5x6

5x6

cstyle643
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cstyle643
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Joined: Aug 12 2018, 06:08 PM

Aug 24 2018, 08:21 PM #1

In my "Getting to Know You" post, I linked to a 5x6 approach by a professor named David Madore.  I really like Madore's system and think it is very straightforward for people interested in pursuing trigesimal.  I thought I would outline it below.

Like other alternating base systems, Madore's numbers have paired symbols.  The "inner" or right-hand symbol uses the standard arabic numerals 0-5.  The "outer" or left-hand symbol uses latin letters.  Since these are in Base 5, you only need A-D, so they're very easy to understand.  (Madore also uses a Z for zeros in this category; I, however, find it easier just to use the standard zero in both columns.) 

It is true that one does not need to use different symbol conventions in an alternating base system.  For example, in 6x10, we often see the numbers represented just as they would in decimal.  But I think the A-D/1-5 convention works well here, as both columns are non-decimal, but are different from another.  I find it is easier to remember that you can't go past D/5 than it would be if the rule were you can't go past 4 in the left of each pair and 5 in the right.  Also, I think the mixed notation helps underscore that you really need to consider each symbol pair as a unit.

The result is a system that requires only ten symbols, just like decimal.

Madore's post has three different multiplication tables - for "ones times ones", "ones times sixes" and "sixes times sixes."  I'm attaching a simplified, combined table.  Note that it only includes 1-5 and exact multiples of 6.  As a result, if you are multiplying "mixed numbers", you need to perform the multiplication operation separately for each of the two pairs and then add them up.  While this is a little more involved, it's pretty easy and the benefit is a very simple table that's easy to learn.  (If you strip out the multiplicative identity row and column, you are left with an 8x8 table, i.e., 36 products, several of which are immediately obvious.)

For example, say you want to multiply B4 by D2 (equivalent to 16x26 in decimal or 14x22 in dozenal).  You would find the products (from the table) for B0xD0 (=A3,C0); B0x2 (=D0); 4xD0 (=3,A0); and 4x2 (=A2).  Summing these up yields B1,D2 = 416 (decimal); 2X8 (dozenal).      
5x6 Table.xlsx (9.87 KiB)
5x6 Table.xlsx (9.87 KiB)
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Kodegadulo
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Kodegadulo
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Joined: Sep 10 2011, 11:27 PM

Aug 26 2018, 02:45 PM #2

Here is your multiplication table rendered into Tapatalk's BBCode, so other members can see it more easily. This way, members won't need to download an .xlsx file, or even own Excel.

1
2
3
4
5
A0
B0
C0
D0
2
4
A0
A2
A4
B0
D0
1,A0
1,C0
3
A0
A3
B0
B3
C0
1,A0
1,D0
2,B0
4
A2
B0
B4
C2
D0
1,C0
2,B0
3,A0
5
A4
B3
C2
D1
1,00
2,00
3,00
4,00
A0
B0
C0
D0
1,00
1,A0
2,B0
3,C0
4,D0
B0
D0
1,A0
1,C0
2,00
2,B0
4,D0
A1,A0
A3,C0
C0
1,A0
1,D0
2,B0
3,00
3,C0
A1,A0
A4,D0
B2,B0
D0
1,C0
2,B0
3,A0
4,00
4,D0
A3,C0
B2,B0
C1,A0


To see the raw BBCode that generates this, try quoting this post and then checkmark the "Disable BBCode" option. (I often find it easier to edit a post that contains a lot of BBCode while this option is checked -- just have to remember to uncheck it before actually posting! ) To generate the BBCode for this, I opened your original file in Excel and then used a formula to surround each of your cells with td, center, and some cases b tags, and then another formula to concatenate the cells in a row and wrap them in tr tags, then just pasted the resulting 9 lines of text here, surrounding them with a table tag.  Note that Tapatalk generally honors line breaks within a section of raw text, but it's got its own opinions about line breaks between tags, so if you look at this after it's posted, you won't see my table rows as the single text lines I actually pasted here.

cstyle643 wrote:Like other alternating base systems, Madore's numbers have paired symbols.  The "inner" or right-hand symbol uses the standard arabic numerals 0-5.  The "outer" or left-hand symbol uses latin letters.  Since these are in Base 5, you only need A-D, so they're very easy to understand.  (Madore also uses a Z for zeros in this category; I, however, find it easier just to use the standard zero in both columns.)

