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).

For completion of the examples shown in the panel, the numbers up to 200 in this variable base are:

1=1
2=10
3=11
4=20
5=21
6=100
7=101
8=110
9=111
10=120
11=121
12=200
13=201
14=210
15=211
16=220
17=221
18=300
19=301
20=310
21=311
22=320
23=321
24=1000
25=1001
26=1010
27=1011
28=1020
29=1021
30=1100
31=1101
32=1110
33=1111
34=1120
35=1121
36=1200
37=1201
38=1210
39=1211
40=1220
41=1221
42=1300
43=1301
44=1310
45=1311
46=1320
47=1321
48=2000
49=2001
50=2010
51=2011
52=2020
53=2021
54=2100
55=2101
56=2110
57=2111
58=2120
59=2121
60=2200
61=2201
62=2210
63=2211
64=2220
65=2221
66=2300
67=2301
68=2310
69=2311
70=2320
71=2321
72=3000
73=3001
74=3010
75=3011
76=3020
77=3021
78=3100
79=3101
80=3110
81=3111
82=3120
83=3121
84=3200
85=3201
86=3210
87=3211
88=3220
89=3221
90=3300
91=3301
92=3310
93=3311
94=3320
95=3321
96=4000
97=4001
98=4010
99=4011
100=4020
101=4021
102=4100
103=4101
104=4110
105=4111
106=4120
107=4121
108=4200
109=4201
110=4210
111=4211
112=4220
113=4221
114=4300
115=4301
116=4310
117=4311
118=4320
119=4321
120=10000
121=10001
122=10010
123=10011
124=10020
125=10021
126=10100
127=10101
128=10110
129=10111
130=10120
131=10121
132=10200
133=10201
134=10210
135=10211
136=10220
137=10221
138=10300
139=10301
140=10310
141=10311
142=10320
143=10321
144=11000
145=11001
146=11010
147=11011
148=11020
149=11021
150=11100
151=11101
152=11110
153=11111
154=11120
155=11121
156=11200
157=11201
158=11210
159=11211
160=11220
161=11221
162=11300
163=11301
164=11310
165=11311
166=11320
167=11321
168=12000
169=12001
170=12010
171=12011
172=12020
173=12021
174=12100
175=12101
176=12110
177=12111
178=12120
179=12121
180=12200
181=12201
182=12210
183=12211
184=12220
185=12221
186=12300
187=12301
188=12310
189=12311
190=12320
191=12321
192=13000
193=13001
194=13010
195=13011
196=13020
197=13021
198=13100
199=13101
200=13110

Note the apparent gap at 24 (4!) and 120 (5!) - apparent for those of us who are used to decimal numbers.

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}}

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