Why is the complete state of a turing machine representable by a finite word?
I'm just wondering why this word is necessarily finite. Why can't the input of the turing machine have an infinite amount of non-zero characters?


No, it can not.

When the Turing machine starts, the tape contains the input (a finite-length word) followed by an infinite amount of blank symbols (often written as #). It is common to simply omit the trailing infinitely many blanks, and just represent a finite prefix instead, adding blanks as needed when the head moves right past the end of the tape.

When the machine runs, in each step it can at most (over-)write over one tape symbol, and move right by at most one position. This is an essential feature of any Turing machine.

Hence, after finitely many steps, we still have a finite amount of non-blank symbols. The tape again contains infinitely many trailing blanks (usually not represented), so it can be represented by a suitable finite prefix.