9717

Factorial Number Base

This Demonstration expands on Stephen Wolfram's Demonstration, called "Numbers in Various Bases", to show a fanciful number system that breaks numbers into blocks of size 1!, 2!, 3!, etc., with the number of blocks of size ! equal to the digit in the number's representation.
For example, "11" stands for one 2! + one 1! = 3. "321" is the largest three-digit number and is three 3!'s + two 2!'s + one 1! = 23, one short of "1000" or one 4! = 24.
Digits past 9 are represented by letters and the conversion function here stops where the letter is the last letter needed, servicing numbers up to about .

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DETAILS

The factorial base, while not very compact for small numbers, wins out eventually as a more compact notation than an ordinary base. The factorial base is the last one in the list, subscripted by and is more compact than any others in the list over the range of exponential numbers in the Demonstration. Furthermore, for any normal base there is easily found some number large enough that and all greater numbers are more compact, containing fewer digits, than the numbers in base and smaller bases.
The base conversion function used for the factorial base here uses an explicit table of factorials up to 36, thus using 10 digits and 26 letters. That represents numbers up to about and is coincidentally near the point where the factorial base starts to appear as more compact than the largest standard base in the list, the hexadecimal.
Furthermore, a font size slider allows longer numbers to fit on single lines, at the expense of readability.
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