Ask A Scientist Mathematics Archive

name         Rumana W.
status       student
age          14

Question -   How would you round off an answer if you had to
multiply/divide and add/subtract measurements to get an answer.
Do you round off your final answer according to the last
procedure/operation you carried out, so for eg.

If you were asked to add 600.34 to 4.3 and then multiply by 100, would you
end up rounding the answer (after multiplying by 100) to the lowest Sig
Fig or would you round off to lowest Decimal point?

And the answer from my example would give an exact answer of 60464.  If I
were to round it to lowest sig fig possible, do I round to the no. of Sig
Fig of 100? (since I used it in the multiplication process to get final
answer) or do I round off to 60000, according to the lowest no of sig fig
(ie: 2) for the whole question due to the measurement given 4.3?
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Rumana,
When rounding numbers, you do it one step at a time.  For adding or
subtracting, it is position that counts rather than number of digits.  For
multiplying and dividing, number of significant figures is the important
thing:

When you add 600.34 to 4.3, you get 604.64.  Because the 4.3 measurement
could really be as large as 4.6 or as small as 4.0, it is the tenths digit
that is uncertain in your sum.  It is therefore the last digit you can keep.
You state your answer as 604.6.

When you then multiply by 100, you need to know the uncertainty of the
number.  If the "1" is uncertain, then the measurement represents a value
that could really be as small as zero or as large as perhaps 200 or 300.
The 100 measurement has only 1 significant figure.  Multiplying 604.6 by 100
is 60460.  This could represent a quantity as small as zero or as large as
12000 or 18000.  You only keep one significant digit.  You state your answer
as 60000, because the six could be different from reality.  If you had
perfect measurements, the real quantity could easily be 20000 or 90000,
maybe even larger.

If the 100 were something like a unit change from meters to centimeters,
then the 100 is exact.  It counts as 100.00000000000....  In that case, you
can keep all four significant figures from 604.6 and write your solution as
60460.

Do your rounding one step at a time.  Use number of digits when multiplying
or dividing.  Use position when adding or subtracting.

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College
==============================================================
It depends upon what it is that the numbers represent. If they are
mathematically exact numbers i.e. 4.3 really means 4.30.... and
60464.0......., you can keep all the digits. If they are the result of an
experimental measurement, you keep only the number of significant figures of
the number with the least precision. The best way to determine how many
digits to keep is to do what is called an error propagation analysis. For
example, suppose you know that the error in the number 4.3 is +/- 0.5, and
the error in the other number is: 600.34 +/- 0.02. Then the worst possible
cases are ANSWER +/- (0.52). You would keep all the digits throughout the
entire calculation, so that you do not round off twice, and you would express
the "ANSWER" with the number of significant figures that represents the
collective uncertainty that the error propagation tells you that your
"window" of uncertainty is.

Vince Calder
========================================================================
When you add numbers, you look for the lowest-valued decimal place in
which all of the numbers have a significant figure.  In this case, that
decimal place is tenths, so the sum 604.64 will have four significant figures.
When you multiply, the product has only as many significant figures as the
number with the fewest.  So, multiplying a four-figure number by a
three-figure number, you round 60464 to 60500.

Tim Mooney
====================================================
Hello,

Rounding off will affect results but not in simple and easily predictable 
ways.

To establish how much of rounding off we can accept, we have to answer two 
questions: First, what is the desired precision and accuracy of the finial 
results.  Second, how much an incremental (i.e., a very small) change in 
each of the input numbers will affect the final result.

For simple calculations such as the example you provide, this is easy, and 
the change in the final answer can be found by trial. But in engineering 
and scientific calculations where millions and billions of calculations 
are performed in a single run, one has to either do what is called an 
"error analysis" ahead of time to know what level of precision is needed, 
or simply increase or decrease the number of digits and decimal places the 
computer keeps and see how the final results are affected.

In summary, there is no simple relations in terms of decimal places 
between input and output numbers in a calculation.  An obvious example of 
this is this:  1/(1.020-1.010) = 100.00;  1/(1.021-1.010) = 90.91. Thus, a 
0.1% change in one of the numbers will change the result by almost 
10%.  When  millions of calculations involving additions, subtractions, 
divisions, and multiplications are involved. it is very difficult and at 
times impossible to predict this type of situations without rigorous error 
analysis or a trial and error process as described.

Ali Khounsary, Ph.D.
Argonne National Laboratory
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