In my lecture notes there is a proof for the division algorithm which sets $S=\{a-xb|x\in \Bbb Z, a-xb \geq 0 \}$ then says $S\subset\Bbb N$ so we can use the well ordering principle.
There's a similar proof here http://www.mathpath.org/concepts/divisionalgo.htm, although the main body of the proof isn't relevant to my question.
My question is how can $S$ be contained in $\Bbb N$, when zero is an element of $S$ ? Surely I'm missing something here ?
Some authors define the set of natural numbers starting with $0$ therefor they consider $0\in\Bbb N $
The proof of the theorem does not change whether you call $0$ a natural number or not.