Calculating With Measurements

Multiplication and Division

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Multiplying & Dividing Measurements

Suppose you need to multiply 10 +/- 1 cmby 20 +/- 2 cm. 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 (2.0 +/- 0.4) x 10^2 cm^2.

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 - 10 +/- 1 cmhas a relative uncertainty of 10%, and 20 +/- 2 cmhas a relative uncertainty of 10% also. Now, notice that (2.0 +/- 0.4) x 10^2 cm^2is a 20% measurement.

The Shortcut is:

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 delta xand delta y, then the relative uncertainty in x is relative uncertainty in xand the relative uncertainty in y is relative uncertainty in y equation. The relative uncertainty in xy (or x/y) is:

relative uncertainty equation

Going Deeper:

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:

formula for s using quadrature

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.

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copyright © 1997, 1998 by Jerry L. Stanbrough. All rights reserved.
last update January 8, 1998 by JL Stanbrough (