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Double sharp 
Posted: Nov 2 2017, 02:20 PM


Dozens Disciple Group: Members Posts: 1,402 Member No.: 1,150 Joined: 19September 15 
Base 180 Base 180 is a “grand base” that lies well beyond our current ability to wield as a number base of general human arithmetic. Its multiplication table is astronomical (16,290 unique products, over 296 times the size of the decimal table) so any notion of memorizing it is beyond the ability of the average kid in school. Why consider such a gigantic number as a base? The number 180 is another step in the sequence {1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, …}, the highly composite numbers (see OEIS A002182). These numbers set records for their number of divisors. These HCNs shower us with divisors: according to the OEIS, “the number of divisors … increases to a record.” We are intersted in patterns in a number base: the number of divisors and regular numbers in general aids arithmetic in that base. Indirect relationships with neighboring integers helps; base 180 suffers in this consideration as it is flanked above and below by primes. Despite the greater number of divisors, 180 has a burgeoning set of totatives, which conspire to resist human use of a base as a tool of arithmetic. If 60 and 120 were a bit beyond parity between divisors and totatives (12/16 for sexagesimal, 16/32 for base 120), the 48 totatives swamp the 18 divisors of base 180. Over half of the digits of base 180 are neutral, neither divisors nor totatives, but just under half of all digits are semitotatives; it is the last HCN to remain so, since base 240 has over half its digits semitotatives. Perhaps the only way to bypass all the mess over digits is to avoid using base 180 in its pure form and use a mixed radix, say based on 18on10, 12on15, or 9on20. We use the number 360 to measure angles in our system of degrees of arc. Using 360, we can divide a unit circle into {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360} parts without the need to resort to fractions; hence the common colloquial use of "180" to mean a Uturn, since it is halfway around the circle. Base 180 serves us better as an auxiliary base, its arithmetic tamed by much smaller bases, but its divisibility minimizing our need to turn to fractions. As a pure number base, its sheer magnitude puts it out of our ken by a long shot; its resistive set of digits ensures that even if we were able to deal with the size of the arithmetic tables, we’d be in for a great deal of difficulty wielding it. This grand base stands as a testimony to the fact that more divisors does not mean a “better base”. Perhaps we ought to make a Uturn back to 60 and 120, indeed! Let’s take a look at 180 as a number base. Note that this examination is not as thorough as those for smaller bases, for obvious reasons. (Please refer to “Icarus’s Standard Nomenclature for Number Bases” for the legend of the digit map below and any terminology. References to elementary number theory books are given in that post and thread to support what is written in this post.) Base 180 has the following properties: Digits of Base 180, using 9on20 notation, primes in boldface type


Double sharp 
Posted: Nov 2 2017, 02:22 PM


Dozens Disciple Group: Members Posts: 1,402 Member No.: 1,150 Joined: 19September 15 
Base 180 Intuitive Divisibility Tests Intuitive Divisibility Tests for Base 180, using 9on20 notation, primes in boldface type
Base 180 is “bone dry” when it comes to intuitive compound divisibility tests. The intuitive divisibility tests do cover all but 12 of the integers between 2 and 30 inclusive, which is an impressive range considering that each of the tests are directly related to the prime divisors. Some of the regular tests for “remote” regular numbers like digits 125 and 128 would prove highly impractical. 

icarus 
Posted: Nov 2 2017, 09:19 PM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Very good! Added. Wrestling with some work and will add some bases as requested soon.

Double sharp 
Posted: Nov 3 2017, 03:38 AM


Dozens Disciple Group: Members Posts: 1,402 Member No.: 1,150 Joined: 19September 15 
Please, do take your time: we all want to see them done perfectly! I have some time this weekend and will try to get another few done. I have {64} in the works and should be able to finish it shortly. 

wendy.krieger 
Posted: Nov 3 2017, 09:00 AM

Dozens Demigod Group: Members Posts: 2,432 Member No.: 655 Joined: 11July 12 
Base 180 joins 10, 45, 117, in that the 'db' scale is also a useful scale covering all of the regulars. The logs of 2, 3, and 5 are 0;24, 0;38 and 0;56. This gives 2 steps per semitone. 180 is 90 semitones.
Base 117 gives the logs of 3 and 13 are ;27 and ;63 resp Base 45 gives 3, 5, 7 at 13, 19, and 23. This scale is quite handy for sorting the 357 numbers, and goes a good deal towards explaining base 105 has quite sizable gaps in the logwheel despite having three prime divisors. Base 10 gives 2, 5 at 3, 7. So log 2 = 0.3, log 5 = 0.7 Base 17 has e at 0;6 b17, which is quite unusual. Among the reciprocal pairs we find 13.60 v 13.90 Yes, 13,75 sq gives 1,00,01,45. cf b12 346, 368 gives 357˛=100121, and base 40 with 6.10 * 6.16, where 6.13 gives 1,0,0,9. 