Question

I'm trying to find out more about an idea I had but I can't find anything online about it. It's about the base of counting.

Answer 1

So specifically, it's kind of like a "factorial" base rather than base 2 or base 10. Every digit only contains as many numbers as it is. So for instance, the "ones" place only has the digits 0 and 1, the "tens" place only has the digits 0,1, and 2, and the "hundreds" place only has the digits 0,1,2, and 3.

So if I count to 7 it looks like this:

0 - 0
1 - 1
2 - 10
3 - 11
4 - 20
5 - 21
6 - 100
7 - 101

So here you can see that in this base every new digit is a factorial, 1 is one factorial, 10 is 2 factorial, 100 is 3 factorial, and 1000 will of course be 4 factorial, etc...

I came across this idea because I wanted to represent "e" as a fraction and by Taylor's Theorem, \[e= \sum_{n=0}^{\infty} \frac{1}{n!}\] So in this weird factorial base, you can represent e as \[e= 1+ 1.1 \bar 1 \] since \[e=1+1+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...\]

Anyone have any thoughts or ideas about this? I've never really seen anything like it before so I don't really even know how to add or multiply very well with it but it might make certain things more convenient.

Answer 2

Have you seen this already?
http://en.wikipedia.org/wiki/Factorial_number_system
Found by looking up "factorial number base" on Google.
I have no comments on it myself, although I was just touring some results and was surprised to find that there was indeed a concept on this.

Answer 3

Woah thanks. That seems sort of similar to what I'm talking about thanks haha.

Answer 4

It's always kind of fun and interesting discovering something new on your own only to find that of course someone else already thought of it first! lol