Talk:Unary numeral system

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Multiplication and Division[edit]

The article asserts that "Multiplication and division are more cumbersome"

Addition and subtraction are particularly simple in the unary system, as they involve little more than string concatenation. Multiplication and division are more cumbersome, however.

They seem pretty trivial to me. There is no citation for this assertion. If Addition is "little more than string concatenation", then surely multiplication is simply "repeated concatenation". Division is a little odd, but I would not call it cumbersome. —Preceding unsigned comment added by Kwerle (talkcontribs) 17:20, 4 February 2010 (UTC)

Unary 8 image[edit]

er... the bottom graphic for '8' seems to actually be 9... the tally on the left has 5 vertical bars and 1 horizontal, instead of 4 vertical and 1 horizontal. Someone might want to fix that.

Image:Unary_numeral_eight.png fixed --Henrygb 03:24, 18 Mar 2005 (UTC)
Whoa. I got to this article just as you made that change. It was five when I got the article, and four when I got to the image page. - Vague | Rant 12:51, Mar 18, 2005 (UTC)
Well, judging by the timestamps, I'd say I actually arrived as the cache was being cleared. Same difference. - Vague | Rant 12:52, Mar 18, 2005 (UTC)

0 in unary[edit]

I am top-posting. Sorry if anyone is offended. The notion of zero is a relatively recent one - see . There is no zero in unary. Kwerle (talk) 03:20, 22 November 2010 (UTC)

Am I correct in assuming the unary numeral system is incapable of showing 0? Negative numbers work by putting a minus-sign in front of the digits, just like with any other system, but 0 doesn't seem possible. Of course you'd sort of expect unary's first digit to be the 0, just like every other system's, but that obviously wouldn't work either. :) So, can anyone with the proper knowhow add something sensible about 0 (and negative numbers) to the article? Thanks! Retodon8 18:19, 13 October 2005 (UTC)

This whole page is inaccurate since following the pattern set by other bases (the first digit in the set of available digits being 0) would lead to any unary system equating to zero (0, 00, 000, 0000, ... all equal zero). ThomasWinwood 12:11, 26 October 2005 (UTC)
Well, since n in all other n-ary bases is represented as 10, it could possibly be 0, 10, 100, 1000. But... that's be retarded. I think a problem here is that, in all other n-ary bases, adding a zero to the end of a number multiplies it by n. In my above attempt at consistency with other bases, adding a 0 would add 1, it wouldn't multiply by 1 (since that gets you nowhere). There doesn't seem to be any way to make it rationally system with the other systems. It seems that, due to the explicitly multiplicative nature of a position number system, there's no way to make a consistent base 1 system; 2 is the smallest integer that can be used multiplicatively. (talk) 03:54, 29 September 2011 (UTC)
Yes, that's (part of) what I said, but how does that make this page inaccurate? I'd call it incomplete, because I'd like to see the 0 thing talked about. It does make unary an exceptional base system. Basically it is another symptom for the problem my question was about... how to show the value 0 in unary. I guess it's really as simple as I assumed... it's just not possible. Retodon8 17:35, 26 October 2005 (UTC)
Unary is basically the system of using strike marks. If you try to write zero in strike marks -- well, you're basically done before you start. "|" is one. "||" is two. || - || = nothing, or null, or the null symbol. But it is not "0" in that you can't do the usual things you can do with a zero. You can not write "||0" and mean twenty; if you do, it really is not unary.

I Don't see why there wouldn't be a zero in unary.

