Somewhat off topic rant, but am I the only one who find mathematical notation unnecessarily obtuse?
The bit that gets me is defining degree as n-1. For someone without a mathematical background, it takes a bit of pondering to figure out that you have to define n as one more than the actual degree, the opposite of what seems natrual. My mind at least just wants to think about n as the degree, and use n+1 as the last index. To me it seems aggressively unintuitive.
I guess you want to align the coefficient numbers but would it be a sin to define another index c = n-1 for that purpose?
But I'm a mathematical lightweight and maybe mathematical thinking is all about this. Perhaps some greater talent can correct my thinking.
Fence post and rail is "off by one", two fence posts are joined by one rail, N fence posts are joined by N-1 rails, and this polynomial order and defining coefficients discussion has that in common.
Two points define a line, a polynomial of degree 1. A polynomial with 2 coefficients, ax + b.
Three points give us a quadratic, a polynomial of degree 2 with three coefficients, ax^2 + bx + c.
N points gives us a polynomial of degree N-1 with N coefficients.
Indexing coefficients by their associated power of X seems natural to some.
A(N-1).X^(N-1) + ... A(1).X^1 + A(0).X^0 (where X^0 == 1)
are the N indexed coefficients of a generic polynomial of order N-1.
By giving it a quick read I saw that the have n data points where a polynomial of n-1 degrees provides a useful fit in the sense of this blog post.
Every field has its own language to speak. And shouting into the field from "outside" that they should change is not polite.
E.g * if you redefine c = n-1 the connection between number of points and dimension is lost. * c ist very often used as a constant Skalar. E.g as the speed of light. Using it as a dimension of a problem is quite unintuitive.