Editing 2835: Factorial Numbers

{{comic
| number    = 2835
| date      = September 29, 2023
| title     = Factorial Numbers
| image     = factorial_numbers_2x.png
| imagesize = 628x481px
| noexpand  = true
| titletext = So what do we do when we get to base 10? Do we use A, B, C, etc? No: Numbers larger than about 3.6 million are simply illegal.
}}

==Explanation==
A {{w|factorial}} is a product of positive integers. For instance, four factorial, written '4!', means 4×3×2×1=24. 

The "base" of a numbering system defines which numbers it uses as digits and what each place value in a number means.  For example, in decimal numbers (base 10), the digits go from 0 to 9, and place values are ones, tens, hundreds, etc.  So "137" means 1×100 + 3×10 + 7×1 = 137.  Numbers can also be written in other bases, such as binary (base 2, using the digits 0 and 1 and place values of 1, 2, 4, 8...) or octal  (base 8, using the digits 0-7 and place values of 1, 8, 64, and so on).  Using different bases is uncommon, but is sometimes useful in computer science.

In the comic, [[Cueball]] proposes a {{w|factorial number system}}, where the base ''changes'' for each place value - the first digit can be 0 or 1, the next digit can be 0, 1, or 2, the third can be 0, 1, 2, or 3, and so on.  Each place value is the factorial of the base.  So the number 137 in base 10 could be written as 10221, meaning 1×5! + 0×4! + 2×3! + 2×2! + 1×1!.  While this numbering system is technically usable and can express any number, it seems excessively complicated, and the only reason Cueball gives for using it is that he thinks large digits like 9 should only be used in vast numbers (9 would not be used unless the number was at least 9 digits long, or over 3.2 million in decimal).  This is a silly reason for using a new numbering system,{{cn}} so the math department thinks this is a prank, and has security throw him out.

In the title text, someone points out that a factorial number system needs more and more digits for each place value.  The tenth digit in a factorial number would be in base 11, which needs 11 possible digits, and 0-9 only provides 10.  In bases higher than 10, you can use letters to represent higher digits.  For example, hexadecimal (base 16) goes from 0 to 9, then from A to F.  It would be reasonable to do the same thing for higher bases in factorial numbers.  Instead, Cueball says that it's simply illegal to write numbers larger than about 3.6 million, the largest you can go without using a base greater than 10. This is an absurd limitation, as other numbering systems can go as high as you like.

The number at the top of Cueball's presentation, 353011, is 3×6! + 5×5! + 3×4! + 0×3! + 1×2! + 1×1! which gives the decimal value of 2835, the number of the comic.

Cueball's examples of numbers written in factored appear as sequences [https://oeis.org/A007623 A007623] in the OEIS.

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon. - De-tabled the Transcript (thanks for reminding me), but definitely still needs work...}}
:[Cueball is standing in front of a large presentation poster. There are two uniformed officers (a Ponytail and a further Cueball, wearing badged hats) approaching Cueball.]
:[Poster:]

::[Title:]Variable-base Factoradic™ numbers

::[Diagram in the form of a slip of paper, left edge torn jagged to indicate indefinite continuation]
:::[Several stacked elements. Following the general pattern of having "Base" at the top a base-number below that, and a square below that holding a digit (less than the given base-number)]
:::[Rightmost stack:] Base 2 [square:] 1
:::[2nd rightmost stack:] Base 3 [square:] 1
:::[3rd rightmost stack:] Base 4 [square:] 1
:::[4th rightmost stack:] Base 5 [square:] 0
:::[5th rightmost stack:] Base 6 [square:] 5
:::[6th rightmost stack:] Base 7 [square:] 3
:::[Leftmost stack has no obvious text, its square is cut across by the 'torn' edge of the diagram.]

::[Two column-pairs, with idential paired-headers, above (generally) a 'number', a linking dash and another 'number']
:::[Left headers for both:] Base 10
:::[Right headers for both:] Factoradic
:::[First column of number pairs:]
::::1 &mdash; 1
::::2 &mdash; 10
::::3 &mdash; 11
::::4 &mdash; 20
::::5 &mdash; 21
::::6 &mdash; 100
::::7 &mdash; 101
::::[Gap]
::::21 &mdash; 311
::::22 &mdash; 320
::::23 &mdash; 321
:::[Continuation in second column of number pairs:]
::::24 &mdash; 1,000
::::25 &mdash; 1,001
::::[Gap]
::::5,038 &mdash; 654,320
::::5,039 &mdash; 654,321
::::5,040 &mdash; 1,000,000
::::[Gap]
::::999,998 &mdash; 266,251,210
::::999,999 &mdash; 266,251,211
::::1,000,000 &mdash; 266,251,220
::::1,000,001 &mdash; 266,251,221

:Cueball: Small numbers like seven or nineteen shouldn't use big numerals like "7" or "9".
:Cueball: I mean, "9" is the biggest numeral we have! It should be reserved for '''''big''''' numbers.
:Cueball: Small numbers should be written with small numerals like "1" or "2".
:Cueball: That's why my variable-base system uses...Hey! No, listen!
:[Caption under the comic:] Factorial numbers are the number system that sounds most like a prank by someone who's about to be escorted out of the math department by security.

{{comic discussion}}

[[Category:Comics featuring Cueball]]
[[Category:Characters with hats]]
[[Category:Math]]
[[Category:Self-reference]] <!-- Comic number encoded in image 'example' -->
[[Category:Comics featuring Ponytail]] <!-- Hatted 'security officer' -->
[[Category:Multiple Cueballs]] <!-- If including otherwise cueball-like hatted 'security officer' of no other distinction -->