# Talk:Factorial number system

## Factorial inventor is FRENCH

You said The origins of the term 'factoradic' are obscure. see an article from 1888 http://www.numdam.org/item?id=BSMF_1888__16__176_0 zorg724

The main article states that the origins of the term factoradic are obscure, not that the origins of the concept are obscure. The construct itself goes back as least 200 years (well before the 1888 reference you mention, which by the way does not contain the term factoradic as far as I can see). What is somewhat new are computer implementations of the factoradic and the connection between factoradics and combinatorial indices.

## Others

Have received information from Peter Cameron to the effect that he first heard about factoradics from a talk attended in the mid-1980's, when the speaker used the term as if it was already standard. So without a source from that era, the best one can say is the origin is simply obscure. --J. W. McLeod 20:57, 29 Mar 2005 (UTC)

I think the problem may be that this is obvious, and has probably been invented several times. I once described it in Usenet as "factorial base" in January 2002[1], and I did not think it was anything special. Nor can I remember being told about it by anyone else before I used it myself to solve a minor problem. --Henry Bottomley 13:13, 3 Jun 2005 (UTC)
Donald E Knuth's The Art of Computer Programming provides additional information. In Volume 4, Fascicle 2: Generating All Tuples and Permutations the answer to question 7.2.1.2-3 on page 101 cites H. A. Rothe, Sammlung combinatorisch-analytischer Abhandlungen, 2 (1800), 263-264. He refers to inversion tables in question 5.1.1-7, and notes that he described the factorial number system in equation 4.1-(10). It also shows up earlier as the Algorithm P in section 3.3.2 Empirical Tests (for random numbers), pp 65-66 of Volume 2. Knuth also refers to a "Narayana cited in Section 7.2.1.7", which would be the history of combinatorics that is currently being drafted. So factoradics would seem to have been kicking around for over two centuries in one form or the other. --J. W. McLeod 15:05, 4 Jun 2005 (UTC)
Iverson was using it, presumably with the associated database applications, as a running example by the early 1970's. You may find Cantor Expansion to be a more productive term for investigating the history than factoradic. 83.76.113.61 23:57, 13 January 2007 (UTC)

## Rightmost zero digit

At first look the last always zero digit in the definition seems redundant, but it's used in the algorithm for making a permutation from factoradic, so either the algorithm should be corrected or the last always zero digit should be present in the definition. Besides it fits nicely because you can just start counting from zero: the factoradic dn...d2d1d0 is equal to 0!d0 + 1!d1 + 2!d2 + ... + n!dn, where every digit di is in [0, i].

There is a major contradiction on this page, and I don't know how to solve it, so I'm hoping someone else will. At the beginning, and at the first table of numbers, a notation is used such that the nth radix is n!, but after that a notation is used such that the nth radix is (n-1)!. At the beginning 719 would be 54321, but after that it would be 543210. This is EXTREMELY confusing! Someone who actually knows about factoradic, please change this! Soon! —Preceding unsigned comment added by 70.104.133.164 (talkcontribs)

