Talk:Mixed radix

Here are some subtopics that might help the exposition along once the proper references are dug up …

Why each radix must be an integer

For example Dollars/Quarters/Nickels/Pennies makes a perfectly good MRN system, but Dollars/Quarters/Dimes/Nickels/Pennies doesn't because the radix Dimes -> Quarters is 2.5, and 0₁₀0₄2_2.50₂4₅ (2-dimes,4-pennies) doesn't know whether to roll over to 0₁₀0₄2_2.51₂0₅ (2-dimes,1-nickel) or 0₁₀1₄0_2.50₂0₅ (1-quarter).

There is a perfectly legitimate 'base by' notation, which does support fractional integers. Binary-coded decimal is an example. In the case of us money, one would consider the quarter (1s 3d) to be 2 dimes (6d) + 1 nickel (3d), You could have a dime and a nickel, or two dimes, but no other combination. (I put money equivalents in so to follow what is going on.) The quarter is then 5 nickels, counted by intermediate dimes.

In the usual form, one might count in decimal to say, 71, and then roll over to 1,00. This would be 'base 71 by 10s'. Computers are quite at ease handling BCD, and one can express any base in a decimal, or other form, encoding, even when this does not divide. Bases like 2.61803398875 etc work by having a broken carry, and you can have complex bases by having a double column negative carry.

--Wendy.krieger (talk) 07:33, 17 November 2011 (UTC)[reply]

A radix doesn't necessarily have to be an integer. You just have to make sure that the number that you use to count by divides your radix evenly. One divides every integer, so that's why bases are typically integers. But if you count by fourths you can have a radix of 1/2. I've been using the term "step" to define what number you count by. For instance, counting with base 3/2 step 1/2 gives the following sequence, where each number is separated by a comma, and certain digits within a number are separated by parenthesis:

0, 1/2, 1, (1/2)0, (1/2)(1/2), (1/2)1, 10, 1(1/2), 11, (1/2)00, (1/2)0(1/2), (1/2)01, (1/2)(1/2)0, (1/2)(1/2)(1/2), (1/2)(1/2)1, (1/2)10, (1/2)1(1/2), (1/2)11, 100...

This sequence translates to the more familiar quantities:

0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9....

So the number two in the familiar base ten step one system is written in the base 3/2 step 1/2 system as (1/2)(1/2). And the number four would translate to 11. Four and a half (4.5) would be represented as (1/2)00, and nine would be 100.

So you can have a non-integer radix, but only if you are counting by a number less than one. It's interesting how having a non-integer radix creates fractional digits though.

Theboombody (talk) 17:44, 8 May 2013 (UTC)[reply]

A practical example of a non-integer radix is a system based on poles/rods/perches and yards - there are 51⁄2 yards in a pole/perch/yard. An example using irrational numbers is radians and revolutions - 2π radians in a revolution.Martinvl (talk) 18:03, 8 May 2013 (UTC)[reply]

1 is a perfectly good radix

Radix 1 digits can act as place holders, but don't otherwise affect the MRN system.

Proper and improper MRNs

"Improper" MRNs have digits equal to or greater than their radices. They still specify integers, but these may not be limited to the product of the radices minus 1.

I don't know. A base of 3, with digits 3, 1, -1, works quite nicely. Wendy.krieger (talk) 08:32, 30 October 2011 (UTC)[reply]

Carry

Procedure to generate the proper MRN from an improper one if the integer represented is within range. Uses truncating division and the modulo operator. Overflow results if the integer is not in range.

Addition and scalar multiplication

Addition defined in the obvious way on two MRNs with the same base produces in general an improper MRN, which is rendered into a proper one using carry (with the possibility of overflow). Scalar multiplication on an MRN is just multiplying all the digits by a (non-negative) integer. Again carry is applied and overflow is a possibility.

long division, long multiplication are both implemented by scalar multiplication (or division), and addition. The method of practice allows one to multiply lengths in chains, yards and feet to get a result in acres, roods, and perches, with no reduction of unit. You can do things like find 3 3/8% of £21 10 6 with no reduction

   100%  costs  £ 21 10 6       0@    =   0
    10%  costs  £  2  3   .5    0     =   0
     1%  costs        4  3.65   3     =   1s   10.95d
   1/8%  costs           0.65   3     =         1.95d
                                          2s    0.90d  =   2s 3/4d  or 2s 1d

One sees in this example, a multiplication of mixed radicals, with little confusion. This is what i call 'invoice arithmetic' but i seen a reference in an old 'shilling arithmetic' as the method of practice. Wendy.krieger (talk) 08:42, 30 October 2011 (UTC)[reply]

Conversion between MRN systems

Obvious algorithms for doing this using the above tools are well known (so says I, research to be done :-) Doing these conversions efficiently seems to be a serious issue in Computer Science.

