hes@ecsvax.UUCP (Henry Schaffer) (08/30/85)
We have a clash of conventions here. There is a convention that a measurement should be reported in a format that also portrays the possible degree of rounding. If I measure and round to the nearest meter I can report the measurement as 3. x 10^0 (where the decimal point can be omitted.) This is a measurement to 1 significant digit in decimal arithmetic. There is no trouble so far - but: 1)Someone may read it as "3 meters" which then obscures the rounding point, especially since one would not normally call it "3 meters" if it was more than a centimeter off. By the convention, a measurement to the nearest centimeter would have been reported as 3.00 x 10^0 (where the x 10^0 can be omitted.) This problem is *just* a misunderstanding of notation or conventions. 2)The significant digits convention is a good guideline, but it does not completely substitute for range arithmetic (or other more complete error analysis.) If we add four measurements of 3. meters each, what is the total. We only know the total to 1 significant digit, and to 1 significant digit the total is 1. x 10^1. However, the true range of the total is [1.0 x 10^1, 1.4 x 10^1] and *not* the [.5 x 10^1, 1.5 x 10^1] implicit in 1. x 10^1. The problem here is in trying to use a general (and useful) guideline as a detailed solution to an error analysis problem. I have sometimes tried to extend the notion to fractional significant digits, e.g. "a floating point number has 6 1/3 digits of significance", but there is no good way to represent this notationally. The usual convention does not do anything like this. 3)Change of units or number bases. What happens if I want to represent 3. x 10^0 meters in centimeters. No problem, we retain 1 significant figure in the product 3. x 10^2 cm. But what if we want it in inches. Just multiply by 39.370, which is the conversion factor. There is no way to represent this with 1 significant digit *and* to have it represent the original half meter rounding scheme (unless we allow nonintegral powers of 10 - which, by convention, isn't done.) Even though significant digit notation is not completely adequate in this context, and understanding of the principle would stop absurdities like the following - which is often found in scientific literature "a length of wood with a cross-section 5.08 cm x 10.16 cm was used to ..." Did they mean that the wood was planed very accurately to the nearest hundreth of a cm? I think they bought a "two by four" at a lumberyard - and then used a 2.54 conversion factor - on the nominal (not actual) measurement - and reported it as an actual measurement. Back to the original interview. Asking a question with no clue as to the convention used in the notation is usually thought of as a sophomoric trick. It may be fun to discuss - but can you imagine the response of the company if they ordered a "3 meter long" cable and you delivered one 2.6 meters long, and claimed that you met their spec? --henry schaffer (gee, this is almost long enough to go in net.philosophy)