Related idea which I don't see mentioned yet: William Bricken's iconic arithmetic, with regards to what he calls James Algebra.
I don't have my copy of the material handy but it comes down to using different containers to represent logarithms and powers such that
(x) ~> #^x
[x] ~> log_#(x)
<x> ~> -x
where # is an arbitrary base.
Writing expressions next to each other is implied addition. Whole numbers can be written e.g.
0 ~> _
1 ~> ()
2 ~> ()()
3 ~> ()()()
etc.
Operations, like addition, read
A+B ~> A B
and subtraction
A-B ~> A<B>
where <<x>> ~> x,
on to multiplication
A*B ~> ([A][B])
and division
A/B ~> ([A]<[B]>)
and exponentiation
A^B ~> (([[A]][B])).
There are a few axiomatic equations (maybe 3?) that are used to establish the general properties of the system, and from which the rest of it can then be deduced.
It also introduces an interesting construction, which he simply calls J ~> [<()>], analogous to the imaginary number i.
I'd recommend taking a look at this if TFA tickled your fancy.
I don't have my copy of the material handy but it comes down to using different containers to represent logarithms and powers such that
(x) ~> #^x
[x] ~> log_#(x)
<x> ~> -x
where # is an arbitrary base.
Writing expressions next to each other is implied addition. Whole numbers can be written e.g.
0 ~> _
1 ~> ()
2 ~> ()()
3 ~> ()()()
etc.
Operations, like addition, read
A+B ~> A B
and subtraction
A-B ~> A<B>
where <<x>> ~> x,
on to multiplication
A*B ~> ([A][B])
and division
A/B ~> ([A]<[B]>)
and exponentiation
A^B ~> (([[A]][B])).
There are a few axiomatic equations (maybe 3?) that are used to establish the general properties of the system, and from which the rest of it can then be deduced.
It also introduces an interesting construction, which he simply calls J ~> [<()>], analogous to the imaginary number i.
I'd recommend taking a look at this if TFA tickled your fancy.