Counter-intuitive conventions

(f . g) x = f(g(x))

Or in general, math notation tends to be right-associative, which to me is quite counterintuitive in contrast to natural language. Natural language usually reads left to right, but math reads right to left.

Make sense nonetheless but unconventional, in a manner of right to left is the writing style, just like this sentence.
Wouldn’t it make sense for 0! to be 1, as you’re basically multiplying nothing, so you want the identity element of multiplication.
0\^0 being undefined in most, but not all, cases.
Intersection over an empty index set
1 is not a prime number
The degree of the zero polynomial is by convention either -1 or -infinity. The least upper bound of the empty set is -infinity and the greatest lower bound is infinity. An intersection over an empty index set is sometimes defined to be the universe of discourse.
Direct limits actually being colimits. And not limits.
The fact that we use pi instead of tau leads to a lot of unnecessary 2s all over the place.
I remember when first studying compositions, I found the notation quite counter-intuitive. Having studied function compositions in a more formal way has since resolved this for me, but I think function composition is often unnatural when someone runs into it for the first time!
The zero power giving result one is based on exponent rules, the quotient rule, the difference gives zero for exponent and also the first step one is equal to anything divided by itself.

So

1
=a^n/a^n
= a^(n-n)
= a^0
by

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