The Atlantic-Pacific Rule

In preparation for this weeks Global Physics Department meeting, Andy has set up a online corkboard for people to begin sharing how they teach error propagation/analysis.  After briefly posting how I teach it, I thought it might be worthwhile to share in greater detail.

As an engineering major, I had my fair share of error anaylsis.  I think many of us High School teachers would agree that to get our students proficient in proper error analysis and propagation is not likely.  Although I feel the engineering way is a better method (ie: $4.2 \pm 0.3\, cm$), from my experience, HS students get lost in the numbers and miss the point.  Getting them to then use something like this is basically impossible:

$\large \sigma_f = \sqrt{\left(\frac{\delta f}{\delta a}\sigma_a)\right)^2+\left(\frac{\delta f}{\delta b}\sigma_b)\right)^2}$

Moreover, I don’t think a HS physics student needs that level of sophistication for their error analysis.  A practicing engineer, with lives in his or her hands, absolutely, but a student just being introduced to the idea, I don’t think so. 
 
So instead, I build off what my school’s chemistry teachers do.  One of their fundamental aspects is the beginnings of error analysis and propagation: writing the measured value with the correct significant figures (SigFigs), and knowing how to determine the number written by others. For this, instead of the traditional rules for reporting numbers with SigFigs, they use the Atlantic-Pacific Rules, which goes something like this:

Imagine the measured number is written such that the outline of the United States is surrounding the number:

You then ask yourself a series of questions:
1. Is the decimal point Present or Absent?
     – If the decimal point is Present, start from the Pacific side; if it is Absent start from the Atlantic side.
2. Once you have identified from what “side” of the number you start,
     – begin by counting the first non-zero number.  Count that number and every number that follows it.
3. The right-most SigFig is the uncertain digit that was estimated.

For the number above, since the decimal is Present, start from the Pacific side and count left to right.  The first non-zero number is “1” and if you follow through, there are 4 significant digits with the 0 (to the right of the 7) as the estimated digit.

For the number “$549,030,000” the Atlantic-Pacific Rule would tell you that there are 5 SigFigs with the 3 being the uncertain digit (decimal point absent, so you begin from the Atlantic Ocean side and count from right to left).

From my experience, this Rule gives all the same results of the typical rules found in most texts.  However, I find that students can remember this much easier, many liking the visual aspect of the rule.  If you’ve never seen this before, maybe give this a try.