# Evaluating rational functions

Let $k$ be a field. How do we define evaluation of rational functions $k(x_1, \ldots, x_n) \to k$? For example, consider a function $$f = \frac{(x_1 - x_2)(x_3 - x_4)}{(x_1 - x_3)(x_2 - x_4)}$$ over a field of characteristic zero. We want to evaluate it at $(0, 1, 0, 0)$. The answer that I expect is $-1$, but I can't see a straightforward way to define this without topology or first factoring by some ideal (in this case generated by $x_1 - x_3$, $x_1 - x_4$), then reducing the fraction and then evaluating. Is there a way or do we have to steer clear of the points where the denominator is zero? Is there a way to make evaluation a morphism between some objects?

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You don't without either adding an $\infty$ point to $k$ or excluding points where the denominator evaluates to $0$. – Thomas Andrews Nov 20 '12 at 15:34