Well, I don't know another way to explain. Maybe... try figuring out how you would write zero if you only have the "1" character" (or "|" or whatever), and you'll reach the same conclusion. You can't. You can just not write anything at all, but not having a character for zero disallows a lot of useful stuff in calculations. Retodon8 23:07, 4 January 2006 (UTC)
For example? Kwerle (talk) 03:20, 22 November 2010 (UTC)

The thing is, 'the unary numeral system' is a misnomer. It's something that doesn't exist. For any numeral system, you need an origin (zero) and a unit (one). Your 'digits' go from the origin to the number of the base minus one. A number is the summation of each digit times the base to the power of the position of the digit, where the last digit has position zero. So '||||' in unary is 0 * 1^3 + 0 * 1^2 + 0 * 1^1 + 0 * 1^0. Therefore, the smallest possible base is 2. Talking about zero or negative numbers doesn't make sense, because the 'unary' system doesn't make mathematical sense to begin with. SeverityOne 21:06, 6 February 2006 (UTC)

You are off by one - though I don't know that your definition is required for unary. So '||||' in unary is 1 * 1^4 + 1 * 1^3 + 1*1^2 + 1*1^1 + 1*1^0 = 5 (base 10). So by your definition, unary works just fine. -1 in unary is just "-|". Nothing in unary is null or nothing. "" Kwerle (talk) 03:20, 22 November 2010 (UTC)

Further to your point, if "unary" is defined as a base system with only one mark, what is the correct name for a positional base-1 system following the normal rules of positional notation?

  • Decimal: value of digit n in position x = n×10^x
  • Binary: value of digit n in position x = n×2^x
  • Base 1: value of digit n in position x = n×1^x
  • Phinary: value of digit n in position x = n×φ^x

If Phinary can allow 0 and 1 as standard digits, why can't base 1 have them? --Slashme (talk) 01:31, 18 July 2011 (UTC)

Perhaps the difficulty in representing zero in a system like this can be a possible explanation of the late arrival of the concept of zero. Non-positional systems don't need a zero. it wasn't invented until the numbers being counted grew so large that a non-positional system was unfeasible. Any body know of a similar conjecture in a reliable source? Cliff (talk) 04:25, 29 September 2011 (UTC)

These are the 1-adic numbers![edit]

The numeral system described here as "unary numeral system", but without 0, is rather the 1-adic numeral system. Consider

  • in the p-ary numeral system there are p number symbols, the first one with value 0, the last one with value p-1. As SeverityOne points out, it does not make sence below base 2. In the true unuary numeral system, the only number symbol would be 0. The only representable number would be zero as well, represented by all possible strings: 0, 00, 000, -0, -00, -000, etc.
  • in the p-adic numeral system there are p number symbols, the first one with value 1, the last one with value p. This would mean the only symbol in the 1-adic numeral system has value 1. This (and all the other properties of 1-adic numbers) are in accordance with the system described in this article.

Adhemar 13:01, 21 February 2007 (UTC)

I don't think that's quite accurate, p-adic numbers are usually considered to be expressions
where are taken over the set {0, 1, ... p-1}. Thus there is still only 1 1-adic number, 0. Cheers, — sligocki (talk) 05:59, 26 October 2009 (UTC)
Cheers. But nobody is talking about p-adic numbers here. 1-adic numbers are k-adic or bijective numbers. — Preceding unsigned comment added by (talk) 18:23, 2 December 2012 (UTC)

Non-standard positional numeral systems[edit]

I have addressed certain issues by creating the article Non-standard positional numeral systems, and making related changes to Unary numeral system, Golden ratio base, Quater-imaginary base, Positional notation, Base (mathematics), and Category:Positional numeral systems. I suggest further discussion of these issues takes place at talk:Non-standard positional numeral systems.--Niels Ø 14:30, 26 February 2006 (UTC)

Unary is not a positional system, even a nonstandard one (though the others you listed are). The "digits" in unary do not depend on position: five tally marks represent the number five, and will still unambiguously represent the number five if you write the same five tally marks in a different order. -- Milo

I initially held the same point of view, but others did consider unary a pos.num.syst., which basically is what made me create the non-std. article. I will now revert your changes; I think the matter should be discussed here first.--Niels Ø 07:41, 2 November 2006 (UTC)
The unary numeral system (base-1, a non-standard positional numeral system) ...

