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Double sharp 
Posted: Nov 5 2017, 05:54 AM


Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 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


Double sharp 
Posted: Nov 5 2017, 06:00 AM


Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
Intuitive Divisibility Rules
As is evident, there are relatively few intuitive divisibility tests in base 480 considering the large scale of the base. This said, most common 5smooth 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. 
