# Exercise 2.13

Let’s take two intervals defined by their center and percentage tolerance: - interval \(A\) with center \(c_1\) and percentage \(p_2\) - interval \(B\) with center \(c_2\) and percentage \(p_2\)

We can easily find the lower bound and upper bound of each interval using the ideas of the `make-center-percent` procedure defined in the previous exercise:

\(A=(c_1-,c_1+)B=(c_2-,c_2+)\)

Assuming that the bounds are positives (as proposed by the exercise), we can determine the product of the two intervals :

\(A.B=((c_1-)(c_2-),(c_1+)(c_2+))\)

\(\ \ \ \ =(c_1c_2+--,c_1c_2+++)\)

Now we can easily see the center and the with of our new interval.\ Next we can evaluate the percentage of our new interval following the ideas of our `percent` procedure:\ \[(AB)=\] You can verify that this percentage is correct with substitution using the `percent` procedure.

If the percentages \(p_1\) and \(p_2\) are small then \(\) is negligible and we get: \[(AB)p_1p_2\]

We just proved that the percentage tolerance of the product of two intervals is approximately the sum of the two percentage tolerance of the factors if these percentages are small.