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==
{{incomplete|Created by a VARIABLE-BASED BOT BEING ESCORTED OUT OF THE COMPUTER SCIENCE DEPARTMENT BY SECURITY - Please change this comment when editing this page. Do NOT delete this tag too soon.}}

This comic is about the {{w|factorial number system}}, which are based on {{w|factorial|factorials}}. No digit in a number will be larger than the position of that number (starting with 1, in the rightmost position, which can be 0 or 1 as base or {{w|radix}} 2) and the positional factor for any digit is the factorial of that same position, rather than the more traditional (constant) radix to the power of the position minus one (or to the power of the positional index, which starts at zero).

This 'works' because each additional digit is required when the prior digit is 'full'. The second digit is needed when the value goes beyond 0 or 1 units/ones (factorial 1, which is 1) and you start to need to know how many twos you might need. 0, 1 or 2 twos (factorial 2, or 2x1) can be invoked. This suffices up until the point where sixes (factorial 3, =3x2x1) are necessary, having exceeded the point at which 2 twos and 1 one are sufficient. Beyond 3 sixes, 2 twos and 1 one (which is 23), the fourth digit must represent the number (0 to 4) of twenty-fours (factorial 4, =4x3x2x21), etc.

In the comic, the top example represents 3x(720) + 5x(120) + 3x(24) + 0x(6) + 1x(1), after calculating each factorial accordingly, which gives the decimal value of 2835, [[2835: Factorial Numbers|this comic's number]].

The title-text 'addresses' the issue of which digits are used once any individual digit's radix goes beyond base-10 by... not allowing numbers to get so big that you'd try to use digits with a place-value greater than 9, which would limit any use to below factorial 10.

In the xkcd version of this number system, the rightmost digit has a value of 1!, the second one 2! and so on (that is, the i-th digit has a value i!). That can be compared with the usual decimal system where the i-th digit has value 10^(i-1) or the binary system where the i-th digit has value 2^(i-1).

The title text discusses a "problem" with this system, in that numbers above 3,628,800 (10!) have ambiguous notation, as it can be difficult to know whether the number in this system is (10)000000000, or (1)0000000000. Some use the letters A-Z to denote such larger numbers, e.g. A000000000. However, Cueball in this comic just announces that an number above 987654321 in this number system (or 3,628,799) is illegal.

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}
:Variable-base Factoradic™ numbers

:{|
|Base 7||Base 6||Base 5||Base 4||Base 3||Base 2
|-
|3||5||3||0||1||1
|}

:Left side

:{| class="wikitable"
|Base 10||Factoradic
|-
|1||1
|-
|2||10
|-
|3||11
|-
|4||20
|-
|5||21
|-
|6||100
|-
|7||101
|-
| 
|-
|21||311
|-
|22||320
|-
|23||321
|}

:Right side

:{| class="wikitable"
|Base 10||Factoradic
|-
|24||1,000
|-
|25||1,001
|-
| 
|-
|5,038||654,320
|-
|5,039||654,321
|-
|5,040||1,000,000
|-
| 
|-
|999,998||266,251,210
|-
|999,999||266,251,211
|-
|1,000,000||266,251,220
|-
|1,000,001||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!

: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]]