扩展：部分分数分解有理数*

"Partial fraction decomposition" for rational numbers

This is an example of the analogy between integers and polynomials: rational numbers (quotients of integers) have partial fraction decompositions, not just rational functions/expressions (quotients of polynomials)!

(The usual partial fraction decomposition is covered elsewhere in the map: theory, networking, and algorithm. But we won't actually need that polynomial stuff in our discussion here. :)

Let's just dive into some examples. We can start with something simple, like $\frac{7}{9}$ or $\frac{5}{12}$.

We can't decompose $\frac{7}{9}$ further (except for boring things like $\frac{7}{9} = \frac{1}{1} - \frac{2}{9}$, where the denominators do not decrease; why?), but $$\frac{5}{12} = +\frac{3}{4} - \frac{1}{3} = -\frac{1}{4} + \frac{2}{3} = \cdots,$$ for instance.

If you'd like, try to figure out a partial fraction decomposition for $\frac{1}{2015}$. (This was Algebra #2 in the HMMT 2015 competition. :)