I still find the concept of significant digits to be useful in helping students understand that there are uncertainties in our measurements. And the general scheme gives one way to estimate this for beginning students. To say that it is not used by real scientists is a stretch.
As to the way that I teach the concept, or have had it taught to be in chemistry and physics classes, is that the rules only apply to the final results. One should not apply it to intermediate results on the way to a final answer. This is also true forr the more formal derivative method. One should look at the whole procedure not just at parts of it.
Richard
Richard L. Bowman, PhD | Department of Physics | Professor of Physics
BRIDGEWATER COLLEGE, Bridgewater, VA 22812, USA
phone: 540-828-5441 | online: www.bridgewater.edu/~rbowman
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From: phys-l-bounces@carnot.physics.buffalo.edu [phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of Philip Keller [PKeller@holmdelschools.org]
Sent: Wednesday, March 07, 2012 8:56 AM
To: 'Forum for Physics Educators'
Subject: [Phys-l] Significant figures -- again
My apologies to all who are not interested in any further discussion of this topic. As a high school teacher, I would support a nation-wide agreement to excise this from the curriculum. But my son is now in chemistry. As he has had to contend with this, I have tried to sort it out. Here's what I have so far...
The chem book seems to say AFAICT:
Say you mass a sample. Reporting a mass of 1.2345 kg rather than say 1.2 kg is an implicit claim that you used a more precise instrument. And by convention, we all agree that the last digit your report is one that you guessed on.
Say that having massed the sample to be 1.2 kg (you only had access to the less precise balance) and measuring its volume to be 3.56 liters, you then calculate the density. Your calculator says .3370787 kg/L -- but you don't want to be immodest. You only want to claim precision to the same number of sig figs as your least precise datum. So in this case, knowing only 2 sig figs for mass, you round your density to 2 sig figs as well and report .34 kg/L.
All that seems OK to me -- even if it is not the way it is done by real scientists, I don't see evil or mystery in it.
The weirdness begins when you add:
I have a 4500 lb car. I load it with an 8.5 lb can of paint. Now what does it weigh? By the rules of sig figs, I round to the largest uncertain digit -- in this case the hundreds place. And I still get 4500 pounds. This seems weird but I will say it is defensible: 4500 was a guess -- it could have been 4600 or 4400. The can of paint won't matter.
But some student (I hope) is sure to ask: what if you keep adding cans of paint? A car with 50 cans of paint does not weigh the same as an empty car. Still, I think I can weasel out of this problem-- I would say calculate the total mass of the paint cans first: 8.5 x 50 = 425 -- which I will round to 430 b/c 8.5 has 2 sig figs and the 50 cans is an exact count -- and then the total mass is 4930 which I report as 4900.
But that stubborn student continues: OK, say you have an empty bus that weighs 12000 pounds and 50 different passengers with weights varying from 50 to 250 pounds, all different but you do have a list of each weight. Now you add them one at a time to the bus weight and you still get 12000 pounds -- when the combined weight is more like 17000 pounds.
I would say that yes, this sure seems to make the sig fig rules look silly. And your chemistry book does not address this. But we can still try to make it work. We just have to recognize that this silliness arises when you are adding quantities that have been measured by two different instruments that have different precisions. We measured the bus with a device that gave use answers to the nearest thousand pounds. Then we measured the passengers with a device that measured to the nearest pound. To make this work, we now have to agree to add up all the quantities that were measured with the same device first before adding quantities that were measured with different devices. In math class, order of addition never mattered. In this strange case, it does.
So does this scheme rescue the sig fig notion? I welcome any feedback, but remember that I am not free to reject the whole notion (though I can think of many better uses of the class time).
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