letter in Mathematics Teacher, December 1995, p. 725, “Gary Larson investigation”:

Clyde Paul’s article on using a scene from a Matt Dillon episode to initiate a mathematical investigation (March 1995, 192-95) included the highly appropriate Gary Larson cartoon, “I asked you a question buddy ... What’s the square root of 5,248?” I wonder how many of your readers noticed that sqrt( 5248) can be solved mentally to two places.  Begin by factoring out 4s, giving sqrt(5248) = 2 sqrt(1312) = 4 sqrt(328) = 8 sqrt(82).  Using elementary calculus, we can approximate sqrt(82) as sqrt(81) + (1/ (2 sqrt(81) )) =
9 1/18.  So sqrt(5248) = 8 sqrt(82) is about 72 8/18 = 72 4/9 = 72.444...  The correct answer to five places is 72.44308 ....

Most four-digit numbers would hardly give such a happy result.  I cannot help but wonder whether Larson chose sqrt(5248) deliberately, hoping that mathematics teachers would notice that the problem is more tractable than it appears and use his cartoon to illustrate a lesson in differential calculus.

J. Mark Thompson

letter in Mathematics Teacher, September 1998, p. 512, “More about 1998”:

Monte Zerger gives some interesting lore on 1998 and its relation to 666 n the February 1998 issue of the Mathematics Teacher (Reader Reflections:  666 + 666 + 666”), including that 1998/636 is approximately pi, whereas 1998/735 is approximately e.  I had noticed these results last summer and also that the proportional error in the pi approximation is about e x 10^-5, whereas the proportional error in the e approximation is about pi x 10^-5.

Another interesting tidbit arises when you program a calculator to look for other numerators that approximate pi and e with such exactness, to give an idea of how extraordinary is 1998.  The most recent year with a combined error lower than 1998’s is 1775:  1775/565 gives pi with a proportional error of only 8.5 x 10^-8, and 1775/653 is e with error 2.1 x 10^-5.  The next number that improves on 1998 is 2441:  2441/777 = pi, with proportional error 7.2 x 10^-6, and 2441/898 = e, with error 7.0 x 10^-6.  Therefore, 1998 is the best numerator for approximating pi and e in a span of 666, or 2441-1775, years.

Mark Thompson

Questions, corrections, comments:  Send me e-mail at  markthom@flash.net