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Re: [Phys-L] Sig Figs homework from my 7th grader



For fear of opening Pandora's box I submit the following reply on the question of sig figs and uncertainties.

I would take the stance if there are no uncertainties stated then one should simply use the rules for sig figs - round the most precise number to match the lsf of the least precise number:

10.53 + 0.62 = 11.15

That's a good enough number for our purposes - a practice problem who's purpose is anything other than teaching how to calculate uncertainties.


That said one can beat a dead horse and proceed as follows:

If one assumes, or has been told, that the least significant figure truly indicates the uncertainty then we have

10.530 +/- 0.005 + 0.615 +/- 0.0005

= 11.145 +/- sqrt(.005^2 + 0.0005^2) = 0.0050249….

= 11.145 +/- 0.0050 (the big one wins)

Per g.u.m. we write the uncertainty to two digits though one could argue that one is sufficient in this case. So our final answer using this method and as best we can state it given llittle information is

11.1450 +/- 0.0050

And yes, you can end up with more digits than you started with. What you do with them, whether you keep more than one is dependent on an understanding of where the numbers came from and what you need them for. Since this is an academic exercise we can drop the least significant figure with impunity to give

11.145 +/- 0.005

Now if we use the "standard rules for sig figs then we round the digits of the most precise number to match the digits of the least precise number. In this case, since we have a 5 we round up (according to my rules - others may have different rules).

10.53 + 0.62 = 11.15

Which, for all practical purposes, is essentially the same number. I would argue that we really don't care which way we write the number. We started with an ill-constrained problem and ended up with an ill-constrained answer.

In general and in my humble opinion, using significant figures to determine an uncertainty is a bad practice. There are too many divergent sets of rules on how to do the calculations. And we usually apply the rules in situations where we either don't care about the uncertainty(e.g. in a lecture teaching the ideal gas law) or we don't have time to deal with a full blown uncertainty calculation (when you just want a ballpark number). I know we all like to use them because it saves time in lectures but sadly it reinforces the idea that a number without a stated uncertainty means something. It doesn't. And it suggests one can blindly use hard and fast rules to measure something when one should be using thought and intelligence instead. Sadly this is not true. But I am not going to get on my high horse because the practice of using significant figures is pervasive. I am not Don Quixote. I rarely ride horses. I don't like to fight windmills and I don't speak the proper language.

As for a 7th grader, the simple rules of significant figures for combining numbers should be sufficient. If the student is in a concrete operational mode he /she is going to complain about not being given explicit enough information. I leave it to the education experts to figure out how to cope witth this. If he/she is in a formal operational mode then he/she will likely go "ok" and we all will go on to more significant things to study.

So my short answer is 11.15 for as much as we care about any answer.

Dan


On Oct 9, 2013, at 12:00 PM, phys-l-request@phys-l.org wrote:

Message: 4
Date: Wed, 9 Oct 2013 09:42:41 -0500
From: Paul Nord <paul.nord@valpo.edu>
To: Phys-L@Phys-L.org
Subject: Re: [Phys-L] Sig Figs homework from my 7th grader
Message-ID: <F9610926-A216-46E7-9663-495EDB7B576C@valpo.edu>
Content-Type: text/plain; charset=us-ascii

True, then they are exact with zero error. For this exercise the student is intended to presume that these are scientific data where the uncertainty is indicated by the number of significant digits.

But what would if you if these were the height of a block and the thickness of a coat of paint?

Paul

On Oct 9, 2013, at 9:26 AM, Chuck Britton <cvbritton@mac.com> wrote:

If these numbers were representing Currency, then no rounding or traction would be called for.

i.e. Counting Numbers don't get treated like Measurement Numbers.


On Oct 9, 2013, at 9:49 AM, Paul Nord <paul.nord@valpo.edu> wrote:

I know the answer the teacher wants, but i find it troubling...

10.53 + 0.615 = ?

Share and enjoy,
Paul
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