How is unary a positional system? -- Jao 11:43, 27 February 2006 (UTC)

Read the article Non-standard positional numeral systems. One can argue either way, hence non-standard. If you can find an elegant way of putting it, perhaps something like unary, arguably a non-standard positional..., please go ahead and edit accordingly.--Niels Ø 16:52, 27 February 2006 (UTC)
If it is truly "arguably" a positional system, then this is the place to start arguing for it. I have not seen any such arguments, so at the moment the statement is completely unfounded. Incidentally, Non-standard positional numeral systems argues that unary is a non-standard system (which is quite clear), but not that it is a positional system. -- Jao 14:23, 2 November 2006 (UTC)
From Non-standard positional numeral systems:
In a standard positional numeral system, the base b is a positive integer, and b different numerals are used to represent all non-negative integers. Each numeral represents one of the values 0, 1, 2, etc., up to b-1, but the value also depends on the position of the digit in a number. The value of a digit string like in base b is
So what applies to unary, and what not? (Surely, not everything must apply; that's the whole point of considering it a "non-std." system.) The base b=1 is a positive integer; b different numreals are used to represent all non-negative integers (or at least the positive ones; zero only if you allow an null string to represent it); the numeral represents 1 instead b-1=0 as it "should"; the value does not depend on the position, but the equation above applies all the same - because all powers of b are 1. So, many of the standard features do apply, including the equation , but of course, your objections are caused by the fact that a feature that does not apply is the one implied by the word "position" in the name "positional numeral system". This is a strong argument, but consider that unary undeniably is the bijective numeration system base 1, and all the other bijective numeration systems are positional numeral systems. This proves nothing, but it means that Unary should remain closely linked to Positional numeral system and other related articles. Suppose pos.num.syst.s were called "weighted digit numeration systems" instead (as the digits are assigned weights depending on their position); then we could all agree to include unary as a special case where the weights happen to be all equal.
I suggest we leave unary in the non-std. category, but I also think it's a good idea to mention the doubts one may have about this classification in the article. If unary is not a positional system, it is at least intimately connected to the positional systems, and that must be reflected in our articles. Entirely deleting the section on unary from the article Non-standard positional numeral systems and deleting references to positional systems from Unary is not the way.--Niels Ø 15:00, 2 November 2006 (UTC)

One can consider uniary as a standard number system by noting that it does not require the number value zero. In other standard number systems the sybmol 0 is used for the value zero because it is used as a place holder (e.g. in the decimal number 105). This place holder requirement does not exist in unary so the glyph 0 is free to be used for the value 1. This yields a system using only the glyph 0 and is different from the marking system. 05:39, 29 January 2007 (UTC)

Unary in theoretical computer science[edit]

The article makes out the use of unary in theoretical computer science to be sneaky. I'm sure the reasons unary is used is nothing of the sort. It is used to make formal models of computation manageable. Try writing a Turing machine that multiplies together two numbers given in binary. It is not easy.

Incidentally, in the theory of computation, a modified unary system is often used. In this modified system, "1" means 0, "11" means 1, and so on, so that 0 is representable. CyborgTosser (Only half the battle) 14:04, 17 February 2006 (UTC)


Proposed merge of Tally mark into this article[edit]

Someone added merge tags to Unary numeral system and Tally mark, but failed (I think) to provide arguments or start a discussion, which I hereby do:

  • Oppose: While the things are closely related, I think it's reasonable to have separate articles. Tally marks are unary numeral systems all right, but they are more than that. They have a cultural history, and take different forms in different cultures.--Niels Ø (noe) (talk) 13:34, 19 November 2007 (UTC)
  • Oppose They are distinct, although related. Bubba73 (talk), 17:42, 11 January 2008 (UTC)

Does the Unary System even exist?[edit]

