What, and that's exactly the part I dislike about both Matlab and Julia! The amount of "+ 1" and "- 1"s in matlab indices you need when subdividing matrices into multiple equal sized parts is horrible.
Weird, I thought mathematicians would know better what makes sense and what not. Starting with an offset of 1 is clearly what does not make sense, you start at distance 0 from your starting point.
A matrix is a collection of elements and nothing more. Indexing starts at 1 because that's the first element in your matrix, and numbering the first element 1 makes sense. Numbering from 0 makes sense when it represents an offset into something, but a matrix isn't that. 0-based indexing just always feels to me like letting the implementation details leak out. (I don't feel this way when an array actually represents a chunk of memory, rather than a math object.)
The proliferation of +1 and -1 depends on the application. Some work better with 1-indexed, some work better with 0-indexed. Personally I get annoyed at having to use `len-1` too often when working with 0-indexed arrays. This is why some languages like Julia (and FORTRAN apparently?) let you choose your index-base, which... has trade-offs.
Why?
Mathematicians of course label objects however suits the problem at hand. It's extremely common to write say c_0 + c_1 x + c_2 x^2 + ... for a taylor series starting with a constant, or start at -1 or -n if that's tidier.
I'm not going to say "zeroeth" in spoken language, but for any serious computation, you're going to be indexing variable sized things and need to compute the indices themselves. And I don't want to make code or formulas less readable due to shifts by 1.
The number line concerns real numbers, not elements.
(There are perfectly reasonable arguments why computer languages might be zero-indexed, whereby objects start at the zeroth offset. But counting from 1 upwards is definitely more natural for counting, it's disingenuous to pretend otherwise).
[1]https://docs.julialang.org/en/latest/devdocs/offset-arrays/
I'm no mathematician, but when I studied linear algebra, I remember index starts from 1 in matrix. I think MATLAB just follows that convention, it's "Matrix lab" after all.