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 Base 480, A Traditional Ream Of Paper, though it'll take more for the tables
Double sharp
Posted: Nov 5 2017, 05:54 AM


Dozens Disciple


Group: Members
Posts: 1,401
Member No.: 1,150
Joined: 19-September 15



Base 480

Base 480 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 (115,440 unique products, nearly 2100 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 480 is another step in the sequence {1, 2, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, …}, the largely composite numbers (see OEIS A002201). These numbers set or equal previous records for their number of divisors. If we are interested in producing a highly patterned number base so we creatures sensitive to pattern can wield said number base, we want to maximize the divisors in a base. Indirect relationships with neighboring integers helps; base 480 has 13 times 37 as its neighbor upstairs; at the scale of 480, 13 is fairly small, but still a rather obscure prime for most people. The number 480 does not have the sort of convenient neighbors on both sides like 120 does.

Despite the greater number of divisors, 480 has a burgeoning set of 128 totatives, which conspire to resist human use of a base as a tool of arithmetic, along with the vast quantities of semitotatives that are the rule once one follows the largely composite numbers from 240 onwards. The number 480 serves us better as an auxiliary base, its arithmetic tamed by that of decimal, but its divisibility minimising our need to turn to fractions. Indeed, in this role 480 served as a grouping in the form of a traditional ream of paper, seen as 20 times 24, until decimalisation replaced it with the less convenient 500.

Perhaps 480 has too large a multiplicity of two for decimal, being less common than 240 or 360, and larger without many distinct benefits. However in octal it would be even more ideal, since it is then written "740", close to the third octal power, making fractions have a familiar appearance: compare octal "1/2 = 0.4, 1/3 = 0.2525..., 1/4 = 0.2, 1/5 = 0.14631463..., 1/6 = 0.12525..." with base-"740" "1/2 ~ 360, 1/3 ~ 240, 1/4 ~ 170, 1/5 ~ 140, 1/6 ~ 120". It thus behaves analogously to its relatives 120 and 960 in decimal.

Let’s take a look at 480 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 480 has the following properties:

Digit Map

  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
1 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
2 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89
3 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119
4 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149
5 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179
6 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209
7 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239
8 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269
9 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299
10 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329
11 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359
12 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389
13 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419
14 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449
15 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479
  • Of the 92 primes p less than r = 480, only {2, 3, 5} are divisors of r. The remaining 89 primes q are coprime to r (i.e., the greatest common divisor (highest common factor) gcd(q, r) = 1).
  • There are 24 divisor digits d | r: {0, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 80, 96, 120, 160, 240} (the numeral “0” signifying an arbitrary integer n ≡ 0 (mod 360).) Read this brief description of the divisor.
  • There are 128 totatives t such that gcd(t, r) = 1. Four-fifteenths or 26.7% of the digits of base 480 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 480 has 40 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 480 that does not divide 480 evenly. Together with the 24 divisors, there are 65 regular digits in base 480; just over 1/8 (13.3%) of the digits of base 480 are regular. This post defines regular numbers, and their significance.
  • There are 329 neutral digits in the scale of 480, including the 40 aforementioned semidivisors, the remaining 289 are semitotatives, products pq of at least one distinct prime divisor p and at least one prime q that is coprime to 480. Over 2/3 (68.5%) of the digits of base 480 are neutral; the preponderance are semitotatives, making up over 3/5 (60.2%) of all the digits of base 480. The superior highly composite number 480 represents the case for largely composite numbers regarding semitotatives. As the largely composite numbers gets larger, a vast amount of semicoprime numbers are admitted as digits, such that they dominate at base 240 and beyond.
  • Let the integers ω = (r − 1) and α = (r + 1). The ω-totative of base 480 is 479, which is prime and does not distribute the omega divisibility rules to smaller digits. The “digit-sum” rule would be an academic curiosity in base 480 as it applies to digit-479, a large prime. The α-number of base 480 is 481 = 13 * 37. Digit-thirteen and digit-37 enjoy the “alpha” divisibility tests, i.e., the “alternating sum” rule, along with base 480’s “11” = 481. Multiples of either 13, 37, and any of the divisors of 480 enjoy intuitive compound divisibility tests. Thirteen is a fairly large prime, exceeding the decimal 11 for which the decimal alpha test applies. Observing that the decimal alternating-sum test is fairly obscure, one can say that any test for divisibility by 13 or 37 will be more so. See this post for maps and descriptions of the alpha divisibility rules for bases less than 30.
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Double sharp
Posted: Nov 5 2017, 06:00 AM


Dozens Disciple


Group: Members
Posts: 1,401
Member No.: 1,150
Joined: 19-September 15



Intuitive Divisibility Rules

  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
1 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
2 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89
3 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119
4 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149
5 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179
6 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209
7 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239
8 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269
9 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299
10 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329
11 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359
12 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389
13 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419
14 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449
15 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479

As is evident, there are relatively few intuitive divisibility tests in base 480 considering the large scale of the base. This said, most common 5-smooth numbers have intuitive divisibility tests, and the intuitive divisibility tests do cover all but 10 of the numbers between 2 and 30 inclusive. Some of the regular tests for “remote” regular numbers like digits 128, 243, and 256 would prove highly impractical.

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