Editing 2835: Factorial Numbers
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==Explanation== | ==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 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. | |
− | + | A factorial is a product of 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. | |
− | + | 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. | |
− | The number | + | In a "factorial base," instead of each place value being an escalating power of some constant base, each place value is an escalating factorial. The amount to multiply each place value by to get the next place value increases by 1 each time. So that same number (137 in base 10) 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. |
− | + | 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}. Any of these digits can be used in any position. But in factorial base, each position needs an increasing number of different digits to express all n-digit numbers. The comic labels each position with the equivalent base that would allow the same digits, e.g. the place value 3! is "base 4" because it uses the digits 0 to 3. | |
+ | |||
+ | For instance, with just two digits, we can express some numbers with the digits 0, 1, and 2, like 21 = five. But we can't express 30 = six. 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, but he may wishes to stay away from the mere possibility of representing the digit that ''ought'' to use another symbol. The first number that actually cannot be represented with our usual ten symbols {0,...,9} comes right after 9987654321, which in decimal equals 36,287,999. | ||
+ | |||
+ | In the comic, the top example represents 3×720 + 5×120 + 3×24 + 0×6 + 1×2 + 1×1, after calculating each factorial accordingly, which gives the decimal value of 2835, [[2835|this comic's number]]. | ||
+ | |||
+ | |||
+ | 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. | ||
==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. - 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}} | ||
− | :[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. | + | :[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. |
:[Poster:] | :[Poster:] | ||
− | : Variable-base Factoradic™ numbers | + | :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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|} | |} | ||
− | : Left side | + | :Left side |
:{| class="wikitable" | :{| class="wikitable" | ||
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|} | |} | ||
− | : Right side | + | :Right side |
:{| class="wikitable" | :{| class="wikitable" | ||
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|} | |} | ||
− | :Cueball: Small numbers like seven or nineteen shouldn't use big numerals like "7" or "9". | + | :[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:] 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:] 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. | :[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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[[Category:Comics featuring Ponytail]] <!-- Hatted 'security officer' --> | [[Category:Comics featuring Ponytail]] <!-- Hatted 'security officer' --> | ||
[[Category:Multiple Cueballs]] <!-- If including otherwise cueball-like hatted 'security officer' of no other distinction --> | [[Category:Multiple Cueballs]] <!-- If including otherwise cueball-like hatted 'security officer' of no other distinction --> | ||
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