# Calculating With Measurements

# Multiplication and Division

##

Multiplying & Dividing Measurements

Suppose you need to multiply by
.
The best estimate of the product is 10 cm x 20 cm = 2.0 x
10^{2} cm^{2 }(remember significant digits). The
smallest reasonable value would be 9 cm x 18 cm = 1.6 x
10^{2} cm ^{2}, and the largest reasonable value
would be 11 cm x 22 cm = 2.4 x 10^{2} cm^{2}. 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.

##

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 and
,
then the relative uncertainty in `x` is and
the relative uncertainty in `y` is .
The relative uncertainty in xy (or x/y) is:

##

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:

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.

copyright © 1997, 1998 by Jerry L. Stanbrough. All rights
reserved.
last update January 8, 1998 by JL Stanbrough (jstanbro@venus.net)