Kaba
Posts:
289
Registered:
5/23/11


Re: Is there a name for this notation?
Posted:
Oct 9, 2011 11:37 AM


Kaba wrote: > Kaba wrote: > > In the case of "base 2", this can be seen as a sparse form of the > > traditional base2 representation. Let I subset NN. Then the value of I > > is given by > > > > V(I) = sum_{i in I} 2^i, > > A straightforward generalization to represent all positive real numbers > is to allow I subset ZZ, with I countable.
Let b in NN, b >= 2. A _number_ in sparsebaseb representation is a set A subset [ZZ times (0, b) subset NN], such that
(i, v_1) in A and (i, v_2) in A ==> v_1 = v_2.
The value of A is given by
V(A) = sum_{(i, v) in A) v 2^i.
Examples follow.
In sparsebase2:
0 = {} 1 = {(0, 1)} 2 = {(1, 1)} 3 = {(0, 1), (1, 1)} 4 = {(2, 1)} 5 = {(0, 1), (2, 1)} 6 = {(1, 1), (2, 1)} 7 = {(0, 1), (1, 1), (2, 1)}
In sparsebase3:
0 = {} 1 = {(0, 1)} 2 = {(0, 2)} 3 = {(1, 1)} 4 = {(0, 1), (1, 1)} 5 = {(0, 2), (1, 1)} 6 = {(1, 2)} 7 = {(0, 1), (1, 2)}
In sparse base4:
0 = {} 1 = {(0, 1)} 2 = {(0, 2)} 3 = {(0, 3)} 4 = {(1, 1)} 5 = {(0, 1), (1, 1)} 6 = {(0, 2), (1, 1)} 7 = {(0, 3), (1, 1)}
 http://kaba.hilvi.org

