question about precision of arithmetic involving empirical, measured quantities Re: Unum 4.0 beta

[Tim Roberts]

"John Benson" <jsbenson at bensonsystems.com> wrote:
>
>Will this package handle approximate quantities, too?
>
>Here's some background for the question:
>
>Let's say you have measured the side of a square area to be 10 meters, +/- 1
>meter. We want to compute the area of the square.
>
>We are squaring a range of 9 to 11 meters, and get a resulting area range of
>81 to 121 square meters. If we take the midpoint of this range as the most
>representative value, we end up with (81 + 121)/2 or 101 square meters,
>plus-or-minus 20 square meters!

Right.  Algebra would have told you this:

  (a-1)**2  + (a+1)**2  = a**2 - 2a + 1 + a**2 + 2a + 1 = 2a**2 + 2

Divide that by two, and you get a**2 + 1.

>But what about the discrepancy between 100 and 101?

The discrepancy is that you chose to take the center of the range as the
"most representative value".  There is no mathematical basis for that
decision.

What you're talking about here is interval arithmetic.  Although it is
useful for working with approximate measurements, it actually increases the
precision of your knowledge.  For example, we KNOW that the actual area is
in the interval [81,121].  We do NOT know what the exact area is, or even
what the best estimate is, and there is no analytical method for extracting
that information from the interval.
-- 
- Tim Roberts, timr at probo.com
  Providenza & Boekelheide, Inc.