hard minimum/soft minimum/exact/approximate notation

Factual Questions
1

Hi,

I was trying to figure out if there was a standard notation for describing:

(1) hard minimum/hard maximum

For instance, 11 inches = 27.94 cm, which rounds to 27.9 cm, but you would want to write 11 inches (28 cm) because 27.9 cm = 10.98 inches, which is too small. You don’t want to write 27.94 because you figure people can’t measure any finer than 1 mm.

(2) soft minimum/maximum. In general if you have a hard minimum on one end of the range, you will have a soft maximum on the range below. So you would want to write 10.9 inches (27.9 cm) even though 10.9 inches = 27.686 cm.

(3) exact – say 10 inches = 25.4 cm.

(4) Approximate – here you are happy with an approximation. So you have no qualms about writing 11 inches (27.9 cm).

The best I could find was that NIST underlines case 3.

2

All I can say is that the significant digits Nazi in me strongly objects to writing “10 inches = 25.4 cm”

This might actually answer parts of your question. If you do mean exactly 11 inches, then write 11.00 inches and go ahead and use 27.94 cm. Most people can’t measure a hundredth of an inch or a cm, but using additional digits makes it clear that you’re not just talking 11 +/-0.5 inches.

3

For nonsymmetric error bounds, I’ve seen notation like 11.0 +0.0/-0.01 to mean no more than 11, and no less than 10.99. That would address 1).

For 4), I’ve seen something like 11 (nom) (for nominal).

4

Your example is a difficult one because you are expressing two different accuracies at once: the accuracy of some measurement, like “I would say that bolt is about 11 inches long but I didn’t bring a tape measure”, and the accuracy of the conversion, which is (trivially) perfect. I haven’t heard any notation to do this, other than language, as in “it’s about 11 inches, or about 28 cm”.

There are specific notations that deal with things like 25.4 cm in 10 inches by saying they are accurate to 0.1 cm because there is one place reported after the period. This is fairly common in computer languages and applications, for example. Meanwhile if you said there were 254 cm in 100 inches, that would be exact. I find it maddening that as fancy a tool as Mathematica automatically reads 2.54 as approximate and 93000000 (the distance to the Sun in miles) as exact.

Scientific notation already supports expressing the accuracy in a way practically unrelated to the size of the value.

5

Back when I taught chemistry, I was also a significant figures “Nazi.” That being said, there is actually nothing wrong with writing “10 inches = 25.4 cm (exact),” although I usually write “1 inch = 2.54 cm (exact),” which of course means the same thing.

This is because exact equalities are taken to have an infinite number of significant figures. Similar example include:

1 yard = 3 feet (exact)
1 foot = 12 inches (exact)
1 cm =10 mm (exact)

…as well as direct counts (e.g. 5 apples). All of these numbers are assumed to have an infinite number of significant figures.

6

I would probably write 11 inches (28 cm), because 11 has two significant figures, and so does 28.