OTHER SYSTEMS OF ARITHMETIC OFFER OTHER MEANINGS
The statement 1+2+4+8+...=−1 is meaningless in ordinary arithmetic. But who says we have to stay with ordinary arithmetic? Is there an extraordinary way to view matters?
We tend to view numbers as spaced apart on the number line additively. Walk one step to the right of 0 and we end up at position 1. Now add to that two steps and we end up position 3. Now add four steps, position 7. And so on. The sum 1+2+4+8+..., in this viewpoint, takes infinitely far to the right of 0 on the number line.
1+2+4+8+...=∞ in ordinary arithmetic. It does not equal −1.
But let’s think of numbers multiplicatively. In particular, since we are focusing on the sum 1+2+4+8+..., let’s think of factors and multiples of powers of two.
Now 0 is a highly divisible number. It is the most divisible number of all. With regard to just two-ness it can be divided by 2 once, in fact twice, in fact thrice. In fact, you can divide 0 by two as many times as you like—and still keep dividing.
With regard to two-ness, the number 8 is somewhat zero-like: you can divide it by two three times. But 32 is even more zero-like: you can divide it by two five times. And 2100 is even more zero-like still.
So in this sense, 2100 is a number very close to 0. The number 32 is somewhat close to 0. The number 8 is less close. The number 1 is not very close to zero at all: it cannot be evenly divided by two even once.
So, in this context, could 1+2+4+8+... possibly be −1?
These finite sums grow to become “a number very close to zero, minus one.” In the limit, the infinite sum thus has value 0−1=−1, just as our formal arguments said it would be.
So in this multiplicative view of arithmetic, 1+2+4+8+... is a meaningful quantity, and it does indeed have value −1. The geometric series formula is meaningful and correct for x=2 in this context.