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+====== Exercise 2.16. ======
+Explain, in general, why equivalent algebraic expressions may lead to
+different answers. Can you devise an interval-arithmetic package that
+does not have this shortcoming, or is this task impossible? (Warning:
+This problem is very difficult.)
+====== Solutions ======
+As seen in Exercise 2.14, most of the algebraic identities that hold on
+operations over real numbers do not hold on the analogous operations over
+intervals.
+A - A ≠ 0
+A / A ≠ 1
+A / B ≠ 1 / (B / A)
+and so on.
+Furthermore, the correct interpretation of an expression involving
+intervals requires distinguishing those intervals which represent a
+single physical parameter (and thus can only have one value at a time)
+and those which represent different parameters (though they may have
+the same upper and lower bounds).  This implies an interval arithmetic
+package which does not preserve identity between different intervals
+cannot avoid this shortcoming.
+However, we may be able to build such a package if we use a different
+interface.
+Instead of raw operations on intervals, we would like to have
+operations which manipulate expressions.  An expression would represent
+a mathematical algebraic statement, possibly including variables.
+The user would construct an expression, such as R₁R₂ / (R₁+R₂), and
+then would call an evaluate function with the expression and a set of
+bindings giving for each variable a measured value and an error
+tolerance.  The package would then need to compute the upper and lower
+bounds, using only one value at a time for each variable.
+If intervals can only be expressed as center-point and percentage
+error tolerance, then intervals spanning zero cannot exist, and we
+may assume that all intervals are either fully positive, fully
+negative, or equivalent to an exact zero.
+If we then show that multiplication, division, addition, and
+subtraction are all monotonic over all such intervals, we can write
+such a package simply by calculating the expression once with every
+possible combination of upper and lower bounds of each variable, and
+taking the minimum and maximum results as the bounds of the result.
+We could also extend the package to optimize these calculations,
+at least in some cases, by determining that some combinations of
+upper and lower bounds for particular variables will fall within
+the bounds of the results given by some other combinations, and so
+avoid calculating combinations that cannot contribute to the final
+bounds of the result.