreciated. -R. S. Shaw 22:55, 16 September 2006 (UTC)

I watch this article as well and have noticed the "binary blocks" links being inserted and deleted, however I never investigated them. I'll delete them as well from now on. capitalist 03:55, 17 September 2006 (UTC)

FYI, the editor serially spamming the "physical binary blocks" link (User:Derouch) violated the 3-revert rule and was blocked for 24 hours, but he's back at it again. -R. S. Shaw 00:44, 6 October 2006 (UTC)

[edit] Other binary numeral systems

Should we say something about the other ways that numbers are commonly represented with binary digits? Gray codes, one-hot codes, thermometer codes, BCD codes, etc? Dicklyon 16:34, 10 November 2006 (UTC)

[edit] Instant Converter

www.permadi.com/tutorial/numConvJs/index.html

This link converts hexadecimal-decimal-binary, all ways. (So you can insert a number into any catagory and have it converted.) Should it be added to External links? --Ryudo 23:15, 22 November 2006 (UTC)

[edit] Binary Alphabet

Something about binary as used to represent letters should be added, with a link towards ASCII which has a complete table of ASCII printable characters with their binary equivalents. -- hibou

Character codes are a quite separate idea. The numbers for the codes can be represented in any numeral system (like any numbers), and ASCII is in fact quite often represented in octal, hexadecimal, and decimal. I don't think it (or EBCDIC, Unicode, etc.) should should have a link from this article -R. S. Shaw 19:19, 1 December 2006 (UTC)

[edit] Cultural References

Referenences in culture related to the binray numeral system by myself and others was recently removed, as "Not relevent to the numeral system." However, I do think that since it was under the proper cultural references category it had some worth and should be rvt'd. For the time being I will leave it removed, as I would like to hear some other people's opinions. DeniabilityPlausible 23:53, 28 March 2007 (UTC)

I removed the paragraphs about Family Guy and Futurama because the role of the binary numeral system in them was not notable (with respect to this article). I similarly think having a paragraph about 24 (TV series) in the Decimal article would not be a good idea even though 24 is a decimal number, nor would having a description of 60 Minutes in Hour be good. (Also, the first paragraph talked about "binary for the word water" which does not make sense; maybe ASCII was meant). The article on Atom also probably should not have a list of all movies with "Atomic" in their title (much less those simply mentioning the word). A reader about number systems is very unlikely to want to hear about every mention of the name of the number system in all contexts throughout history. -R. S. Shaw 03:03, 29 March 2007 (UTC)

[edit] Decimal Section Examples

Those examples in the decimal section aren't consistent. Some of them are images, other are just text formatted with <math>. Someone should fix that to be all images, or all text with <math> tags. --SomeoneElse699211 18:06, 1 July 2007 (UTC)



[edit] What does a Binary Number represent? Or... What is a Binary Number?

A Binary Number represents the Exponent in a Base 2 Exponential Operation / Equation!

In other words, if a Binary Number is the Exponent in a Base 2 Exponential Equation, then it defines the Binary Conversion for the result from a Base 2 Exponential Operation... Hence, if Binary '00' = '1', then the '0' in the Numerical expression represents an Irrational Number, and any Binary representation containing any Combination of Zeros and Ones must also be equal to Some Irrational Number! {Note: This does not include Leading Zeros!}

Given that For Every Binary Number, which represents a Whole Number or Integer, the Binary Number is said to define the Exponential Cardinal of the Base X, in a Base 2 Exponential Equation...

That is, this Number or Integer, which equals the Result from an Exponential Base 2 Operation, has a Binary Translation / Conversion that is equal to the value of the Exponent in the Equation.


 Hence, Enumerating, or Counting using only the Exponent Proves this Fact:



1)  An 8 Bit-Mapped Length  = 2^8 = 256 IP Addresses = 256 = 11111111



2)  A 7 Bit-Mapped Length   = 2^7 = 128 IP Addresses = 128 = 1111111



3)  A 6 Bit-Mapped Length   = 2^6 = 64  IP Addresses = 64  = 111111



4)  A 5 Bit-Mapped Length   = 2^5 = 32  IP Addresses = 32  = 11111



5)  A 4 Bit-Mapped Length   = 2^4 = 16  IP Addresses = 16  = 1111



6)  A 3 Bit-Mapped Length   = 2^3 = 8   IP Addresses = 8   = 111



7)  A 2 Bit-Mapped Length   = 2^2 = 4   IP Addresses = 4   = 11



8)  A 1 Bit-Mapped Length   = 2^1 = 2   IP Addresses = 2   = 01



9)  A '0' Bit-Mapped Length = 2^0 = 1   IP Address   = 1   = 00


Needless to say, while there are many ways to Represent an Irrational Number, the above conclusions, does not imply that the Current Method of Combining Zeros and Ones is wrong... In fact, it actually represents the Binary Method for representing an Irrational Number... while there is No such method defined by the Unary Set!