I suggest that the rightmost always-zero digit should be suppressed from the definition. I give the following motivations:
1. Its useless.
2. It's confusing.
3. No serious reference is given supporting this convention; the OEIS reference which IS serious supports the other notation (without -0).
4. No other numbering system includes a digit which has the same value for all numbers.
5. In any positional numbering system using arabic digits, the number 1 is represented by 1.
6. In any positional numbering system using arabic digits, the sucessor of a number having least significant digit 0 is the number obtained by replacing that digit by 1.
7. No positional numbering system has (nor should have) two distinct places with the same weight, here d0=d1.
8. The only *vague* motivation for including the -0 is the "coding of permutations" issue, but for this one could easily adopt the definitions/algorithm to always add the -0 "explicitely".
9. The "explanation" that d0=0! since d0=b^0 for the usual positional system, is not valid. In fact, in the usual numbering system the weight of the least digit is 1 (which by accident happens to be equal to b^0), and then the weight of the other positions is obtained by sucessively multiplying the preceding weight by a number bigger than one.
I'm waiting for other people's opinion concerning this convention. Please don't hesitate to add a line below with remove (= remove trailing 0) or don't change (leave as is with trailing 0) or any other comment. — MFH:Talk 21:24, 27 March 2007 (UTC)
For the record, I think think the final 0 is useful for the same reason that one usually defines n!=n×(n−1)×...×2×1 including a final 1, even though it is not necessary. With the final 0 digit the bases of the digits (for a number up to n!) are n, (n−1), ..., 2, 1, rather than n, (n−1), ..., 2. Recall that the digits must be strictly smaller that the base for their position, and that the place value is the product of the values strictly to the right of the current position (this is the correct statement for any mixed radix system). The 0 digit at the end should be written but need not be stored in implementations (similarly to the initial bit 1 in binary floating point number mantissas), or rather, its storage can be done in log2(1)=0 bits, since only 1 value is possible. The OEIS sequence is not an argument, because it is not an actual infinite sequence: it is based on converting factoradic digits into decimal digits, which is not possible once the numbers get sufficiently large, namely at base 11 (and one can understand Sloane wanted to avoid a sequence with all terms multiples of 10). Points 4-7 of the above list are all false for mixed radix systems which have a base 1 (final) digit; though not particularly useful, this is not forbidden in such systems (as long as not all positions are base 1). Marc van Leeuwen (talk) 21:29, 2 February 2010 (UTC)

## Non-integers

I was at the 0.999... page and it gave an example that "In the factoradic system, 1 = 1.000… = 0.1234…" . but the current Factoradic article doesn't seem to allow for a 'decimal' point. I would think that either the 0.999... article should have the factoradic example removed or the Factoradic article should be added to. It seems to me like there are different ways to implement a 'decimal' point though, and that having one could make there be multiple representations for the same number (1100.0011 would be another way of representing 2, right?) --Sgt. Muffles 18:56, 27 November 2006 (UTC)

I vaguely remember reading a magazine article about a factorial fraction number system. Integers were represented normally, but numbers after the fraction point a.(b)(c)(d)(e)(f)(g)(h)... were interpreted as ${\displaystyle a+{\frac {b}{2!}}+{\frac {c}{3!}}+{\frac {d}{4!}}+{\frac {e}{5!}}+{\frac {f}{6!}}+{\frac {g}{7!}}+{\frac {h}{8!}}...}$

This factorial fraction system can represent any rational number exactly in a finite number of places after the fraction point (unlike the decimal system, which requires endlessly repeating patterns for things like 1/7).

For example,

 1/2 = 0.(1)
1/3 = 0.(0)(2)
1/4 = 0.(0)(0)(6)
1/5 = 0.(0)(0)(0)(24)
1/6 = 0.(0)(1)
1/7 = 0.(0)(0)(0)(0)(0)(720)
e   = 2.(1)(1)(1)(1)(1)(1)(1)(1)....
1/e = 0.(1)(-1)(1)(-1)(1)(-1)(1)....


However, in this system, 0.(1)(2)(3)(4)... is significantly less than one. So perhaps the 0.999... page is referring to some *other* factoradic system? Or is it a typo? -- DavidCary --68.0.120.35 05:41, 17 December 2006 (UTC)

Let's take your definition for the time being (Sgt. Muffles seems to prefer an extra 0 to the right of the point but so long as the definition is clear it hardly matters much). I suspect there is a rule that fractional numbers should have the nth digit less than or equal to n and your examples should be:
1/2 = 0.1 or 0.023456789...
1/3 = 0.02 or 0.013456789...
1/4 = 0.012 or 0.011456789...
1/5 = 0.0104 or 0.010356789...
1/6 = 0.01 or 0.003456789...
1/7 = 0.003206 or 0.003205789...
e = 2.111111111...
1/e = 0.020406080...
and ${\displaystyle {\frac {0}{1!}}+{\frac {1}{2!}}+{\frac {2}{3!}}+{\frac {3}{4!}}+{\frac {4}{5!}}+{\frac {5}{6!}}+{\frac {6}{7!}}+{\frac {7}{8!}}+{\frac {8}{9!}}+{\frac {9}{10!}}={\frac {3628799}{3628800}}}$ is indeed close to 1 --Henrygb 18:15, 17 December 2006 (UTC)

