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 Base 180, Time to turn around?
Double sharp
Posted: Nov 2 2017, 02:20 PM


Dozens Disciple


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Joined: 19-September 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 18-on-10, 12-on-15, or 9-on-20.

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 U-turn, 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 U-turn 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 9-on-20 notation, primes in boldface type

  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c _d _e _f _g _h _i _j
0_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1_ 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
2_ 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
3_ 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79
4_ 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
5_ 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119
6_ 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139
7_ 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159
8_ 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179
  • Of the 41 primes p less than r = 180, only {2, 3, 5} are divisors of r. The remaining 38 primes q are coprime to r (i.e., the greatest common divisor (highest common factor) gcd(q, r) = 1).
  • There are 18 divisor digits d | r: {0, 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90} (the numeral “0” signifying an arbitrary integer n ≡ 0 (mod 180).) Read this brief description of the divisor.
  • There are 48 totatives t such that gcd(t, r) = 1. Four-fifteenths or 26.7% of the digits of base 180 are totatives (the totient ratio φ(r)/r = 4/15 for all bases r having the distinct prime divisors {2, 3, 5}). Read this brief definition and description of the significance of the totative.
  • Base 180 has 26 semidivisors (regular numbers gr that do not divide r evenly). This set includes any integer 0 < gr that is the product strictly of one or more of the distinct prime divisors of 180 that does not divide 180 evenly. Together with the 18 divisors, there are 44 regular digits in base 180; just under a quarter (24.4%) of the digits of base 180 are regular. This post defines regular numbers, and their significance.
  • There are 115 neutral digits in the scale of 180, including the 26 aforementioned semidivisors; the remaining 89 are semitotatives, products pq of at least one distinct prime divisor p and at least one prime q that is coprime to 180. Nearly 2/3 (63.9%) of the digits of base 180 are neutral; the preponderance are semitotatives, making up just under half (49.4%) of all the digits of base 180. The highly composite number 180 lies just before the line where semitotatives are the dominant type of digit. As the HCN gets larger, even more semicoprime numbers are admitted as digits, such that they dominate highly composite 5-smooth bases more intensively.
  • Let the integers ω = (r − 1) and α = (r + 1). The ω-totative of base 180 is 179, which is prime and does not distribute the omega divisibility rules to smaller digits. Likewise, the α-number of base 180 is 181, which is also prime and does not distribute the alpha divisibility rules to smaller digits. Both the “digit-sum” and “alternating-digit” rules would be academic curiosities in base 180, flanked by two large primes. Base 180 is even more bereft of these indirect divisibility tests and brief recurrent periods for unit fractions than dozenal. See this post for maps and descriptions of the omega and alpha divisibility rules for bases less than 30.
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Double sharp
Posted: Nov 2 2017, 02:22 PM


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Base 180 Intuitive Divisibility Tests

Intuitive Divisibility Tests for Base 180, using 9-on-20 notation, primes in boldface type

  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c _d _e _f _g _h _i _j
0_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1_ 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
2_ 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
3_ 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79
4_ 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
5_ 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119
6_ 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139
7_ 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159
8_ 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179

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.

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icarus
Posted: Nov 2 2017, 09:19 PM


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Very good! Added. Wrestling with some work and will add some bases as requested soon.
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Double sharp
Posted: Nov 3 2017, 03:38 AM


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QUOTE (icarus @ Nov 2 2017, 09:19 PM)
Very good! Added. Wrestling with some work and will add some bases as requested soon.

Please, do take your time: we all want to see them done perfectly! happy.gif

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


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Joined: 11-July 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 3-5-7 numbers, and goes a good deal towards explaining base 105 has quite sizable gaps in the log-wheel 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.
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