Duotrigesimal (Base 32) Numbers
Date: 06/11/99 at 08:17:55 From: Jan A. de Boer Subject: Base 32 numbers, duotrigesimal Dear Dr. Math, By chance I read several of the base [any number] pages on your site. I would like to mention a problem we had in finding a condensed way of writing numbers. In the financial software of our university we had to represent six- digit customer project numbers in a four-position alphanumeric database field. The traditional way. by maintaining conversion tables with the numbers and their four position code, seemed cumbersome. I thought that increasing the base number sufficiently would result in four-position representations of decimal numbers up to 1000000. Clearly hexadecimal numbers (Greek hexa-kai-deka = 16) were not the answer, since FFFF represents decimal 65535. So I tried base 32. Each position in a number would have values from 0 up to 31, and four positions would result in 31*32^3 + 31*32^2 + 31*32 + 31, which is of course equivalent to 32^4 - 1 = decimal 1048575. That would do the trick. I baptised these numbers "duotricesimal" (Latin duotriginta = 32). For the position values 0-9 one uses of course 0-9, and for 10-15 the hexadecimal notation A-F. For 16 and up we excluded the letters I and O to avoid mixups with 1 and 0. So following 0-9 we have in use: A 10 G 16 N 22 U 28 B 11 H 17 P 23 V 29 C 12 J 18 Q 24 W 30 D 13 K 19 R 25 X 31 E 14 L 20 S 26 F 15 M 21 T 27 Next thing was to write short function procedures to convert decimal to duotricesimal and vice versa. Because base conversions are so simple, they turn out to be much faster than look-ups in conversion tables, and we avoided an extra table to maintain. Best regards, Jan A. de Boer Office of the Faculty of Mathematics and Natural Sciences University of Groningen, The Netherlands
Date: 06/11/99 at 13:03:06 From: Doctor Peterson Subject: Re: Base 32 numbers, duotrigesimal Hi, Jan. It's always nice to hear that the math we discuss has real uses! I think the proper term would be "duotrigesimal"; there is a "vigesimal" system, the base 20 used by the Maya and others, and a "sexagesimal" (base 60) system used by the Babylonians, which suggest how to make such names from the Latin. But I didn't find any variant of your name mentioned anywhere on the Web. I've been told about some weird bases used in strange situations, but this is a first for 32. I'm curious about several things. First, why did you have to fit a number into a four-character field? Is this a case, similar to the Y2K problem, of having to change ranges without affecting existing stuff, or is it simply that the database is not designed with flexibity in mind? Second, why did you stop at base 32, and not go all the way to base 36 using all the letters? If conversion to or from binary is important, I can easily see why you'd do this; otherwise, I suppose it's as much aesthetic as anything. Thanks for the information. Maybe we'll put together a list of "unusual bases in unusual places" someday. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 06/14/99 at 04:28:01 From: Jan Allerd de Boer Subject: Re: Base 32 numbers, duotrigesimal Dear Dr. Peterson, Yes thanks, I think duotrigesimal is like the word the Romans would have used. I will change our documentation and use duotrigesimal. >I'm curious about several things. First, why did you have to fit a >number into a four character field? Is this a case, similar to the >Y2K problem, of having to change ranges without affecting existing >stuff, or is it simply that the database is not designed with >flexibity in mind? It is the everlasting story of practical limitations. The University's financial department is not keen on introducing peculiarities in the software it bought from BaaN. Everything is flexible, but changing table structures has a high price that is billed again if a software upgrade is implemented. So "below" the level of faculty projects that act as supplier we only have that table with a four character key (indexed, very important) to store customer project numbers. Other fields of the table are non-indexed character fields for descriptions, not involved in the referential integrity of the database. We negotiated changing the key to six characters but we lost. >Second, why did you stop at base 32, and not go all the way to base >36 using all the letters? If conversion to or binary is important, I >can easily see why you'd do this; otherwise, I suppose it's as much >aesthetic as anything. In the conversion function procedures we did not use the property of 32 being a power of 2. So the choice seems aesthetic, but it is also instinctive. Considerations were: First, 32 is the minimum base size that counts up to a million in four positions: 31^4 - 1 is only 923520. Second, I wanted to exclude letters I and O because of their resemblance with 1 and 0. So that would make only base 34 possible. Third, I am of the lazy type, fond of things one could use in another way. Base 32 might be handy for a binary problem (like base 8 and 16). Base 34 has no advantage there. Best regards, Jan de Boer
Date: 06/14/99 at 09:01:36 From: Doctor Peterson Subject: Re: Base 32 numbers, duotrigesimal Hi again, Jan - Thanks for indulging my curiosity. It does sound as if you came up with a beautiful solution to your problem. The fact that 2^10 is close to 10^3, and therefore 32^4 = 2^20 is close to 10^6, is what made it work. Incidentally, if you want to see one of my "weird bases", take a look here: http://mathforum.org/dr.math/problems/behymer8.20.98.html where someone actually uses base 48 as part of her job, and not for such pragmatic reasons as your base 32. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 05/12/2000 at 14:28:03 From: Tomer Subject: About Bases (64 & 85) As a follow-up comment on your response about base32, I'd like to add two more bases that I know are being used in the computer world. The first is base 64, which is used when transferring files via e-mail in SMTP, the second is base 85 which is used in the notation of new IP addresses. Thought you'd be interested to know.
Date: 05/12/2000 at 17:10:57 From: Doctor Peterson Subject: Re: About Bases (64 & 85) Hi, Tomer. Thanks for adding to my collection. I looked for some good references to these two applications and found "A Compact Representation of IPv6 Addresses" at: http://www.landfield.com/rfcs/rfc1924.html but nothing yet that gives details on SMTP MIME base 64 encoding, though I found some references to it. Do you have a good source I can add to my list? - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 05/13/2000 at 11:51:36 From: Levinboim Family Subject: Re: About Bases (64 & 85) Try "Mechanisms for Specifying and Describing the Format of Internet Message Bodies " at this URL: http://www.ietf.org/rfc/rfc1341.txt in Section 5.2
Date: 05/13/2000 at 20:11:06 From: Doctor Peterson Subject: Re: About Bases (64 & 85) Hi, again. Thanks. If you run across any other odd uses of bases (especially outside the computer world, where bases are familiar), let me know! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 07/13/2001 at 21:35:37 From: Sean Riddle Subject: More bases used in the "real world" I've got a couple more bases that I've run into in the "real world." They are both computer uses, though. I bought an EPROM programmer many years ago (EPP-1 from Applied Reader Technology in Holland) that used a format called "four packed code." The manual wasn't too informative, and there was no example of its use, but the idea was that 85^5 is very close to 256^4, so five printable characters can represent 4 bytes. This is more efficient than using the ASCII representation of the binary data, which takes two printable characters for each byte. WorldCom uses base 36 in filenames. That lets them use 1 character for the day of the month. This is important since their filenames are restricted to 8 characters with a 3-character extension. They also use base 36 in the extension as a sequence number. That lets us get 12 files a day and not repeat a sequence number in 10 1/2 years. Sean
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