Earlier quoted context omitted.
> Array indexing is such a core thing and I don't understand why anything mathematical or scientific would start with 1. Because starting with 0 is neither math nor array indexing in general. It's just how the base addresss of an array pointer memory block was referenced in C (and it spread from there). Which is why all math focused languages use 1-based (fortran, apl, matlab, r, mathematica, etc.)
> Because starting with 0 is neither math nor array indexing in general. It very, very much is. Polynomials all start at a zero "index", as does just about every expansion I can think of (Fourier, Bessel, Legendre, Chebyshev, Spherical Harmonic, etc.) Combinatorics, too, make lots of use of zero indices and zero-sized sets. As for arrays, I'll leave it to Dijkstra[1] to explain why zero indexing is most natural. Zero…
Notice how you had to put index in quotes.
Because it's not an index, it's the degree of each polynomial term, which is a power.