I think this is only because you don't care dividing by zero. But you  may as well define the value at zero to be 0, and any other value at least 1. See [this][1] expository paper by Keith Conrad for interesting remarks on euclidean domains and functions.


  [1]: http://www.math.uconn.edu/~kconrad/blurbs/ringtheory/euclideanrk.pdf