Right. But while I'm just trying to do my work as a programmer, which of those two identities is going to be easier to work with, reason about, and come out of it with correct code? Hint: It doesn't have a "1" in it.

Your identity perfectly illustrates why 1-based arrays are worse rather than better.

That's true for C/C++, but in most languages used today where we don't use pointers, where is the advantage?

I'm hundreds of times more likely to have to write lastElement = array[array.length - 1] (there's the 1 again!) than I am to have to reason about the equivalent pointer arithmetic.

Even in a language without pointers, in a multidimensional array you can index the data one-dimensionally by array[index0 + N * index1] with 0-based indexing, as opposed to array[index0 + N * (index1-1)]. Where N is the length of the inner nested array.

And for that matter all algorithms for which zero-based indexing makes more sense are simpler, such as working with polynomials and expansions where the zeroth power is included in the expansion.

Having said that, I too prefer 1-based indexing because the list of algorithms that are written more cleanly (i.e. without constantly causing bugs when you implement them) is far larger for me personally, including all of linear algebra and related numerical methods.

In a language that has one-based indexing, can you treat a multidimensional array as one-dimensional? Or will it give you an array index out of bounds error?

That is, are there any languages that are both one-based, and will let you do that trick?

You would need it to support one-dimensional views (unless your data is just stored as one-dimensional to begin with). Certainly matlab and Julia do. Don't know of any new 1-based languages either way.

I thought we settled on lastElement = array[-1] for this problem (a little, but not entirely, tongue-in-cheek).