2.13|

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SICP Exercise 2.13

Let $C_a\(be the center of an interval and\)T_a$$ be the tolerance of ‘a’. Then the interval ‘a’ spans:

\[a = [C_a \times (1 - 0.5 T_a), C_a \times (1 + 0.5 T_a)]\]

Similarly for ‘b’:

\[b = [C_b \times (1 - 0.5 T_b), C_b \times (1 + 0.5 T_b)]\]

The interval of ‘a*b’:

\[\begin {align} a * b = [&C_a C_b (1 - 0.5 T_a)(1 - 0.5 T_b), \\ &C_a C_b (1 + 0.5 T_a)(1 + 0.5 T_b)] \\ a * b = [&C_a C_b (1 - 0.5 T_b - 0.5 T_a + 0.25 T_a T_b), \\ &C_a C_b (1 + 0.5 T_b + 0.5 T_a + 0.25 T_a T_b)] \\ \end{align}\]

The term \(T_a T_b\) can be ignored when \(T_a\) and \(T_b\) are sufficiently low.

Therefore:

\[\begin{align} a * b = [&C_a C_b (1 - 0.5 (T_a + T_b)), \\ &C_a C_b (1 + 0.5 (T_a + T_b))] \end{align}\]

Thus, for sufficiently small tolerances, the resultant interval of the product of two intervals has a tolerance of the sum of each tolerance.