I occasionally come across a new piece of notation so good that
it makes life easier by giving a better way to look at
something. Some examples: Iverson introduced the notation [X] to
mean 1 if X is true and 0 otherwise; so for example Σ1≤n<x [n
prime] is the number of primes less than x, and the unmemorable
and confusing Kronecker delta function δn becomes [n=0]. (A
similar convention is used in the C programming language.) The
function taking x to x sin(x) can be denoted by x ↦ x sin(x).
This has the same meaning as the lambda calculus notation λx.x
sin(x) but seems easier to understand and use, and is less
confusing than the usual convention of just writing x sin(x),
which is ambiguous: it could also stand for a number. I find
calculations with Homs and ⊗ easier to follow if I write
Hom(A,B) as A→B. Similarly writing AB for the set of functions
from B to A is really confusing, and I find it much easier to
write this set as B→A. Conway's notation for orbifolds almost
trivializes the classification of wallpa