Thursday, July 5, 2018

A proof in limericks

     The word "transcendental" is an adjective that refers to an abstract or supernatural noun.  In mathematics, the term's meaning is specified more precisely -- a transcendental number is one that cannot be a root of any algebraic equation with rational numbers as coefficients. The number π (ratio of the length of the circumference of a circle to its diameter) and the number e (base for the system of natural logarithms) are the best known examples of transcendental numbers.
     Retired Arkansas law professor (and former math teacher) Robert Laurence has fun with this pair of transcendentals using limerick stanzas.  Get out your pencil and graph paper -- and enjoy puzzling through his rhymes.

A Transcendental Proof in Six Stanzas     
by Robert Laurence   © 2018
       They are transcendent you see:
       eπ and πe.
       The prize you’ll win when,
       With pencil or pen,
       You prove which is smaller to me. 

       “I’m sure against Gauss it’s a sin,
       But I don’t know where to begin.”
       No need to squirm,
       Take the ln of each term.
       And I’ll show you the way you can win.

       e ‧ ln π is the key.
       Is it smaller than π? Well, let’s see.
       f is e ‧ ln x,
       And g is just x.
       From there it’s simple.  Trust me.

       “Simple? Is that what you meant?”
       Yes, the curves are at one point tan-gent.
       The point is (e,e),
       So now do you see,
       That f below g is all bent?

       “All bent?” you ask with a frown.
       “You must think I’m some sort of clown.”
       Forgive me my crime.
       I was trying to rhyme.
       And the curve is concave down.

       Forget my original plea.
       These rhymes have been all about me.
       What’s done is what’s done;
       I’ve spoiled all the fun:
       The smaller is πe.

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