It is true that one does not need to use different symbol conventions in an alternating base system.  For example, in 6x10, we often see the numbers represented just as they would in decimal.  But I think the A-D/1-5 convention works well here, as both columns are non-decimal, but are different from another.  I find it is easier to remember that you can't go past D/5 than it would be if the rule were you can't go past 4 in the left of each pair and 5 in the right.  Also, I think the mixed notation helps underscore that you really need to consider each symbol pair as a unit.
Well, no one can argue with what works for you personally. But for my part (and of course I can only speak for myself), I'd find using two sets of symbols for this, with the second set being letters, distinctly disadvantageous, for a couple reasons.  First of all, in dealing with transdecimal bases, one of the conventions some of us here have adopted is what I like to call the "Computerese" convention, where A=10A, B=11A, C=12A=10C, D=13A=11C, etc ... all the way up to Z=35A=2BC (here using the same letters as subscripts indicating the bases themselves). That convention can thus support a "pure-base" scheme for bases from binary up to base 36A=30C. In fact, both Excel and Google Sheets have a BASE function that supports this.  And of course it's quite familiar to folks in the mainstream computer industry as the digit representation for base G=16A=14C=10G.  And of course base thirty can fit into this as base U, with "pure" digits from 0 to 9 and A to T.  Being accustomed to this convention, my natural tendency is to map letters to transdecimals, so I find a usage such as yours quite jarring.

Second, what you have here is a custom solution for a particular favorite base; but having studied many kinds of bases in this forum, I prefer more general solutions that cover a lot more ground, so my poor overtaxed brain 🙂 might be able to take skills it's already mastered in one base and re-apply them to other situations, with minimal effort.  I'd describe this base you're using as a "two-column superbase". It can be characterized as either of these cases:
  1. Base [5×6] = quinary-on-senary alternating superbase, i.e., a superbase with quinary subbase in the high column, and senary subbase in the low column. (This seems to be the way you're looking at it.)
  2. Base [506] = base "fifsen", i.e. senary-encoded base thirty (or senary-encoded base twenzy-six).  In other words, the subbase is simply senary, but instead of "saturating" every two digits of senary with values from zero to fifsen-five (thirty-five, or twenzy-leven), we would only make use of values up to foursen-five (twenty-nine, or twenzy-five). The big advantage is that, if you're already versed in senary, then you can immediately transfer some of your skills in that to tackle this base. For instance, if you already know that two times four is eight, three times three is nine, two times five is ten, and three times four is two-sen or "dosen", and if you're already accustomed to spelling "eight" as 126, "nine" as 136, "ten" as 146, and "dosen" as 206, then you can immediately apply that knowledge to doing arithmetic in base fifsen.
In either case, I would represent every superdigit of this base as being composed of two subdigits, ranging from 00 to 45, annotated as the case may be, as either 005×6 to 455×6, or 00506 to 45506. (Nifty that Tapatalk BBCode lets you nest subscripts and superscripts. At least that much was an improvement over InvisionFree.)

Note that this way, a 1 is a 1 is a 1, no matter what base it's in, be that quinary or senary or decimal or dozenal or whatever. No needing to learn that A might also mean 1, but only in a subbase that happens to be in the high column of some superbase.  So if you've already mastered arithmetic in quinary, and for instance already know the quinary multiplication table:

1
2
3
4
2
4
11
13
3
11
14
22
4
13
22
31

then you can immediately make use of that for the high subdigit column here.

cstyle643 wrote:The result is a system that requires only ten symbols, just like decimal.
Eleven, actually. Because you are also making use of commas as superdigit separators.  Some kind of separator between superdigits is completely understandable, because we do want to reinforce that this is a large base, each of whose "digits" is being encoded as a pair of "subdigits" in a smaller subbase (or two smaller subbases depending on your interpretation), and not, let's say, a string of hexadecimal.  I wouldn't choose to use commas for that, since I'd still like to use commas in the same way we use them in decimal, that is, as "grouping" separators (for instance, break up long numerals every three "digits", or in this case every three "superdigits"). Some of us here have opted to use apostrophe as superdigit separators and keep comma for its original purpose. For example:

