Re: [Phys-l] Significant figures -- again

[David Bowman <David_Bowman@georgetowncollege.edu>]

Regarding:

> One of my favorite problems, before I retired, was the "drop a
> stone down a well and listen for the splash."  Calculate the
> well depth from the time delay.   Following the normal sig figs
> rule eliminates the answer.  Students did not know what to do
> next! 
>
> "Expand" the square root (from the quadratic formula).  This
> shows precisely what is happening.  The "big" terms cancel out
> - exactly - leaving the answer in the (quickly converging)
> series.  Sig figs takes on a whole new meaning in this problem
> after the expanding algebra is complete.
>
> Do students learn how to expand a function any more?  Will a
> computer do this for them??
>
> Oren Q.

Any problem such as this one where a quadratic equation has one of its roots nearly cancel out from a subtraction involving the square root hardly requires an expansion of the square root to get a easily calculable result without the subtraction error.  All that is needed is to rewrite the "small" root in a way that does not involve the subtraction.  In particular, consider the quadratic equation:

x^2 - 2*B*x = C

where the magnitude of C is multiple orders of magnitude smaller than B^2.  The way to write the 2 solutions to this equation so no subtraction of nearly cancelling terms is done is to write and evaluate them as:

x_1 = B + sgn(B)*sqrt(B^2 + C) and

x_2 = -C/(B + sgn(B)*sqrt(B^2 + C)) = -C/X1.

In this case the x_2 root is the "small" root and no subtraction of nearly equal terms is done to evaluate it, nor is an expansion of the square root is needed to accurately evaluate it.

Of course sometimes it *is* necessary to expand a function to preserve significance in the difference involving a subtraction of nearly equal quantities.  It is just that in the case of the quadratic equation example, such a drastic step is not really needed, and it is easier to just evaluate the expression in a sensible way that avoids the subtraction error.  In general, when faced with a problem that has a significant loss of significance due to subtraction, I would recommend that one first look for ways rewrite the expression to avoid the subtraction.  I would recommend the series expansion method only *after* one has decided that there is no discernible way to rewrite things to avoid the subtraction.

David Bowman