I'm not sure if it is possible to have a unary system, for when you try the unary system it always relies on a nothing to differentiate it between the something hence using 2 numerals. Does that make sense? It does to me anyway. —Preceding unsigned comment added by (talk) 07:28, 4 June 2008 (UTC)

Unary is not a positional notation but a bijective notation. The base-1 positional notation doesn't exist. The base-10 bijective goes like 1, 2, 3, 4, 5, 6, 7, 8, 9, A, 11, 12, ..., 19, 1A, 21, ..., 99, 9A, A1, ... - TAKASUGI Shinji (talk) 08:14, 4 June 2008 (UTC)
See Non-standard positional numeral systems.--Noe (talk) 16:00, 4 June 2008 (UTC)
I also agree on the fact that Base 1 is not even possible. Since Base 1 does not have a character "0", any number in Base 1 would equal to infinity because the number would continue on both the left and right sides of the number. For example, the number 410 is equal to the number 00000000000000004.0000000000010 because the zeros go on infinitely on both the left and right sides of the number however, the extra zeros are cut off because they are irrelevant. However, in Base 1, there is only one kind of number so therefore the number 111111111 in Base 1 is equal to 111111111111111111 and is also equivalent to 1 and is also equal to 1.11111111111(etc.). If all numbers are equal to infinity then you cannot do math using Base 1. If you cannot do math or represent numbers in a finite way than you really do not have a base at all. Also, another way to think about Base 1 is to think of a blackboard. A blank blackboard, in reality, is Base 1. There is only one kind of color and therefore there is nothing you can do with a blank blackboard. However, once you start to use chalk then you have two colors, black and white.(aka binary)--JavaBinaryUser (talk) 04:21, 15 August 2013 (UTC)


Really? From the number of confused asking questions here, it seems that many people find the system they learned in (usually denary) to be the simplest. I suggest removal of this term. Objections?Cliff (talk) 18:26, 14 February 2011 (UTC)

Of course it's the simplest; it is one-to-one: Three cows, three cats, three dollars, three points, three whatever, is represented by three (often indentical) glyphs. People may misunderstand or have questions about the explanations and other things in the article; they are not misunderstanding unary numbers. Does the simplicity have to be mentioned in the first sentence of the article? Perhaps not, but it certainly makes sense to me.-- (talk) 22:07, 14 February 2011 (UTC)

Base 1 Redirect[edit]

I feel Unary system is different from base 1 in many ways, for example not being base 1, therefore the redirect from Base 1 should be removed and an article created describing a numeral system in which only 0 may exist because the maximimum is 1. Does anyone object? Does anyone agree?--Sonez1113 (talk) 18:43, 15 March 2011 (UTC)

That does not sound like a notable system. The whole point of having number systems in the first place is to represent all numbers, why would you have one that represented only one number? Although this is not a positional number system, it is a base 1 bijective number system and AFAIK the only system commonly called base 1. Cheers, — sligocki (talk) 01:07, 16 March 2011 (UTC)

The redirect from Base 1 needs to be removed for several reasons. First, mentioned specifically in the Unary page is that Unary is a bijective base-1 numeral system. That is not the same thing as a base otherwise it would be mentioned that Unary was a base and not a bijective numeral system. Since Unary is not a base, then most definitely Base 1 should not be redirected to Unary. Second of all, I would argue that Base 1 does not exist, whereas, Unary does. A Base-1 chart is shown in the same fashion of a binary chart:









Given the Base 1 chart and using one's reasoning skills, it can be seen that in Base 1 one cannot represent 0 in decimal or any other number in decimal that is not an integer. All other bases have an equal counterpart to any random number in another base. Following are some examples in case you are not following this: The number 10 in binary is equal to the number 2 in decimal. The number 10 in decimal is equal to "a" in hexadecimal. The number 20 in Octal is equal to the number 10 in hexadecimal. All bases have a number that is an equal counterpart to any random number of any other base. Yet, Base 1 fails to do so. For the number 10.5 in decimal, there is no Base 1 counterpart. Again, for the binary number 101.10110, there is no Base 1 counterpart. Once again, for the hexadecimal number 41.245, there is no Base 1 counterpart. Recalling upon the event upon which the now dwarf-planet Pluto was no longer accepted as a planet, what was the foundation reason for why Pluto was not longer accepted as a planet? The reason was that Pluto did not meet the requirements to be a planet. What were the requirements modeled after? They were modeled after our own planets. I take it that essentially the reason why it was not considered a planet any longer was because it was substantially different from the other planets. Is not Base 1 the same? In addition to Base 1 not being able to have a number equal to any random number of another base, Base 1 is clearly symmetrical based off the Base 1 Chart diagramed before, and Base 1 is infinite. I will further expound upon the other two reasons now to demonstrate my point. First of all, Base 1 is symmetrical. The Base 1 Chart clearly dictates that the Base 1 number 1111 is equivalent to the Base 1 number .1111, therefore making it symmetrical. Both numbers are equal to 4 in decimal because the addition of 1^1,1^2,1^3, and 1^4 is equal to 4 in decimal as is the addition of 1^-1,1^-2,1^-3, and 1^-4. Calling again upon the reasoning of the dwarf planet Pluto, no other base has numbers that when reflected about the "." are the same equivalent in decimal still. Secondly, Base 1 is infinite. In all other bases there is a number zero that can be left off when redundant, and at least another number that is greater than 0 making them numbers with value rather than just place holders. As mentioned in the "Does the Unary System even exist?" section later down on this page, if there are not any zeros, then the number would be forced to go on forever as each number greater than zero will never be redundant except to the point in saying that each number in Base 1 is infinite. (Seriously, if you have any arguments for me, post them on this page or cause there to be some way for me to see them. I have pondered this whole concept through immensely and I would rather hear counterarguments than have readers brush this off thinking that I don't know anything.) --JavaBinaryUser (talk) 22:25, 28 October 2013 (UTC)

Having a redirect from Base 1 does not mean it is equivalent to 'base 1': it simply means that this is the article that best covers that term. Any mention of 'base 1', even one explaining that it is not proper for indicating 'unary' makes it a good candidate for the redirect. I don't think any better target exists, nor that one is needed. Bcharles (talk) 22:32, 15 June 2014 (UTC)

trying to be too clever, you've introduced an inconsistency into these wiki pages[edit]

Bijective-unary is not "base 1". This business about "bijectivity" screws up the meaning of the word "radix / base". Bijectivity seems to remove "the symbol for zero" from the set of symbols, yet the definition of "radix / base" is itself based on including the symbol for zero. It's gotta be one way or the other; without rewriting all the definitions, you are going to have a problem if you "change the rules for base 1" and try to pretend it's consistent. Bijective-unary is not "base 1". (talk) 19:15, 11 May 2013 (UTC)

Empty References section[edit]

Hey, i'm new to WP here. Is it proper to have an empty references section? Should it be removed or kept? Perhaps does the page deserve a missing refs templates? Scimonster (talk) 16:32, 5 March 2014 (UTC)

The article has references now so no need to remove the References section. Blackbombchu (talk) 03:50, 13 November 2014 (UTC)

Positional unary numeral system[edit]

@David Eppstein: By regualar numeral system, I meant Positional numeral system where you have a digit for all numbers from 0 to 1 less than the base as opposed to Bijective numeral where you have a digit for all numbers from 1 to the base. Maybe the sentence you removed should get added back in except for the phrase "regular numeral system" getting replaced by "positional numeral system". Blackbombchu (talk) 03:46, 13 November 2014 (UTC)

I don't think so. It seems a confusing and longwinded way of saying "this is not a positional number system". Since it isn't a positional number system, what do we care about how a nonexistent positional number system would behave? —David Eppstein (talk) 04:12, 13 November 2014 (UTC)