--J. W. McLeod 19:04, 27 Mar 2005 (UTC)

Factorial radix system

the table shows 0! as 0 whereas it should be 1. There is probably a good reason for this, but it's not obvious. Would s/o care to elaborate ? — Preceding unsigned comment added by 88.172.186.106 (talk) 23:36, 28 September 2011 (UTC)[reply]

The product of zero numbers leaves the original undisturbed, and is so 1. Even something like a × x⁰ = a, for all values of x, including x=0. Elsewise, one can argue that any number is equal to any other, by virtue that

  a = a.0⁰,    by virtue that the LHS has zero values = 0
    = 0                   by placing 0^0 = 0

It's pretty much simple to demonstrate that whatever you set 0^0, like all x^0, equals 1, as is the product of an empty list.

So it's

  0! =    product()     = 1
  1! =    product(1)    = 1
  2! =    product(1,2)  = 2
  3! =    product(1,2,3)= 6
       &c.

Wendy.krieger (talk) 08:30, 30 October 2011 (UTC)[reply]

Huh? The table shows 0! as 1. Double sharp (talk) 10:18, 23 June 2014 (UTC)[reply]

Primorial number system

This subsection was removed by User:remsense. Before starting an edit war, I will list my arguments here:

The removal broke existing WP links in Combinatorial number system, Primorial and List of numeral systems.
The term is, for example, used in these published articles: Survey of RSA Vulnerabilities, Efficient Inner-Product Encryption over NTRU Lattices
The term is also used in well known mathematical resources like Wolfram language reference.
The argument that the On-Line_Encyclopedia_of_Integer_Sequences is user maintained applies only to a certain degree. The first 5488 entries of the OEIS were published in Neil Sloane's book The Encyclopedia of Integer Sequences (ISBN 0-12-558630-2), and it would be merely impossible to publish the currently 377,205 entries in a medium different from an Internet database. All entries are peer reviewed by two editors (many of them university professors) before they are approved. So in my eyes the OEIS is - among mathematicians - a well reknown, reliable, published source.

BTW, a good overview of such number systems can by found in Weird Number Bases.

For me, there are enough reasons to restore the section. -- Gfis (talk) 12:09, 21 October 2024 (UTC)[reply]

Editorial decisions for one article are not made based on whether other articles also have issues, cf. WP:OTHERCONTENT. Feel free to go and fix them instead of breaking this one to ensure they're all equally broken, is what I say.
These aren't really about the subject, they are technical works in a related discipline that just mention it briefly while pursuing other ends. Critically, there's nothing in these sources that actually verifies anything of interest for the article.
Again, this is a brief note in documentation for a programming language. The mention serves other ends, it is not meaningfully discussing your addition.
OEIS does more meaningfully discuss your addition. However, the fact remains that it is user-generated and self-published. That doesn't make it a bad resource, it just makes it one we don't use on Wikipedia to ensure things are worth talking about.

Again, a peer-reviewed paper discussing the properties of primordial numbers would be ideal. There is presently no case according to site policy among your points whatsoever to restore the material as it stands. Remsense ‥ 论 12:18, 21 October 2024 (UTC)[reply]

References to primorial base (not yet using that name) go back almost 100 years, with Pillai proving nontrivial results about it in 1930:

S. S. Pillai, "An arithmetical function concerning primes", Annamalai University Journal (1930), pp. 159-167.

It's not obvious that it's sufficiently notable for its own article, although I would argue that it is. But definitely it's important enough to merit a subsection on a page as on-point as Mixed radix. I'm going to be WP:BOLD and restore it, adding this reference and perhaps a few newer ones.

CRGreathouse (t | c) 12:35, 21 October 2024 (UTC)[reply]

Cheers! Thanks for the research. Remsense ‥ 论 12:37, 21 October 2024 (UTC)[reply]