Partial Fraction decomposition of rational functions plays a role in calculus. Do the ideas extend to rational numbers?
Let me try to ask it precisely, though the answer I'm looking for may ultimately be to a different phrasing.
Given a rational number (in lowest terms) $a/b$ where $b$ factors as a product of distinct primes $b=p_1p_2\cdots p_n$ and $a<b$, does there always exist a decomposition of the form $$\frac{a}{b}=\frac{q_1}{p_1}+\cdots+\frac{q_n}{p_n}$$ where the $q_i$s are integers (and probably satisfying some sort of condition, perhaps only $|q_i|<p_i$).
Easy example: 1/6 = 1/2 - 1/3
Better example: 29/70 = 1/2 - 4/5 + 5/7
Of course it would be extra nice if we could extend to $b$ being composite and handle prime powers in the factorization the same as in the rational function case.