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$ \mathbb{Z}_p$ , $ \mathbb{Q}_p$ , and the ring of Witt vectors

Yoshifumi Tsuchimoto

\fbox{Playing with \lq\lq digits in base $n$''}

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

$\displaystyle (531)_{10}=5\times 10^2 +3\times 10^1+1\times 10^0.

Similarly, given any integer (``base'') % latex2html id marker 505
$ b\geq 2$ , we may write a number as a string of digits in base $ n$ . For example, we have

$\displaystyle (531)_{10}=1\times 7^3+3\times 7^2 + 5 \times 7 +6 \times 1=(1356)_7.

Similarly, we have

$\displaystyle (531)_{10}=(1356)_7=(1023)_8=1000010011_2=(213)_{16}.

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

$\displaystyle (531.79)_{10}=5\times 10^2 +3\times 10^1+1\times 10^0+ 7\times 10^{-1}
+9\times 10^{-2}.

$\displaystyle (531.79)_{10}=(1356.\dot{5}34\dot{6})_{7}

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

      $\displaystyle [97.135]_{10}=(531.79)_{10}$
      $\displaystyle [0.135]_{10}=(531)_{10}$
      $\displaystyle [123.456]_{10}=(654.321)_{10}$
      $\displaystyle \dots$

Let us do some calculation with the above notation:

      $\displaystyle [0.1]_{10}+ [0.9]_{10}=[0.01]_{10}$
      $\displaystyle [0.1]_{10}\times [0.9]_{10}=[0.9]_{10}$
      $\displaystyle [0.01]_{10}\times [0.09]_{10}=[0.009]_{10}$

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

$\displaystyle [0.12345]_8+[0.75432]_8

with our curious notation. Then do the same computation in the usual digital notation in base $ 10$ .

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