Tally marks are just a base-1 numbering system
and every fifth digit is just put in an odd place
and every fifth digit is just put in an odd place
It's a unary numeral system , but it doesn't really have a base, because it's not a positional number system.
Another similar thing to think about is the Roman numeral system. it starts out as unary system, but quickly turns into a mix of base 5 and 10 and even without a fixed position. That's also not a positional number system even if can be considered to have a base and a secondary base..
I think it's reasonable to say it completes that pattern of basal numbers. Saying that it's not positional is like saying that the base10 number 5555 isn't positional - it's just that all the digits happen to be the same.
Now base zero....
I'm not familiar with what basal numbers means.
Anyway, 5555 is just one number in the decimal system that fulfills the requirement that the position of digits is irrelevant, whereas most decimal numbers do not. In the tally mark system all numbers fulfill this requirement.
However, the thing I like most about it is that you'll never need to prove that I+I=II. It literally is II.
I remember reading the book called "Gödel, Escher, Bach" which is about Gödels incompleteness theorem. At some point it comes across this kind of thing and demonstrates how any natural number is the successor of the previous number, basically defining numbers as tally marks. From there it goes on to demonstrate why math itself is incomplete. It's kinda a fat book, but if you're into numbers, logic and coding it's a must read.
Sorry "basal number" is something I just made up, I was trying to describe "numbers of a number system that uses a base". I thought I could save myself some writing, but I got too cute with it lol.
To me, unary seems like just the special case. For all positional number systems, as the base approaches 1, the number of irrelevantly-positional numbers predictably increases until it reaches 100%. It fits a pattern. And in a more meta view, 1 is a pretty common "special" or "trivial" case, along with 0, and infinity. I think it's a bit strange to say it doesn't belong in the set.
I'm not quite at the point where I'm gonna read a math book for fun, but there are these little pieces of math that are fascinating.
the thing I like most about it is that you’ll never need to prove that I+I=II. It literally is II.
I hated discrete structures class in college. Nearly half the class dropped out, me included. Not because I was failing. I just couldn't give a damn. 1+1=2 is true for the same reason I+I=II is true. That's the whole concept of 2.
Base 1 does make a huge difference though. In base 1, number length is linear with respect to the size of the number - i.e. the number 10 needs 10 tallies. For any other base, the number length is logarithmic, i.e. the number 10 needs just 2 numerals in base 10 and still only 4 in base 2.
This is actually important in theoretical computer science, since computers would be much slower if you don't assume an "efficient encoding" of numbers, i.e. a logarithmic sized encoding. Base 1 is not efficient.
It is logarithmic size, it's just the special case of log1. The implication being the higher the base, the more efficient the encoding.
Edit: 1 is supposed to be subscript, but it's not showing properly in my app. Idk what it looks like for others