ℤ_{p}, ℚ_{p}, AND THE RING OF WITT VECTORS

YOSHIFUMI TSUCHIMOTO

Playing with “digits in base n”

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

(531)_{10} = 5 × 10^{2} + 3 × 10^{1} + 1 × 10^{0}.

Similarly, given any integer (“base”) b ≥ 2, we may write a number as a string of digits in base n. For example, we have

(531)_{10} = 1 × 7^{3} + 3 × 7^{2} + 5 × 7 + 6 × 1 = (1356)_{7}.

Similarly, we have

(531)_{10} = (1356)_{7} = (1023)_{8} = 1000010011_{2} = (213)_{16}.

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

(531.79)_{10} = 5 × 10^{2} + 3 × 10^{1} + 1 × 10^{0} + 7 × 10^{−1} + 9 × 10^{−2}.

(531.79)_{10} = (1356.˙534˙6)_{7} = (1023.62˙4365605075341217270˙2)_{8} Now let us reverse the order of digits. Namely, we employ a notation

like this^{1}:

[97.135]_{10} = (531.79)_{10}
[0.135]_{10} = (531)_{10}
[123.456]_{10} = (654.321)_{10}
…

Let us do some calculation with the above notation:

[0.1]_{10} + [0.9]_{10} = [0.01]_{10}
[0.1]_{10} × [0.9]_{10} = [0.9]_{10}
[0.01]_{10} × [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

[0.12345]_{8} + [0.75432]_{8}

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

1This is our private notation.