1. **State the problem:** We need to find two numbers whose product is $-36$ and whose difference is $13$. 2. **Set variables:** Let the two numbers be $x$ and $y$. 3. **Write the equations:** $$xy = -36$$ $$x - y = 13$$ 4. **Express one variable:** From the difference equation, express $x$ as: $$x = y + 13$$ 5. **Substitute into product equation:** $$ (y + 13) y = -36 $$ 6. **Expand and form quadratic:** $$ y^2 + 13y = -36 $$ 7. **Bring all terms to one side:** $$ y^2 + 13y + 36 = 0 $$ 8. **Factor the quadratic:** $$ (y + 9)(y + 4) = 0 $$ 9. **Solve for $y$:** $$ y = -9 \quad \text{or} \quad y = -4 $$ 10. **Find corresponding $x$ values:** - If $y = -9$, then $x = -9 + 13 = 4$ - If $y = -4$, then $x = -4 + 13 = 9$ 11. **Write the pairs:** The two possible pairs are $(4, -9)$ and $(9, -4)$. **Final answer:** The pairs of numbers are $\boxed{(4, -9)}$ and $\boxed{(9, -4)}$.