Definition:Factorial Number System
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Definition
The factorial number system is a mixed radix positional numeral system in which the digit in the $i$th column from the right is of base $i$.
Also known as
The factorial number system is also known as factoradic.
Some sources use the misnomer factorial base, despite the fact that the factorials do not act as a number base.
Examples
2000
$2000$ can be expressed in factoradic as:
- $2000_{10} = 243 \, 110_!$
Multiplication Table
The multiplication table in factoradic can be presented as follows:
$\begin {array} {r|rrrrrrrrrrrrr} \times & 1 & 10 & 11 & 20 & 21 & 100 & 101 & 110 & 111 & 120 & 121 & 200 & 201 \\ \hline
1 & 1 & 10 & 11 & 20 & 21 & 100 & 101 & 110 & 111 & 120 & 121 & 200 & 201 \\ 10 & 10 & 20 & 100 & 110 & 120 & 200 & 210 & 220 & 300 & 310 & 320 & 1000 & 1010 \\ 11 & 11 & 100 & 111 & 200 & 211 & 300 & 311 & 1000 & 1011 & 1100 & 1111 & 1200 & 1210 \\ 20 & 20 & 110 & 200 & 220 & 310 & 1000 & 1020 & 1110 & 1200 & 1220 & 1310 & 2000 & 2020 \\ 21 & 21 & 120 & 211 & 310 & 1001 & 1100 & 1121 & 1220 & 1311 & 2010 & 2101 & 2200 & 2221 \\ 100 & 100 & 200 & 300 & 1000 & 1100 & 1200 & 1300 & 2000 & 2100 & 2200 & 2300 & 3000 & 3100 \\ 101 & 101 & 210 & 311 & 1020 & 1121 & 1300 & 2001 & 2110 & 2211 & 2320 & 3021 & 3200 & 3301 \\ 110 & 110 & 220 & 1000 & 1110 & 1220 & 2000 & 2110 & 2220 & 3000 & 3110 & 3220 & 4000 & 4110 \\ 111 & 111 & 300 & 1011 & 1200 & 1311 & 2100 & 2211 & 3000 & 3111 & 3300 & 4011 & 4200 & 4311 \\ 120 & 120 & 310 & 1100 & 1220 & 2010 & 2200 & 2320 & 3110 & 3300 & 4020 & 4210 & 10000 & 10120 \\ 121 & 121 & 320 & 1111 & 1310 & 2101 & 2300 & 3021 & 3220 & 4011 & 4210 & 10001 & 10200 & 10321 \\ 200 & 200 & 1000 & 1200 & 2000 & 2200 & 3000 & 3200 & 4000 & 4200 & 10000 & 10200 & 11000 & 11200 \\ 201 & 201 & 1010 & 1211 & 2020 & 2221 & 3100 & 3301 & 4110 & 4311 & 10120 & 10321 & 11200 & 12001 \\ 210 & 210 & 1020 & 1300 & 2110 & 2320 & 3200 & 4010 & 4220 & 10100 & 10310 & 11120 & 12000 & 12210 \\ 211 & 211 & 1100 & 1311 & 2200 & 3011 & 3300 & 4111 & 10000 & 10211 & 11100 & 11311 & 12200 & 13011 \\ 220 & 220 & 1110 & 2000 & 2220 & 3110 & 4000 & 4220 & 10110 & 11000 & 11220 & 12110 & 13000 & 13220 \\ 221 & 221 & 1120 & 2011 & 2310 & 3201 & 4100 & 4321 & 10220 & 11111 & 12010 & 12301 & 13200 & 14021 \\ 300 & 300 & 1200 & 2100 & 3000 & 3300 & 4200 & 10100 & 11000 & 11300 & 12200 & 13100 & 14000 & 14300 \\ 301 & 301 & 1210 & 2111 & 3020 & 3321 & 4300 & 10201 & 11110 & 12011 & 12320 & 13221 & 14200 & 20101 \\ 310 & 310 & 1220 & 2200 & 3110 & 4020 & 10000 & 10310 & 11220 & 12200 & 13110 & 14020 & 20000 & 20310 \\ 311 & 311 & 1300 & 2211 & 3200 & 4111 & 10100 & 11011 & 12000 & 12311 & 13300 & 14211 & 20200 & 21111 \\ 320 & 320 & 1310 & 2300 & 3220 & 4210 & 10200 & 11120 & 12110 & 13100 & 14020 & 20010 & 21000 & 21320 \\ 321 & 321 & 1320 & 2311 & 3310 & 4301 & 10300 & 11221 & 12220 & 13211 & 14210 & 20201 & 21200 & 22121 \\
1000 & 1000 & 2000 & 3000 & 4000 & 10000 & 11000 & 12000 & 13000 & 14000 & 20000 & 21000 & 22000 & 23000 \\ \end {array}$
Also see
- Results about the factorial number system can be found here.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $24$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $24$