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In doing estimation problems, we often round to one or two significant figures, since we often do not have the detailed information to establish a more accurate number -- and quoting excessive significant figures that we don't actually know gives a misleading impression. This permits some useful mnemonics.* For each of the mnemonics below, calculate when possible or look up the accurate numbers on the web when not, and state how good the correct number is. State your result as an approximate "percentage error", that is

percentage error = 100 x (True value - mnemonic value)/(True value)

rounded to one or two significant figures. (If a number is good to 11.5% it's better to think of it as "good to about 10%".)

Show your work. If you obtain any numbers from a website be sure to cite the site.

a. The radius of the earth
is approximately (2/ π ) x 10^{7} m. (This is not
an accident. The original meter was defined during the French revolution
so that a longitudinal line drawn through Paris would have a length of 10,000
km when measured from the north pole to the equator. Show why this definitiion
gives the result stated, as well as giving the accuracy requested earlier.)

b. A year is approximately π x 10^{7} s.
(This is an accident, as far as I can tell)

c. The density of air
is approximately 1 kg/m^{3} and that of water is approximately
1000 kg/m^{3}.

d. Measure the length of the first joint of your thumb, your handspan, and your height in cm. Round each of these to a conveniently remembered number and indicate the percentage error in each of our mnemonics.

e. The conversion of linear measures from miles to kilometers can be aided by one of three common mnemonics:

- 1 mi = 1.6 km
- 1 km = 0.6 mi
- 5 mi = 8 km.

Indicate the percentage error of each of these.

f. The conversion from "pounds" (a force) to "kg" (a mass) is really only relevant on the surface of the earth. Given this restriction, how good is this mnemonic?

- 1 kg "=" 2.2 lbs

* "mnemonic" =
a device used as an aid in remembering (*American
Heritage Dictionary)*

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Page last modified January 18, 2008: G29