11'22'33,44'00'12,23'34'45506

At any rate, this scheme only requires seven symbols rather than eleven (or eight symbols rather than twelve if you also count commas -- or for that matter nine symbols rather than thirteen if you also count a period used as a radix point introducing fractional digits).  The presence of distinct superdigit separators, combined with some annotation somewhere of the base in question, is quite adequate reminder that something special must be done at the boundary from one superdigit to another, without requiring redundant symbols for the same values.

So here's how I'd prefer to render your multiplication table:

Base: [506]
01
02
03
04
05
10
20
30
40
02
04
10
12
14
20
40
01'10
01'30
03
10
13
20
23
30
01'10
01'40
02'20
04
12
20
24
32
40
01'30
02'20
03'10
05
14
23
32
41
01'00
02'00
03'00
04'00
10
10
30
40
01'00
01'10
02'20
03'30
04'40
20
40
01'10
01'30
02'00
02'20
04'40
11'10
13'30
30
01'10
01'40
02'20
03'00
03'30
11'10
14'40
22'20
40
01'30
02'20
03'10
04'00
04'40
13'30
22'20
31'10


But given this digit mapping (rows=high subdigit, columns=low subdigit):


0 1 2 3 4 5
0 0 1 2 3 4 5
1 6 7 8 9 A B
2 C D E F G H
3 Ï J K L M N
4 Ö P Q R S T

(using a diaeresis on Ï and Ö to distinguish them from 1 and 0), then the base 506 table above just maps to this excerpt from the base U table:

Base: [ U ]
1
2
3
4
5
6
C
Ï
Ö
2
4
6
8
A
C
Ö
16
3
6
9
C
F
Ï
16
2C
4
8
C
G
K
Ö
2C
36
5
A
F
K
P
10
20
30
40
6
6
Ï
Ö
10
16
2C
C
Ö
16
20
2C
4O
76
Ï
16
2C
30
76
EC
Ö
2C
36
40
EC
J6
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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SenaryThe12th
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SenaryThe12th
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Joined: Mar 1 2018, 02:03 PM

Sep 6 2018, 06:57 PM #3

 senary-encoded base thirty (or senary-encoded base twenzy-six).  In other words, the subbase is simply senary, but instead of "saturating" every two digits of senary with values from zero to fifsen-five (thirty-five, or twenzy-leven), we would only make use of values up to foursen-five (twenty-nine, or twenzy-five). 
Kodo, this is an interesting idea.  
The big advantage is that, if you're already versed in senary, 
which--lets face it--we all should be 😎
then you can immediately transfer some of your skills in that to tackle this base. 
Heretofore, if I were dealing with lots of, say, lengths, in which divisibility by 5 was important, I'd choose a unit, e.g. the archaic term "fist" for "five inches" and do my calculations in terms of those units.  This is more general however.  Not all of the numbers we use have units.  I'll be thinking about this for quite some time.
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Kodegadulo
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Kodegadulo
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Joined: Sep 10 2011, 11:27 PM

Sep 7 2018, 01:10 AM #4

SenaryThe12th wrote:
 senary-encoded base thirty (or senary-encoded base twenzy-six).  In other words, the subbase is simply senary, but instead of "saturating" every two digits of senary with values from zero to fifsen-five (thirty-five, or twenzy-leven), we would only make use of values up to foursen-five (twenty-nine, or twenzy-five). 
Kodo, this is an interesting idea.  
Mapping a larger superbase to a smaller subbase is quite an old idea.

For instance, in the computer field there is such a thing as "binary-coded decimal" [10102], where each decimal digit is mapped onto four binary digits (bits). In principle four bits could represent sixteen hexadecimal digits, but for decimal they are left unsaturated. There have been special processor chips designed to do decimal arithmetic in this format.

But going much farther back, Babylonian sexagesimal could be viewed as decimal-encoded base sixty [60d].
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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