From: Ronald Wong (ronwong@inreach.com)
Date: Sun Oct 15 2000 - 23:24:47 PDT
Message-Id: <l03102800b604fed286da@[209.209.18.208]> Date: Mon, 16 Oct 2000 00:24:47 -0600 From: Ronald Wong <ronwong@inreach.com> Subject: Re: sig Figs and multiplication/division
"Damon Jansen" <dkjansen@yahoo.com> said:
>...This activity is a great lead-in to the sig fig
>rules on adding measured values--we can actually
>derive the rule. (I still haven't figured out a way
>to make the multiplication/division rule intuitive, so
>if you have ideas, let us know.)
The rule for addition and subtraction is easily arrived at by way of a
simple, numerical example. Find the maximum and minimum sum (or difference
in the case of subtraction) between two numbers that are uncertain and
notice the difference between the average value and maximum or minimum
values. The difference turns out to be the sum of the uncertainties of the
two numbers.
You can use the same technique to figure "...out a way to make the
multiplication/division rule intuitive" (well, maybe not AS intuitive).
1. Pick two numbers: A = 25 +/-5 (20% unc.) and B = 24 +/- 6 (25% unc.)
We know that the product of the actual numbers, 25 and 24, is 600. The
question is what is it's uncertainty?
2. Determine the maximum product of A and B (30X30): 900.
3. Determine the minimum product of A and B (20X18): 360.
4. The average of these two products is 630. It differs from the
Max. and min by 270. This represents the uncertainty and can
be thought of as the average uncertainty for this range of
values based on the uncertainty of the original factors.
5. The 600 mentioned earlier -the product of the actual numbers -
is the product of the average of two measurements for which
360 and 900 represent the minimum and maximum values. The 600
represents the actual value for the average value of these
maximum and minimum and the 270 is its average uncertainty.
(Your astute students may notice that, unlike the derivation of
the rule for the sum and difference of uncertain numbers, the
average value for the maximum and minimum values does not equal
the product. Unlike the sum/difference where the average value
of the maximum and minimum was equal to the sum/difference of
the values given, the average value of the maximum/minimum
products will not equal the product of the actual values.
This has to do with the fact that sums/differences are "linear"
in their operations, whereas products generate higher order
terms. As a result, the actual product will not lie midway
between the maximum and minimum values.)
6. So the answer for the product is 600 +/- 270.
7. Notice that 270 is 45% of 600 and that 20% + 25% (the percentage
of uncertainties for A and B) = 45%.
8. Since the uncertainty has only one significant figure, (it
determines the last significant figure in the answer and
its range of variation), the final answer is 600 +/- 300
(50% uncertainty!). The answer is only good to the 100's place.
We've lost a little precision due to the large uncertainties
in the original factors. We started out with two significant
figures in the beginning and ended up with only one in the end.
Try this with a number of other simple examples and you'll get similar results.
Since division is actually the multiplication of one number by the
multiplicative inverse of another (in other words, division is really
multiplication), the rule applies to division as well.
If your students have a background in algebra, you may want to consider a
second, more general approach to the derivation of the rule.
Have them derive the rule using algebra where one number is A + UA and the
other is B + UB - where UX is +/-uncertainty in X.
Point out to them that any term like (UA)(UB) has considerably less
precision than A(UB) or B(UA) and can be ignored (students are always
surprised at how cavalier physicists can be a times but this too is an
important lesson for them to learn).
They'll quickly come up with:
Product = AB + AB(UA/A +UB/B) = AB +/-(unc.A/A + unc.B/B).
And thus the rule: The uncertainty in any operation involving
multiplication or division can be found by multiplying the answer by the
(sum of the percentage of uncertainty of the factors)/100.
So, as far as the uncertainties are concerned:
When you add and subtract: add the uncertainties.
When you multiply and divide: add the PERCENTAGE of uncertainties.
--------------------------------------------------------
Many teachers find that dealing with uncertainties in their labs is too
involved and not worth the trouble.
The problem isn't with addition and subtraction of uncertain numbers (the
application of the rule is straight forward); it's with the multiplication
and division of uncertain numbers (including comparable operations like
square, cubes, square roots, etc.). Adding up uncertainties is one thing
but dealing with percentages, their sum and then multiplying the answer by
that sum was/is just "...too much trouble").
I have to admit that this was true in the days of slide rules but there's
no excuse for it nowadays where just about everyone in a science class has
an electronic calculator with 6 or more memory registers, loads of
mathematical operators, and, in some cases, programmability.
