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==Explanation==
 
==Explanation==
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A {{w|factorial}} is a product of positive integers. For instance, four factorial, written '4!', means 4×3×2×1=24.  
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{{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.}}
  
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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.
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This comic is based on the {{w|factorial number system}}, which is a way of writing integers or real numbers using {{w|factorial|factorials}} instead of powers. Unlike the 'proper' version of this system, [[Randall]]'s version does not include the rightmost digit that adds no information, since it is always 0.
  
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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.
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A factorial is a product of the first few positive integers. For instance, four factorial, written '4!', means 4×3×2×1 = 24. These can be used to write numbers in a strange way.
  
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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.
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Normally, numbers are represented in a positional system with a constant base, especially base ten. This means that each digit in a number has a place value based on its position, and that value is a power of ten. For instance, the number 137 usually means 1×10² + 3×10¹ + 7×10⁰, i.e. one hundred, three tens, and seven units. We say that the 1 is in the hundreds place, the 3 in the tens place, and the 7 in the ones place (or units). The same number could be written in base sixteen as 89, meaning 8×16¹ + 9×16⁰, i.e. eight sixteens and nine units. The 8 is in the sixteens place, and the 9 is in the ones place. In a "factorial base," instead of each digit being multiplied by an escalating power of some constant base, each digit is multiplied by an escalating factorial. So that same number could be written 10221, meaning 1×5! + 0×4! + 2×3! + 2×2! + 1×1!. We could say there is a 1 in the 120s place, a 0 in the 24s place, a 2 in the 6s place, another 2 in the 2s place, and a 1 in the ones place.
  
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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.
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In normal base-n notation, n digits are used, running from 0 to n–1. For instance, in base ten, we use the ten digits {0,...,9}. In base sixteen, we need sixteen digits, so we use {0,...,9,A,...,F}. But in factorial base, we need up to n+1 different digits to express all n-digit numbers. For instance, with just two digits, we can express both one-digit numbers 0 and 1. We can also express some larger numbers like 10 = two and 11 = three, but we can't express 20 = four or 21 = five. As a result, Randall jokes that since we only have ten digits {0,...,9}, we can only express numbers with up to nine digits, making larger numbers "illegal." Randall believes that would make the largest "legal" factorial base number 987654321 = 9×9!+8×8!+7×7!+6×6!+5×5!+4×4!+3×3!+2×2!+1×1!, which in base ten is 3,628,799 (which he calls "about 3.6 million"). In fact, adding one to this number gives 1000000000, which still doesn't require any digits larger than 9. The first number that cannot be represented this way with the ten symbols {0,...,9} comes right after 9987654321, which in decimal equals 36,287,999
  
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Cueball's examples of numbers written in factored appear as sequences [https://oeis.org/A007623 A007623] in the OEIS.
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In the comic, the top example represents 3×720 + 5×120 + 3×24 + 0×6 + 1×1, after calculating each factorial accordingly, which gives the decimal value of 2835, [[2835|this comic's number]].
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<!-- REPEATED INFO? 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). -->
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For completion of the examples shown in the panel, the numbers up to 200 in this variable base are:
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1=1
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2=10
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3=11
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4=20
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5=21
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6=100
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7=101
 +
8=110
 +
9=111
 +
10=120
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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
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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
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73=3001
 +
74=3010
 +
75=3011
 +
76=3020
 +
77=3021
 +
78=3100
 +
79=3101
 +
80=3110
 +
81=3111
 +
82=3120
 +
83=3121
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84=3200
 +
85=3201
 +
86=3210
 +
87=3211
 +
88=3220
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89=3221
 +
90=3300
 +
91=3301
 +
92=3310
 +
93=3311
 +
94=3320
 +
95=3321
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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
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113=4221
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114=4300
 +
115=4301
 +
116=4310
 +
117=4311
 +
118=4320
 +
119=4321
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120=10000
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121=10001
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122=10010
 +
123=10011
 +
124=10020
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125=10021
 +
126=10100
 +
127=10101
 +
128=10110
 +
129=10111
 +
130=10120
 +
131=10121
 +
132=10200
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133=10201
 +
134=10210
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135=10211
 +
136=10220
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137=10221
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138=10300
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139=10301
 +
140=10310
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141=10311
 +
142=10320
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143=10321
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144=11000
 +
145=11001
 +
146=11010
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147=11011
 +
148=11020
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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
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Note the apparent gap at 24 (4!) and 120 (5!) - apparent for those of us who are used to decimal numbers.
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<!-- REPEATED INFO, AND WRONG IN THE "above 10!" BIT? 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==
 
==Transcript==
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{{incomplete transcript|Do NOT delete this tag too soon. - Still needs a lot of deconstruction/reconstruction work on the [Poster:] to make it properly Transcripted (no tables, ideally!), but have improved the surrounding markup/descriptions}}
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{{incomplete transcript|Do NOT delete this tag too soon. - Would be best done entirely without wikitables. And actually describe the police/security intervention going on. But there'll be plenty of editors passing this way soon enough...}}
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:[Cueball is standing in front of a large poster. There are two uniformed officers (a Ponytail and a further Cueball, wearing badged hats) approaching Cueball.]
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:Variable-base Factoradic™ numbers
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:[Poster:]
 
  
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: Variable-base Factoradic™ numbers
 
 
:{|
 
:{|
 
|Base 7||Base 6||Base 5||Base 4||Base 3||Base 2
 
|Base 7||Base 6||Base 5||Base 4||Base 3||Base 2
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|}
 
|}
  
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: Left side
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:Left side
  
 
:{| class="wikitable"
 
:{| class="wikitable"
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|}
 
|}
  
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: Right side
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:Right side
  
 
:{| class="wikitable"
 
:{| class="wikitable"
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:Cueball: Small numbers should be written with small numerals like "1" or "2".
 
: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!
 
:Cueball: That's why my variable-base system uses...Hey! No, listen!
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:[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.
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: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}}
 
{{comic discussion}}
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[[Category:Math]]
 
[[Category:Math]]
 
[[Category:Self-reference]] <!-- Comic number encoded in image 'example' -->
 
[[Category:Self-reference]] <!-- Comic number encoded in image 'example' -->
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[[Category:Comics featuring Ponytail]] <!-- Hatted 'security officer' -->
 
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[[Category:Multiple Cueballs]] <!-- If including otherwise cueball-like hatted 'security officer' of no other distinction -->
 
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[[Category:Popular Comics]]
 

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