Editing 2835: Factorial Numbers

Jump to: navigation, search

Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.

The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then save the changes below to finish undoing the edit.
Latest revision Your text
Line 10: Line 10:
  
 
==Explanation==
 
==Explanation==
−
A {{w|factorial}} is a product of positive integers. For instance, four factorial, written '4!', means 4×3×2×1=24.  
+
{{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.}}
  
−
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.
+
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).
  
−
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.
+
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 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.
+
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|this comic's number]].
  
−
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.
+
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.
  
−
Cueball's examples of numbers written in factored appear as sequences [https://oeis.org/A007623 A007623] in the OEIS.
+
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==
 
==Transcript==
−
{{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}}
+
{{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...}}
−
:[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.]
+
:Variable-base Factoradic™ numbers
−
:[Poster:]
 
  
−
: 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
Line 34: Line 239:
 
|}
 
|}
  
−
: Left side
+
:Left side
  
 
:{| class="wikitable"
 
:{| class="wikitable"
Line 62: Line 267:
 
|}
 
|}
  
−
: Right side
+
:Right side
  
 
:{| class="wikitable"
 
:{| class="wikitable"
Line 94: Line 299:
 
: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!
−
:[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.
+
 
 +
: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}}
Line 102: Line 308:
 
[[Category:Math]]
 
[[Category:Math]]
 
[[Category:Self-reference]] <!-- Comic number encoded in image 'example' -->
 
[[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 -->
 
−
[[Category:Popular Comics]]
 

Please note that all contributions to explain xkcd may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here.
You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see explain xkcd:Copyrights for details). Do not submit copyrighted work without permission!

To protect the wiki against automated edit spam, we kindly ask you to solve the following CAPTCHA:

Cancel | Editing help (opens in new window)

Templates used on this page:

Retrieved from "https://www.explainxkcd.com/wiki/index.php/2835:_Factorial_Numbers"