Suppose I have a real number $x$ and a real number $y$. Now it's quite simple to compute their sum $x + y$.
But now, suppose I don't actually know exactly what value $x$ has. But I do know that it's definitely contained in $x_0 \leq x \leq x_1$. Similarly, I know for a fact that $y_0 \leq y \leq y_1$. Now if I'm not mistaken, we have
$$x_0 + y_0 \leq x + y \leq x_1 + y_1$$
I believe this is correct, regardless of sign; adding the two lowest possible values gives me the lowest bound on the correct figure.
What now of multiplication? Well... uh... my first guess would be
$$x_0 \cdot y_0 \leq x \cdot y \leq x_1 \cdot y_1$$
Trouble is, multiplying by a number $k > 1$ makes the product larger, whereas multiplying by $k < 1$ makes it smaller. So if (for example) we have $x_0 < 1 < x_1$... then I'm not sure what happens! Worse, if we had $x_0 < 0 < x_1$, things appear to get even more complicated.
Things become even more... “interesting” if I wanted, say, the sine of $x$. Now $x_0 < x_1$ does not imply that $\sin x_0 < \sin x_1$. OK, so I can easily swap the answers if needed. But, more importantly, $x_0 \leq x \leq x_1$ definitely doesn't imply that $\sin x_0 \leq \sin x \leq \sin x_1$. Even if I swap the two sides of the interval, it's no longer guaranteed to actually contain the true value!
Is there a name for this system of calculation? (I.e., computing upper and lower limits to the true value rather than just a single approximation to it.) Is there somewhere I can read about a general rule for how to handle it correctly?