Ah, clever. How do we know that the nth digit will never need to be greater than n? (Is there something for factoradic, that would be analogous to Zeckendorf's theorem for Fibonacci coding ?) After thinking about this for a bit, I think I could make up a proof ... but that would be original research (WP:OR). -- DavidCary --68.0.120.35 22:14, 17 January 2007 (UTC)

It is a simple inductive proof that ${\displaystyle \sum _{j=1}^{n}{j-1 \over j!}=1-{1 \over n!}}$ using ${\displaystyle {k \over k!}={1 \over (k-1)!}}$ --Henrygb 21:00, 18 January 2007 (UTC)
Actually ${\displaystyle \sum _{j=1}^{n}{j-1 \over j!}=\sum _{j=1}^{n}{1 \over (j-1)!}-{1 \over j!}=1-{1 \over n!}}$ just because it telescopes pma 23:03, 6 January 2016 (UTC)

## Unambiguous?

No number can be represented in more than one way

According to 0.999..., 1 = 0.1234... in factoradic. That said, I notice that this article might not be talking about the same factoradic that the 0.999... article is talking about. The latter has the radix getting larger in the rightward direction, towards the less significant digits. Shinobu 17:27, 29 May 2007 (UTC)

The examples above seem to show that all rational numbers (except 0) have two factoradic representations: one finite and the other infinite. If 1/2 is 0.1 or 0.0234... then clearly adding them together can give 1 = 2/2 = 0.1234...01:50, 5 October 2008 (UTC) —Preceding unsigned comment added by 81.154.230.101 (talk)

## All base indications are (or were) wrong

This article systematically mislabels the base of the factoradic digits. The base at a given position should be strictly larger than any digit in that position. This makes base 0 absurd, and base 1 the right base for an obligatory final 0; any digits that can be nonzero should have base at least 2. The base for a position is not the factor by which its digit is to be multiplied relative to the next digit, it is the factor by which the previous digit must be multiplied relative to this one. So indicated bases are systematically 1 too small. I've corrected this in the lede, but every single explicit factoradic number needs to be relabeled... Marc van Leeuwen (talk) 21:38, 2 February 2010 (UTC)

## Confusing intro

"341010! stands for 364514031201" .. i wonder, what kind of notation is used in the latter one.. the meaning is unclear.

".. whose value is ((((3×5 + 4)×4 + 1)×3 + 0)×2 + 1)×1 + 0" might be also confusing (i'll have to think about how it was generated (: ), intuitive 3×5! + 4×4! + 1×3! + 0×2! + 1×1! + 0×0! should be mentioned at first

anyway, it's great article. Mykhal (talk) 21:19, 22 May 2012 (UTC)

## Permutations

The table in the 'Permutations' section seems to differ from the description following it. My interpretation leads to 110(!) matching (2,0,1) and 200(!) matching (1,2,0). HikingPete (talk) 21:21, 5 September 2014 (UTC)

## Even/Odd double factorial system

The place value of the n-th digit from right of the even (odd) double factorial system is (2n-2)!! ((2n-1)!!), and the maximum value of the digit is 2n-1 (2n).

 decimal binary factorial even double factorial odd double factorial primorial 0 0 0 0 0 0 1 1 10 1 1 1 2 10 100 10 2 10 3 11 110 11 10 11 4 100 200 20 11 20 5 101 210 21 12 21 6 110 1000 30 20 100 7 111 1010 31 21 101 8 1000 1100 100 22 120 9 1001 1110 101 30 121 10 1010 1200 110 31 122 ... ... ... ... ... ... 15 1111 2110 131 100 211 16 10000 2200 200 101 220 ... ... ... ... ... ... 100 1100100 40200 2020 631 3120 ... ... ... ... ... ... 150 10010110 111000 3030 1300 5000 ... ... ... ... ... ... 1000 1111101000 1212200 14620 10331 45120 ... ... ... ... ... ...