Leo Liu said:
The definitions of them seem like arbitrary choices or an abuse of notation. I wonder what the reasons behind the definitions are. Thanks.
View attachment 292853
PS. My instructor said such defs simply the process of solving modular equations.
You can view this from various perspectives. The shortest notation is, that ##\pi\, : \,\mathbb{Z}\longrightarrow \mathbb{Z}/m\mathbb{Z}## is a homomorphism of rings. This requires ##\mathbb{Z}/m\mathbb{Z}## to be a ring, i.e. that we multiply and add the remainders by division with ##m## in such a way, that adding and multiplication commutes with taking the remainders. The numbers ##\{0,1,\ldots,m-1\}## build representatives of the equivalence classes achieved by taking the remainder by division with ##m##.
More prosaic, we could say that taking the remainders is important in number theory, where we often distinguish primes which have the remainder ##1## by division with ##4## and those with remainder ##3##.
Another important example is when ##m## itself is prime. Then those calculation rules define a finite number field, i.e. we can even divide by the equivalence classes ##[a]\neq [0].## In case ##m=2## we get thus the basic rules for computers, or less complicated: the light switch in your room. Finite fields in general (##m## prime) play an important role in cryptography, i.e. the science of codes.
If ##m## is not prime, say ##m=p\cdot q## then we cannot divide, i.e. we have still a ring but no field. This is because ##[p]\cdot [q]=[0]## although ##[p],[q]\neq [0].## An example for such a case is the analogous clock. The clock has ##m=12## for the big hand, and ##m=60## for the small hand.