References in Mathematical Man Friday

So wearisome did Friday become with his quibbling on numbers...
This leads to Carl Friedrich Gauss' method for summing the integers, a problem he was also apparently set as a young schoolboy.

He said that by a process of argument and figuring (he used strange words for these processes) he was able to write down what he called a "truth".
This indicates that Friday understood the concept of mathematical proof.

It was the same when, in setting out the foundations for a rectangular shed...
This spoof echoes the famous comment by Pierre de Fermat, in which he stated that he'd "proved" his famous conjecture, but found that the margin of his journal was not quite big enough to contain the proof.

Friday looked at me queerly and suggested that instead, we might construct a square stockade...
The ancient geometers' paradox of squaring the circle: did Friday know that π was a transcendental number?

I tried to compute π by measurement, by drawing circles as best I could...
This leads to Buffon's method for determining π — see here for more.

I took this for idle nonsense but later that night he came to me and announced that the value of π was 3.14159.
Hmmm. With the original article I wrote a little Monte Carlo simulation to show just how inefficient this method is for determining π. In fact, had Crusoe persevered with his drawing and measurement of circles he would have obtained a satisfactory answer much more quickly. Friday could not possibly have carried out the number of replications necessary to achieve the accuracy implied by 3.14159, so the strong suspicion is that he used another method: involving series perhaps? Top marks to anyone who suggests that he anticipated Ramanujan's wonderful expression

  \[ \frac{1}{\pi } = \frac{2\sqrt {2}}{9801} \sum _{n=0}^{\infty } \frac{(4n)!(1103+26390n)}{(n!)^4396^{4n}}. \]    

which produces an accuracy better than Friday's quoted value using just the zero value for n.

He talked of the special beauty of numbers divisible only by themselves.

He then raved about what I could only comprehend as "numbers of the imagination".
Complex numbers


"A horse! A horse! My kingdom for a horse!"
From Shakespeare's Richard III

"Tomorrow, and tomorrow, and tomorrow, creeps in this petty pace from day to day."
Shakespeare's Macbeth.

"Nothing will come of nothing, speak again!"
Shakespeare's King Lear.


It would go into what he called a "hedge fund"...

As one of the responders to the article has pointed out, hedge funds are a 20th century invention, so Friday was well ahead of his time with this one.

Peter Schabil, February 18, 2014

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