Section 1.1, Question 14

For the following values of \widetilde{p}, which values p are they approximating with absolute error .0005?
a) .2348263818643
b) 23.89627345677
c) −8.76257664363

Since by definition the absolute error is | \widetilde{p} - p |, given the error and \widetilde{p}, we may work backwards to find p. The general equation for finding p is:

It is now a matter of substituting in the appropriate \widetilde{p} values.

a) \widetilde{p} = .2348263818643 yields .2353263818643 \geq p \geq -.2343263818643. The yielded values create an interval of p values that could be approximated with error not exceeding .0005. Thus, we cannot determine without more information the exact value of p, but we do know that p \in [.2353263818643, -.2343263818643].

b) \widetilde{p} = 23.89627345677 yields 23.89677345677 \geq p \geq -23.89577345677. By similar reasoning to the response above, p \in [23.89677345677, -23.89577345677].

c) \widetilde{p} = -8.76257664363 yields -8.76207664363 \geq p \geq -8.76307664363. By analogous reasoning to the previous iterations, p \in [-8.76207664363, -8.76307664363].

(apologies for the earlier confusion with the question number)

I think this is actually section 1.1 number 14, but, hey, I’m down!

I feel, though, like the answer oughtta be more than one number. For example, the set of all numbers that 5.5 approximates with an absolute error not exceeding 0.5 is an interval - [5,6]. Make sense? I suppose if we insist that the error is exactly 0.5, then the result should be the list of two numbers 5 and 6. That’s how I would interpret it, at least.