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.)
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 indexing overwhelmingly makes the most sense in both math and computers because indexing is a different operation than counting.
[1]: https://www.cs.utexas.edu/users/EWD/transcriptions/EWD08xx/E...
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.
When the argument is:
[0] is very, very much is [natural for indexes]
Polynomials all start at a zero "index"
Notice also how there's no ad-hominen in my response (this or the previous one) as you claim. I argue against the case and the choice of example, not against who wrote it.
(For more than you ever wanted to know about this, The Ad Hominem Fallacy Fallacy covers it exhaustively and entertainingly:
> Notice how you had to put index in quotes.
as the thrust of the argument. In saying that, they imply that I, the arguer, 'A' in Bond's article, don't actually know what an index is (so how could I have a cogent argument about 'correct' indexing?). As this is the only argument of merit, it seems as though OP is actually trying to counter the point by suggesting (attacking) something of the arguer (myself).
Now, it may be that this ad homenim is justified -- if I truly don't know what an index is then yes, I probably should not be making claims about them -- but it's still an ad homenim (and, possibly, poor form).
Of course, this is ascribing a lot to 25 words of text with little other context. I would be interested to understand if you see things differently/think I have grossly erred in my analysis.
Yes. That's addressing the point you made.
Ad hominem would be: "You're a bad person/you have this or that flaw/etc (unrelated personal stuff)".
This is: "You put the index in quotes, because even you know that this is not an index. And in any case, this is not considered an index in math, it's a degree, which is a different thing".
I also didn't "merely offered another term", as if I made up some term on my own, or just offered on of several equal alternatives. Instead, I gave the correct math term for the thing described.
>In saying that, they imply that I, the arguer, 'A' in Bond's article, don't actually know what an index is (so how could I have a cogent argument about 'correct' indexing?).
It would rather imply the opposite: that you know what an index is, and you know that the thing you applied it to, is not an index (which is why it was put in scare quotes).
That's not what they are saying. They are saying you know what an index is so well that you correctly put quotes around your usage of the term, because you understood it's not in fact a technically correct usage.
Now calling an argument poor form... that's closer to ad hominem.
There are many similar path-dependent conventions in human culture. E.g. percentages originated before the concept of decimal fractions, base-sixty time units come from ancient Mesopotamia, and conventions about multi-dimensional array memory layout are based on the convention for drawing matrices on paper.
Most common mathematical sequences and series work better (more naturally/clearly) when zero indexing is used instead, and off-by-1 errors are a problem in mathematics just like computing (but less of a problem, because notation errors get silently corrected in readers’ heads, and don’t actually have to be interpreted strictly).
For indices: indices are about referencing elements of finite ordered sets, say of size N. Hence the 'abstract' indexing set for N elements is the ordinal N. The most canonical way to represent it is to take the length-N prefix of the natural numbers (eg 0-based indexing, von neumann ordinals), which happen to have all sorts of additional structure (eg mod-N arithmetic). This is also consistent with the offset view (the i-th element is at offset i). The fact that people tend to start ordinal numbers at 1 doesn't change anything that mathematicians working with ordinal numbers take them to start at 0, for the same reason we start naturals at 0.
See also: notation for higher derivatives https://arxiv.org/abs/1801.09553; a bit further but in the same vein: notations for free variables in programs as de-bruijn indices (or some variant thereof) (it's further because it's practical for doing proofs, but not for writing concrete terms). There are probably other instances.
There was also the famous 1-pager by Dijkstra: "Why numbering should start at zero"
BCPL.