Sunday, 7 February 2010

e-day and Andy Jackson

Adding on to the post about coincidences yesterday, this post is about a value derived from the same hyperbola, y=1/x. For the mathematician, February 7th, (or 2 - 7) is the date we decide to celebrate the constant which is the base of the natural logarithms, appx 2.71828.... (more later, and a way to memorize it). There have been LOTS of sites that explain LOTS of things (such as at Homeschool Math Blog and here at Let's Play Math.) about the value e, so I will try to not be too redundant and throw in something totally different, as the Monty Python folks used to say...

The letter e was first used for the base of the natural or hyperbolic logarithms by Leonhard Euler. Earlier I had mistakenly thought that Euler was the discoverer of the value, but in fact the number was published in Edward Wright's English translation of Napier's work on logarithms in 1618, almost 100 years before Euler's birth. [and in fact, it was known to the English Mathematician Roger Cotes. Cotes is one of those many promising mathematicians who died at a young age and Newton, who seldom said anything good about anyone else, once said "Perhaps if Cotes had lived, we would have known something"..] The number represented by e is approximately 2.718281828459045... Euler actually computed the number to eight more decimal places. This was done in 1727, and would seem almost impossible accuracy for anyone else, but of Euler it was said, "Euler calculates as other men breathe."

It was known from the work of Gregory of St. Vincent and others that the logarithms were somehow linked to the area under the hyperbola f(x)=1/x because the area under the curve matched the logarithmic property Log(AB)= Log(A)+Log(B). The Area under the curve from 1 to x=ab is equal to the areas from 1 to x=a plus the area from 1 to x=b. The value of e is such that the area under the hyperbola from 1 to e is 1 square unit. It has been conjectured that Euler may have used e as an abbreviation of the word Eins, the German word for one.

One oddity that students and teachers may use to remember the first 15 digits of e, given above, is to recognize their relationship with Andrew Jackson's presidency and an isosceles right triangle. Confusing? Just wait, all will be clear. We begin with 2, because Jackson was president for two terms. The 7 tells us he was the seventh president of the US. 1828 is the year he was elected, and we repeat this because of the two terms. Then we give the three angles of an isosceles right triangle, 45, 90, 45, and we have completed 15 digits of the base of the natural logarithms. I am almost 100% sure I picked that up from one of Martin Gardner's Scientific American columns.

Euler was one of the most influential mathematicians of the period and his prestige was sufficient that his use of a variable often marked it for posterity, but there were other symbols that were suggested occasionally. D'Alembert used c for the same constant in 1747, and Benjamin Peirce suggested a symbol that looked like a paper clip, or the @ symbol now used for e-mail addresses instead of pi, and the same symbol reflected in a vertical line for e.

But now I have to bring up the fact that in my new home town(My lovely Jeannie is there right now working on our "Boat House" which will be our retirement home out of the snow of Michigan.
)of Paducah, Ky, e-day is for Engineers Day. The University of Kentucky College of Engineering has a branch campus at Paducah,and they are having their open house on February 21 at Crounse Hall. They have, among other things, an Edible car contest (would I kid you?) as well as an Egg Drop contest, A Popsicle Stick bridge contest, and of course (drum roll please...) A Duct Tape Challenge...... I had a student only a few years ago who was a master of duct-tape-utilization. He would make roses out of duct tape to impress the ladies, (and did) and had a duct tape wallet... and once came to school in a sport coat made entirely of duct tape... I imagine he could have had one in cashmere for about the same price.
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