>>14055820>>14055795>>14055826See
>>14053456https://en.wikipedia.org/wiki/P-adic_number#Positional_notationIf you want a deeper answer: The way you define addition on the reals (or any ring/group) only says what you get when you add two numbers (/elements), and that could be extended to all finite sums via associativity. It says nothing about infinite sums, meaning infinite sums are undefined.
You can literally assign whatever values you want to every single infinite sum, and arbitrarily. But normally we'd like the way we assign a value to play nice with other properties. It's the exact same idea with division by zero: We leave it undefined. We COULD define it to be anything (even "infinity" if you'd like), but then it doesn't play nice with other properties. But there ARE instances when you'd like to have that be defined as "infinity", e.g. on the Riemann sphere.
One way to assign values is through convergence of partial sums (in the reals). So you end up with some infinite sums defined, while others are still undefined. You can extend the way you assign values to infinite sums to divergent ones, and you can require the method to play nice to some extent, and there are many ways to do this.
But convergence in the *reals* isn't the only way to go. You can also assign values to infinite sums using convergence in the *p-adics*. This is an instance of where you'd want 1+2+4+8+... to equal -1, and it's specifically when p = 2. In the 2-adics, the (2-adic) absolute value of 2^n is 2^(-n), and so the sequence of partial sums converges.
This actually pops up in a subtle way in computers. If you look at the representation of -1 in signed integers in binary, it's a sequence of 1's, e.g. for a byte it's 11111111. This is actually 1+2+4+8+16+32+64+128+256. Works the same way with more bits; you just get more terms.
