Normally, we use a place-value system. This uses exponentials and multiplication.
1234
^^^^
||||
|||└ 4 * 10^0 = 4
||└ 3 * 10^1 = 30
|└ 2 * 10^2 = 200
1 * 10^3 = 1000
1000 + 200 + 30 + 4 = 1234
More generally, let d be the value of the digit, and n be the digit's position. So the value of the digit is d * 10^n^ if you're using base 10; or d * B^n^ where B is the base.
1234
^^^^
||||
|||└ d = 4, n = 0
||└ d = 3, n = 1
|└ d = 2, n = 2
d = 1, n = 3
What I came up with was a base system that was polynomial, and a system that was purely exponential, no multiplication.
In the polynomial system, each digit is d^n^. We will start n at 1.
polynomial:
1234
^^^^
||||
|||└ 4^1 = 4 in Place-Value Decimal (PVD)
||└ 3^2 = 9 PVD
|└ 2^3 = 8 PVD
1^4 = 1 PVD
1234 poly = 1 + 8 + 9 + 4 PVD = 22 PVD
This runs into some weird stuff, for example:
- Small digits in high positions can have a lower magnitude than large digits in low positions
- 1 in any place will always equal 1
- Numbers with differing digits being equal!
202 poly = 31 poly
PVD: 2^3 + 2^1 = 3^2 + 1^1
8 + 2 = 9 + 1 = 10
In the purely exponential system, each digit is n^d^. This is a bit more similar to place value, and it is kind of like a mixed-base system.
1234
^^^^
||||
|||└ 1^4 = 1
||└ 2^3 = 8
|└ 3^2 = 9
4^1 = 4
1234 exp = 4 + 8 + 9 + 1 PVD = 22 PVD
However it still runs into some of the same problems as the polynomial one.
- Small digits in high positions can have a lower magnitude than large digits in low positions (especially if the digit is 1)
- The digit in the ones place will always equal 1
- Numbers with differing digits can still be equal
200 exp = 31 exp
PVD: 3^2 = 2^3 + 1^1
9 = 8 + 1
So there you have it. Is it useful? Probably not. Is it interesting? Of course!