Re: Sig Figures

["John S. Denker" <jsd@MONMOUTH.COM>]

At 12:27 PM 2/2/00 -0500, Robert A Cohen wrote:
>I'm a little late getting into this but I have a few questions...
>
>Wouldn't the rules of sig figs say that 0.98 + 0.04 = 1.02?  Why would
>addition imply that the answer must be rounded to two sig figs?

Well, it depends on what version of the "rules" you try to follow.
  a) There are some who hold that if the addends have two sig digs, the
sum should have two sig digs.
  b) There are some who hold that if the addends have two decimal places
right of the decimal point, the answer should also have two decimal places.
  c) A more thoughtful rule says that if you know that your number has an
uncertainty greater than ten counts in the second decimal place, you should
round it off to one decimal place.

In fact, I had rule (c) in mind when I constructed my example, but alas my
note didn't explain this.  Thanks to Robert for pointing this out.

The general point is that all three versions of the rule are -- and indeed
_ALL_ "sig dig" rules -- are for amateurs only.

------------------

Robert also wrote:
> ... equals 0.002 if not rounded or 0.00 if rounded.  Doesn't this mean
> that I should round the intermediate results?

It depends.  In general,
 a) avoid rounding anything (inputs, math constants, intermediate values)
unless you are sure it won't degrade the final answer, and
 b) avoid rounding anything unless carrying the unrounded numbers would be
unduly laborious.

For details, see next message, wherein I present a much more detailed
example of the sorts of trouble you can get into by rounding intermediate
results.

> P.S. FWIW, I emphasize in my class that sig fig's is the "rough" way of
> estimating the effect of resolution on a calculation and it is better than
> nothing but that there are "better" ways.

That's cool.  But it would be even cooler if we could give the students
some guidance as to when the "better ways" are necessary.