Well I finally decided to look a little deeper at my suggested base conversion, and to my surprise, the formula just popped out at me.

given the definition :

we see instantly:

which works for k = 0 as well since is multiplication.

It's easy now to generate a formula which works for base conversion using these operators. The notation for such is very cumbersome however, therefore I'll write it out step by step for 2, 3, 4.

and which I think the process becomes obvious by this point

and which by generality becomes:

where we are sure to evaluate the highest operator first.

this formula is very easily generalized to include conversion from base to any base

given the definition :

we see instantly:

which works for k = 0 as well since is multiplication.

It's easy now to generate a formula which works for base conversion using these operators. The notation for such is very cumbersome however, therefore I'll write it out step by step for 2, 3, 4.

and which I think the process becomes obvious by this point

and which by generality becomes:

where we are sure to evaluate the highest operator first.

this formula is very easily generalized to include conversion from base to any base