this post was submitted on 16 Jul 2023
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i thought this was pretty weird too when i found out about it. i’m not entirely sure why it’s done this way but i think it has to do with conventions on where to start indexing. most programming languages start their indexing at 0 while much of the time in math the indexing starts at 1, so i=0 to n-1 becomes i=1 to n.
My abstract math professor showed us that sometimes it's useful to count natural numbers from 1 instead of 0, like in one problem we did concerning the relation Q on A = N × N defined by (m,n)Q(p,q) iff m/n = p/q. I don't hate counting natural numbers from 1 anymore because of how commonly this sort of thing comes up in non-computer math contexts.
yeah thats a good example and it shows weird the number 0 is compared to the positive integers. it seems like a lot of the time things are first "defined" for the positive integers and then afterwards the definition is extended to 0 in a "consistent way". for example, the idea of taking exponents a^n^ makes sense when n is a positive integer, but its not immediately clear how to define a^0^. so, we do some digging and see that a^m+n^ = a^m^a^n^ when m and n are positive integers. this observation makes defining a^0^=1 "consistent" with the definition on positive integers, since it makes a^m+n^ = a^m^a^n^ true when n=0.
i think this sort of thing makes mathematicians think of 0 as a weird index and its why they tend to prefer starting at 1, and then making 0 the index for the "weird" term when it's included (like the displacement vector in affine space or the constant term in a taylor series).