I would say it's much like f(x) = sqrt(x) over real numbers. sqrt(-16) doesn't exist, but it is still convenient to think about the square root of negative numb

I would say it's much like f(x) = sqrt(x) over real numbers. sqrt(-16) doesn't exist, but it is still convenient to think about the square root of negative number (for say factoring polynomials). In this case, our function happens to have a representation at f(-1) being equivilent to 1 + 2 + 3 + ..., and being able to calculate f(-1) = -1/12 much like sqrt(-16) = 4i when we introduce i = sqrt(-1).