Multiplying & Dividing Measurements
Suppose you need to multiply 
by
.
The best estimate of the product is 10 cm x 20 cm = 2.0 x
102 cm2 (remember significant digits). The
smallest reasonable value would be 9 cm x 18 cm = 1.6 x
102 cm 2, and the largest reasonable value
would be 11 cm x 22 cm = 2.4 x 102 cm2. The
product must be 
.
What could be the shortcut here? The absolute uncertainties don't add in this case, but you certainly don't want to have to do three multiplications for each product you have to calculate!
Here's the deal - has
a relative
uncertainty of 10%, and has
a relative uncertainty of 10% also. Now, notice that is
a 20% measurement.
When multiplying or dividing measurements, add their relative uncertainties.
You should check this rule for division for yourself.
Symbolically, if x and y are two
measurements, and if their absolute uncertainties are 
and
,
then the relative uncertainty in x is 
and
the relative uncertainty in y is 
.
The relative uncertainty in xy (or x/y) is:
Just as with adding & subtracting measurements, the rule given above, while easy to remember and use, usually overestimates the actual probable error in a calculation. It is usually quite unlikely that, assuming that the measurements are independent and random, both measurements will have their extreme maximum (or minumum) values at the same time. A more mathematically sophisticated approach is to use the "addition in quadrature" idea:
You can find mathematical justification for this approach in the references.
Using this notion for the example above, given that the relative uncertainty in each measurement is 10%, the relative uncertainty in the product would be:
This makes the product:
Again, the question "So, which rule do I use?" is answered, "Whichever one you like - just be consistent and clear about how you make your calculations." Certainly the "simple addition" rule is easier to apply - you can do it in your head most of the time - and it is sufficient for this course.
