Every positive real number x has a g-adic representation
Firstly we consider only the case . For such a number x we use the
Gaussian function
i |
. Note that .
| to define recursively the sequence by
and subsequently set
It is easily seen that , so that is valid as well. Therefore we have
.
When proving the following please note that the number is always a member of .
-
Proof by induction:
►
-
The proof is immediate from 1.
- For all the inequality
Proof by induction:
►
► For the inductive step we employ the induction hypothesis with the number !
3. proves the equality
. So we only need to overcome the initial restriction for x. Now let by any positive real. With (note: is unbounded) we have and from that we can set
for and in addition ,
and get: and finally have:
|