Talk:Bijective numeration

Bijective 0-9[edit]

Considering the Excel example, place value still seems important. Typical decimal symbols could still be used (0-9) and succeed in bijection, (based on what I'm seeing used in the article so) counting 0,1...9,00,01...09,10...19; or 1...9,0,11..19,10,21... This would be useful to add to the article to show the difference between bijective 0-9, 1 thru 0, and decimal 0-9. Phil.andy.graves (talk) 19:36, 25 February 2012 (UTC)[reply]

Talk:Decimal without a zero[edit]

I can't find any reference to this peculiar concept on the Web, nor does it make any sense; there really is a zero-equivalent as defined in this article -- it just is indicated with an "X" and functions a bit differently. I think perhaps this is gubbish? --Jpgordon 15:54, 28 Sep 2004 (UTC)

Well if you think ten is a zero-equivalent but functions a bit differently, then I suppose you are correct. Perhaps you haven't looked hard enough. Try "A Logical Alternative to the Existing Positional Number System" Volume 1 (Dec 1995) of SouthWest Journal of Pure and Applied Mathematics http://www.maths.soton.ac.uk/EMIS/journals/SWJPAM/vol1-95.html or try to decipher http://digilander.libero.it/ultimus2001/introd.htm --Henrygb 02:06, 6 Oct 2004 (UTC)

Thanks! Now I see the point (from the SWJPAM article). I wonder if anyone has followed his suggestion and gone back and looked at seemingly incorrect archeological arithmetic? --jpgordon 06:28, 6 Oct 2004 (UTC)

I'd like to merge this into bijective numeration, since decimal without a zero is just the case k=10 of the bijective base-k system described there. Does anyone object? 4pq1injbok 03:30, 6 August 2005 (UTC)[reply]

I have a question. Couldn't you solve the problem of there being an infinite way of expressing 2 when you expand the system to decimal fractions by making the base symbol associate to the left when on the left of the decimal place and associate to the right when on the right of the decimal place? This would make 1.A equal to 1.01. Then, you have .001, which, in the way described, would be, I think, -1.9A1. This simplifies it to .A9. The only problem I see with this is that .A9 x A becomes .A, which adds further confusion to a confusing system. --Some Random Guy 22:08, 16 January 2006 (UTC)

If I'm understanding you, you're wanting to overcome the fact that bijective base-k numeration does not "naturally" extend to numerals with digits to the right of the radix point. That is, if we define the notation in the natural way,

. d₁ d₂ d₃ ... = d₁ / k¹ + d₂ / k² + d₃ / k³ + ... (finite or infinite series)

then the system is not only not bijective, but there are infinitely-many representations of many real numbers; e.g. 1 = .A = .9A = .99A = ..., etc. Then, too, some reals have no such representation at all; e.g. .001, although .001 = 1.001 - 1 = .9A1 - 1 (not -1.9A1 as you wrote).

But the scheme you're proposing is still not bijective (e.g. still 1 = .A). Another problem is that your scheme — as I understand it — cannot represent most reals (e.g. irrationals), because it involves reversing the order of the digits to the right of the radix point and interpreting the result as an integer; e.g., you put .A9 == .001 because (9A)_{bijective decimal} = (100)_{ordinary decimal}. But if the number has infinitely-many digits in its representation, there is then no integer whose digits can be reversed as the scheme requires. --r.e.s. 19:14, 17 April 2007 (UTC)[reply]

It's worth noting that while many real numbers can't be represented in base-A, many real numbers can't be represented in base-10 either. The square root of two, for example. Many other real numbers need dots above the recurring numbers to show their value in addition to the Arabic numerals.

I don't see why ".A" necessarily needs to represent "1". In Roman Numerals, what does "X.X" (with the "." representing a decimal point) mean? Nothing at all, unless you define what it means to have a digit to the right of the decimal point in Roman Numerals. SomeRandomGuy above discussed a definition of base-A where A associates to the right when on the right of the decimal point, not the left. This would make .A equal to 0.01, and the system still appears to be bijective. Any infinite length string in base-10 would still be an infinite length string in base-A, and you would use an ellipsis or dots above recurring decimals in the same way. I don't see why you'd have to reverse the base-10 representation to make the base-A representation. Kaid100 (talk) 20:29, 27 November 2011 (UTC)[reply]

Isn't "ordinary" numeration bijective too?[edit]

The terminology "bijective base-k" seems rather misleading, since it suggests that "normal" base-k numeration is not bijective. Our familiar decimal counting system provides a bijection between positive integers and finite-length strings, while "bijective base-10" provides a different bijection. So why do we call it "bijective base-10"? The introduction to this article implies that k-adic numeration is a particularly notable example of bijective numeration, when in fact it seems a comparatively obscure one. Mtford (talk) 21:28, 7 July 2008 (UTC)[reply]

Looking at the definition again, I notice that ordinary decimal numeration is not bijective if we allow strings beginning with zero (since 1 = 01 = 001, etc.). But it's still bijective over a subset of the possible strings. Intuitively I would say that the difference between "base-10" and "bijective base-10" lies in the definition of the set of strings used to represent numbers, rather than in the property of bijection. Mtford (talk) 22:52, 7 July 2008 (UTC)[reply]

