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Re: Sig Figures



Hi,
Once things get tangled into calculation at this level, one probably
should step back from simple rules and do a propagation of error calculation.
It gets messy fast. An alternative is to numerically see what a variation in
the last digit of the measured (or provided ) values does to the final answer.

Thanks
Roger Haar

*********************************************************
John Denker wrote:

At 03:01 PM 1/31/00 -0500, Carr, John wrote:
When your dealing with angles in problems like vector addition, or
inelastic collision at vector angles, how do you handle angles and
significant figures?

The only correct answer to your question is:
IT DEPENDS.

For example: suppose we have an angle theta that is somewhere around 89.3
or 89.4 degrees. If you care about knowing tan(theta) to even one
significant digit, you need to know theta to more than three significant
digits.

The book I use tends to keeps the angle to about 2
decimal places and does not follow significant figures of the data.
What are the guidelines for this?

1) As Ludwik wrote earlier today, it never hurts to keep extra sig digs in
the intermediate results. The official name for such things is "guard
digits". They can't hurt unless they become unduly laborious -- but now in
the era of hand calculators and spreadsheet programs it's often easier to
carry huge numbers of digits than it is to think.

2) There exist very detailed guidelines for rounding off if that turns out
to be necessary.

2a) The brute-force formulation is to ask the question: If I did this
twice, rounding up in one case and rounding down in the other case, would
both cases give the same answer, to an acceptable approximation? If not,
you're not carrying enough digits.

2b) In the case where the roundoff errors are small, the just-mentioned
brute force formulation can be well approximated by looking at the first
derivative of the final answer with respect to whatever quantity you want
to round off.

2c) In a multi-step or iterated calculation where many roundoff operations
occur, you need to worry about _accumulation_ of errors. There are methods
for formalizing that, too, but I suspect that's beyond the level of detail
you were asking about.