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Modulo arithmetic is taught to American students as clock arithmetic in elementary school as well. Signed integers are better described as truncated 2-adics, which are definitely an advanced topic. It's basically a Greenland/Iceland situation. The complicated, difficult one is given a friendly name to trap naive programmers.

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Signed ints are no more and no less truncated 2-adics than unsigned ints. Signed ints and unsigned ints are isomorphic as rings; they are both the ring of integers modulo some 2^n. It is only on other operations (ordering comparisons, or the / and % of C) where they differ, and in neither case do these operations have anything to do with 2-adics.

It's a much more useful correspondence than just an isomorphism. Take your base-2 2-adic integer. Truncate it to whatever bitwidth you want. That's the two's complement representation of that number. Unsigned pops out as a special case, obviously.

I've mostly seen this used by cryptographers to prove results over arbitrary bitwidths, since you can prove it over the p-adics and deal with truncation/division separately.

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