# , , and the ring of Witt vectors

Yoshifumi Tsuchimoto

You should know that every positive integer may be written in decimal notation:

Similarly, given any integer (base'') , we may write a number as a string of digits in base . For example, we have

Similarly, we have

You may also probably know (repeating) decimal expresions of positive rational numbers.

Now let us reverse the order of digits. Namely, we employ a notation like this1:

Let us do some calculation with the above notation:

You may recognize curious rules of computations. This curious notation will lead you to a new world called the world of addic numbers''.

EXERCISE 0.1   Compute

with our curious notation. Then do the same computation in the usual digital notation in base .