Dealing with uncertainties using today's basic calculator is really a
no-brainer - but only if they take full advantage of what their calculators
have to offer (that was also part of what I had to do as a physics teacher
- teach them how to get the most out of their $15 calculators).
Take a problem like X times Y divided by Z where X = A + UA and Y = B + UB
and Z = C + UC.
Using an algebraic calculator you would just do the following steps:
1. Enter A 8. Press ( 14. Press ./. 20. Press ) - Unnecessary step.
2. Press X 9. Enter UA 15. Enter B 100 X's This is
3. Enter B 10. Press ./. 16. Press + the Total % of
unc.
4. Press ./. 11. Enter A 17. Enter UC and can skipped.
(division) 12. Press + 18. Press ./. 21. Press =
5. Enter C 13. Enter UB 19. Enter C
6. Press X
7. Store (this is the answer but, since you don't know the uncertainty,
you don't know the precision and thus don't want to bother writing
the value down until you do)
After you do step 21, you look at the number in the display and mentally
round this number down to one significant figure (it's the uncertainty),
place a +/- sign to its left, recall the answer stored in memory, and,
keeping in mind the uncertainty, round it off to the proper place, and
write it down to the left of the +/- sign. That's it.
The actual execution of the above steps is not as complicated as it appears.
Basically the students learned to write their measurements down in an
orderly fashion. Then, in the case of problems where the numbers had to be
multiplied, square rooted, divided, cubed, etc. they would:
A. Do the calculations with the average of their measured values and
finish by hitting the "X" key instead of the equal sign.
B. They would store their answer and then
C. Press the "(" key and add up all the ratios of the uncertainties
to their corresponding measurements - introducing a factor 0.5
into the ratio if a square root occurred, or a factor of 2 if the
original measurement had been squared, etc. (Introducing additional
factors did not complicate matters. They just entered them and hit
the times key and then continued with Enter UX, ./., Enter X, +, etc.)
D. Once the "=" sign was pressed. They just wrote down the uncertainty and
then the answer.
The students found that if they wrote their data down in an orderly fashion
and then the equation, the process became very mechanical and they could do
it very quickly and accurately.
--------------------------------------------------------
Every year my physics course started with a week-long "review" of what my
students should have learned in their biology and/or chemistry classes -
"What accuracy and precision are and how do to deal with numbers that are
uncertain.". The review was necessary because most of my students were
unfamiliar with these concepts. Either they hadn't taken a high school
science course or, if they had, the concepts were quickly forgotten
(usually because they were not used in their lab work).
The week ended with a lab on measurement where they applied what they had
learned to a simple task - taking measurements, keeping track of their
uncertainties, making a simple computation (with the uncertainties in mind)
and comparing their results with others to determine to what degree they
had achieved their objective and understood the week's lesson.
One of the benefits of this activity was that many of the students stopped
looking at their subsequent labs as (a) simply an activity that they were
being asked to perform because it was part of a science class and/or (b) an
activity that would confirm what was brought up in their textbook.
Instead of focusing on what the textbook said, their concerns became more a
matter of how their results compared with those of others in the class -
and, on occasion, with groups in my other physics classes as well. They
knew the level of precision of their results so they knew when their
results differed significantly from that of others. Differences frequently
led to a consideration of whether or not there were any systematic errors
in the execution of the lab. Occasionally, they took the time to do the lab
over again - modifying the procedure or setup to see if the differences
could be reconciled in some way. Sometimes the differences couldn't be
reconciled and they found themselves in a situation not uncommon in science
- a fine learning experience if there ever was one.
These issues arose even in the very first lab - the lab on measurement
(it's amazing what you can do with a block of wood).
In any case, they usually felt very confident about their conclusions. They
had pursued the activity as carefully as they felt was possible and, within
the range of uncertainty of the measurements made, they were sure of their
results.
The semester usually ended with a study of periodic motion. By this time,
the students were so confident of their lab work that it came as no
surprise to some when their results seemed to show that the period of a
pendulum did NOT depend on the length alone. They knew the level of
precision in their work and therefore felt justified in making their
unusual claim. There were always detractors but occasionally the proponents
would find supporters in the form of other groups with corroborating data.
A classic demonstration followed that would seem to have resolved the issue
- what it usually did was lead them to another level of discussion.
That's what I've always liked about teaching physics. The entertainment
never seemed to end.
Cheers - ron
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