Yes, the important point is that for bijective numeration there is — as the article states — "a bijection between the set of non-negative integers and the set of finite strings over a finite set of digits". "Bijective numeration" is the standard term in the literature for this kind of system, even though it is slightly misleading in just the way you mention.
r.e.s. (talk) 02:33, 8 July 2008 (UTC)[reply]

Perhaps the opening paragraph should explain why ordinary numeration (for want of a better term) does not fit the definition, in this case? We should perhaps also mention more explicitly there are only 90 decimal integers of length 2, 900 of length 3, etc, as compared to 100 and 1000 respectively in 10-adic numeration. Mtford (talk) 16:46, 9 July 2008 (UTC)[reply]

I think your suggestion is a good one, so I gave it a shot.
r.e.s. (talk) 19:11, 9 July 2008 (UTC)[reply]

APPLICATIONS OF NUMERATION

The removal of zero may provide a mathematical structure that is useful when making physical measurements. If zero is removed then the arithmetical operations associated with it are removed. In particular the limit as an increment approaches zero is removed. A limit still exists;however,the limit refers to the smallest number that can be measured by the corresponding laboratory instrument. 71.227.168.240 (talk) 02:14, 26 May 2012 (UTC)[reply]

Link removed[edit]

I removed the external link:

Giovanni Fraterno (Italian)

since it points to a site containing crackpot mathematics. —Preceding unsigned comment added by 79.53.165.206 (talk) 19:20, 26 February 2009 (UTC)[reply]

Number of digits in a bijective base-k numeral[edit]

The article has a "citation needed" tag attached to the following statement:

if k > 1, the number of digits in the bijective base-k numeral representing a nonnegative integer n is $\lfloor \log _{k}(n+1)+\log _{k}(k-1)\rfloor$

A proof of this statement is given at http://math.stackexchange.com/a/608884/16397. Is this an adequate source for removing the "citation needed" tag?
— r.e.s. (talk) 14:14, 22 July 2015 (UTC)[reply]

The definition seems to produce infinite strings.[edit]

Where the definition defines the production of a bijective base-k numeral from an integer m it does not define n and seems to produce infinitely long strings. For example, if k is ten then decimal 123 is represented by digit string "...999A123".

a₀ = 3, q₀ = 12

a₁ = 2, q₁ = 1

a₂ = 1, q₂ = 0

a₃ = A, q₃ = −1

a₄ = 9, q₄ = −1

a₅ = 9, q₅ = −1

Since the definition produces a composite string by concatenating shorter strings, it seems that the definitions of a₀, a₁, a₂,... should be definitions of strings, not numbers. These strings will never be "0" because 0 is not in the digit set {1, 2, ..., k}, but they could be the empty string. Beyond some threshold (in my example starting at a₃) the string a_i should be defined as the empty string so that its concatenation has no effect. Either that or the definition should say how big n is and so where to stop the construction.

The meaning of q_i is apparently "the digits to the left of column i". The problem occurs when no digits to the left remain and q_i becomes −1 instead of 0. One way to solve this is to let f(x) = max(ceiling(x)−1, 0) so that q_i never drops below zero.

That produces, in my example, the string "...0000123", which is impossible since 0 is not a valid digit. To finish the cleanup we could append text as follows: "The digit-string representing the integer m > 0 is a_na_n−1...a₁a₀ where, as long as a_i is in the digit set {1, 2,..., k},..."

IOLJeff (talk) 16:40, 20 December 2015 (UTC)[reply]

It does specify what n is, in the following line, although perhaps it could be made clearer:

q_{n}=f\left({\frac {q_{n-1}}{k}}\right)=0

This is to say, n is the first value such that q_n = 0. In your working, stop at q₂ = 0; hence n = 2. -- Perey (talk) 13:24, 21 December 2015 (UTC)[reply]

Thanks, Perey. I wrongly interpreted the line beginning "a_n = " as a generalization of the previous lines, i.e., as "a_i =... for i > 0". Because of that I believe I overlooked the "− 0k" in that line, assuming a q_nk there instead.—IOLJeff (talk) 14:27, 22 December 2015 (UTC)[reply]

Why "Non-negative" and not "Positive"?[edit]

The article says a bijective system can represent all non-negative integers. But isn't zero non-negative? I don't see how you could represent zero in this system. Nupanick (talk) 18:42, 14 April 2019 (UTC)[reply]

Perhaps you missed the first bullet point of the definition section: "The integer zero is represented by the empty string." —David Eppstein (talk) 19:08, 14 April 2019 (UTC)[reply]

→ But what is “represented by the empty string” supposed to mean? (If we want to say ′zero times x equals zero′, do we write 'x ='?) (2A02:A445:EB91:1:BD4B:B06C:FB4C:24DD (talk) 08:36, 8 May 2019 (UTC))[reply]

It means a string that has zero characters. You know, like "". —David Eppstein (talk) 15:57, 8 May 2019 (UTC)[reply]

Unary[edit]

It might be worth noting under the examples that the lowest possible base for a bijective numeration system is 1, and that this is the unary numeral system — essentially equivalent to tally marks. 2603:6010:E301:5600:FD08:6942:B377:8E24 (talk) 14:18, 2 September 2022 (UTC)[reply]