Why does $\frac{a}{b}<0$ imply $ab<0$? I'm not sure if this was asked before, but my question is: why does $\frac{a}{b}<0$ imply $ab<0$? How do you prove it both intuitively and rigorously(using math)? I think I understand it intuitively: it's becuase for $\frac{a}{b}$ to be negative, exactly one of $a$ or $b$ has to be negative. For $ab$ to be negative, exactly one of $a$ or $b$ has to be negative. That means that these two imply each other. But how would I prove this rigorously? If I multiply both sides of $\frac{a}{b}<0$ by $b$, first of all, I don't know whether $b$ is positive or negative so I don't know which way the inequality sign is facing, and second, even if we did know that it flipped or didn't flip, we would only get $a<0$ or if the sign didn't flip $a>0$. Do I split it into cases then(case 1: $b<0$ and case 2: $b>0$)? It seems like it would work but there might be a more slicker way of proving it?

If you know one, you can obtain the other by multiplying/dividing by $b^2$, which is possible since that will always be a positive quantity. Hence, it suffices to show that any one of these statements is true.
