[xigoi]

The infinite sum of powers of 2 indeed diverges in the real numbers. However, in the 2-adic numbers, it does actually equal -1.

https://en.wikipedia.org/wiki/P-adic_number

[Dylan16807]

Eh, P-adic numbers basically write the digits backwards, so "-1" has very little relation to a normal -1.

[mike_hock]

Any ring automatically gains all integers as meaningful symbols because there is exactly one ring homomorphism from Z to the ring.

[xigoi]

-1 means that when you add 1, you get 0. And the 2-adic number …11111 has this property.