[See; 'The Mathematics of Quantification, and the Rudiments Of the Ternary Logical States of the Binary Systems' - http://www.ietf.org/internet-drafts/draft-terrell-math-quant-ternary-logic-of-binary-sys-11.pdf]

§ e. terrell: 68.122.125.3 14:27, 13 September 2007 (UTC)

§ e. terrell: 68.122.125.3 15:38, 13 September 2007 (UTC)

[edit] 0 − 1 = 1 (with borrow)

In the Subtraction section, it provides a list of simple binary subtraction equations. One of them is:

0 − 1 = 1 (with borrow)

Is that correct? It seems like the result would be -1, not 1. Is there not any negative in binary? Also, what does "with borrow" mean? If this equation is correct, perhaps it should be explained in more detail, because it seems counterintuitive to me. -kotra (talk) 19:39, 27 March 2008 (UTC)

Do not confuse numbers with digits. Let's first look at decimal subtraction. Working in decimal notation, using the common algorithm for subtraction where you work from right to left, when you subtract the smaller number from the larger, as in 100−1, you may need to subtract a larger digit from a smaller. If you need to subtract 1 from 0, you get the digit 9 with borrow. That means that you must subtract 1 from the neighbouring digit to the left. The idea is that the digit that is too small borrows from its left neighbour (which, it it is too small for that, must borrow in turn from its left neighbour). In general, in N-ary substraction, where N is the base of the system, when you subtract a digit A from a digit B, you get the digit B−A if A is at most B, but if A is larger than B, there is no digit B−A. Instead you get the digit (N+B)−A with borrow. The amount borrowed equals N, since 1 is subtracted in the next column, where the weight of the digits is higher by a factor of N. For the decimal example I gave, N = 10, A = 1, B = 0, so you get the digit 10+0−1 = 9. For the binary case, N = 2, A = 1, B = 0, and you get the digit 2+0−1 = 1.  --Lambiam 06:09, 28 March 2008 (UTC)
Thanks for the explanation. I understand how subtraction works in binary (and how borrowing works). I was just confused because I thought the examples were supposed to be actual valid equations in binary, with real binary numbers. For example, if it said "10 - 1 = 1" I wouldn't be confused. I was, as you say, confusing numbers with digits. Perhaps this could be clarified in the examples, so others don't make the same mistake I made. Maybe by changing "0 - 1 = 1" to "10 - 1 = 1" or "0 - 1 = -1", or by mentioning that the equations describe how subtraction works on a digit level (as part of a larger number), not a complete number level. -kotra (talk) 20:53, 7 April 2008 (UTC)
I have added a clarification, but in doing so I realized the algorithms presented are incomplete. The little tables for addition and subtraction assume there is no carry or borrow coming from the previous column, and don't tell you what to do in the following situations:
             ? *           ? *
             1 1 1         1 0 0
           +   1 1       -   1 1
             -----         -----
               ? 1           ? 1
The case to the left occurs in the example presented for addition and is handled in the running text where it has "1 + 1 + 1 = 112", but his does not correspond to anything in the addition table presented earlier. The case to the right is not dealt with at all. The tables should be extended to eight lines: two bits + one carry/borrow bit as input, one result bit plus one carry/borrow bit as output.  --Lambiam 05:34, 8 April 2008 (UTC)
I like your clarification using "→" and describing how borrowing applies. As for the rest of your comment, it's over my head. I'm sure someone more knowledgeable can provide some input though. -kotra (talk) 17:19, 11 April 2008 (UTC)

[edit] Merge to binary code

I feel that Binary Code is a good place for this, because there's little difference between the two articles content-wise. Also, from a strictly math perspective, binary doesn't really have "numbers". Calling it a code makes more sense to me.--Mblumber (talk) 01:26, 8 May 2008 (UTC)

I disagree, binary code is the way computers work with everything. the binary numeral system is a type of dealing with numbers in a base other than ten. They are deeply related, but one is computer science, the other mathematics. Try not to combine articles from differing fields.--shdw.puppet

"There's little difference"? The only thing that I see the two articles have in common is that both concern a subject that uses only two symbols. You might as well merge English language into Latin alphabet because both use 26 letters.  --Lambiam 21:07, 8 May 2008 (UTC)