The Grey Chronicles

2009.February.3

The Weird Wonderful World of Numbers


Math Wonders

Math Wonders

I become a fan of the T.V. show, NUMB3RS, whereby a genius Ph.D., Charlie Epps, helps the FBI solve crimes using mathematics. Several episodes of the show are intriguing, mind-boggling and just entertaining.

Posamentier (2003) even encourages everyone to collect books on recreational mathematics, read them, and hold them for reference. “There are many books on topics not usually taught in the schools, such as the history of mathematics, problem solving, and on special topics, e.g., magic squares, mathematical entertainments.”


Fermat’s Last Theorem

Pierre de Fermat (1601—1665) claims that the equation xn + yn = zn has no nonzero integer solutions when n is greater than 2. He added that the margin of of his copy of the Arithmetica of Diophantus, at problem 8 in Book II, was too narrow to contain the truly remarkable proof. This note has become known as Fermat’s Last Theorem, written in 1637:

Cubum autem in duos cubos, aut quadrato-quadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere; cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

That is: ”To split a cube into two cubes, or a fourth (biquadratic) power into two fourth powers, or indeed any higher power unto infinity into two like powers, is impossible; and I have a marvellous proof for this. But the margin is too narrow to contain it.” (van der Poorten, 1998).

Fermat’s Last Theorem is the most tantalizing problem in the world of mathematics. Even after more than 370 years, many have been tempted to delve into it and came near, but never close, such as Andrew Wiles (1995) announced in 1993 to have proven it, as several others before and after him, but many mathematicians have yet to find a simple proof (Schorer, 2007) and not circumvent it (Faltings, 1995; Hammond, 1993).


The Quest for the Value of Pi, Π

Most of us know and used the typical value of π as 3.1416, the ratio of any circle’s circumference to its diameter. But mathematicians and computer scientists are not satisfied with, say, the first 50 digits of π. To this day, there exist the quest to calculate π to millions of decimal places, which may seem rather pointless. Borwein and Borwein (1988) explain:

“Several answers can be given. One is that the calculation of pi has become of a benchmark computation; it serves as a measure of sophistication and reliability of the computers that carry it out. . . leads mathematicians to intriguing and unexpected niches of number theory . . . is simply "because it’s there"”

Mathematicians such as Archimedes of Syracuse, Isaac Newton and Gottfried Wilhelm Leibniz, Srinivasa Ramanujan, John Machin, among others attempted to compute pi to several places. With the advent of digital computers since June 1949 and iterative algorithms, it became possible to generate up to more than 100 million places. Yet, no one has succeeded in proving that the digits of pi follow a random distribution.

Astonishingly, pi turns up in all kinds of unexpected places that have nothing to do with circles. Borwein (1988) gives an example: “If a number is picked at random from a set of integers, the probability that it will have no repeated prime divisors is six divided by the square of pi.”


The Geometry of Phi, Φ in Nature

Polar Spiral

Polar Spiral

The Fibonacci sequence (discovered by Leonardo Pisano Fibonacci where each consecutive number in the sequence is derived from the sum of the previous two) approximate the Golden Ratio. The Golden Ratio usually expressed as 1.618 and 0.618 and is known as Phi [Φ] and phi [φ], respectively; phi being the reciprocal of Phi. Phi is equal to [(Square Root of 5) + 1]/2. Phi’s unique properties include: Φ2 = Φ + 1 or (1.618)2 = 1.618 + 1. Phi multiplied by phi, i.e., Φ × φ = 1 or 1.618 × 0.618 = 1 (Knott, 1988).

The Golden Ratio can be seen embedded in many of the ancient monuments, such as The Great Pyramid (debunked by Fischler, 1981; Markowsky, 1992; among others) or the Parthenon. It is also found in ALL living creatures on Earth. This ratio can be found in fingers one’s hand and it is prevalent in the skeletal structure of all creatures, including the features of the human face.


Notes:

Borwein, Jonathan M. and Borwein, Peter B. (1988). Ramanujan and Pi New York: Scientific American, 1988. Also included in The World Treasury of Physics, Astronomy, and Mathematics edited by Timothy Ferris (1991). New York: Little, Brown & Co., 1991. pp. 647-659. back to text

Faltings, Gerd (1995). The Proof of Fermat’s Last Theorem by R. Taylor and A. Wiles, Notices of the AMS, American Mathematical Society July 1995, pp. 743-746. back to text

Fischler, Roger (1981). How to Find the “Golden Number” without really trying Fibonacci Quarterly, 1981, Vol 19, pp. 406 – 410. back to text

Hammond, William F. (1993). Fermat’s Last Theorem After 356 Years A Lecture at the Everyone Seminar University at Albany, 22 October 1993. GELLMU Edition with Retrospective Comments, 21 April 2001 with minor revisions: 15 July 2004. pp. 20. back to text

Knott, Dr Ron (1996). What is the Golden Ratio (or Phi)? accessed 25 August 2005. back to text

Markowsky, George (1992) Misconceptions about the Golden ratio. The College Mathematics Journal. Vol 23, January 1992. pp 2-19. back to text

Schorer, Peter (2007). Is There a “Simple” Proof of Fermat’s Last Theorem? Introduction and Several New Approaches. Berkeley, 04 November 2007. 83pp. back to text

van der Poorten, Alf J. (1998). Fermat’s Last Theorem. Sydney: Enciclopedia Italiana, Appendice 2000. 1998. back to text

Posamentier, Alfred S. (2003). Math Wonders to Inspire Teachers and Students. Virginia: Association for Supervision and Curriculum Development, 2003. 295pp. back to text

Wiles, Andrew (1995). “Modular elliptic curves and Fermat’s Last Theorem”, Annals of Mathematics, (second series) Vol. 141, pp. 443–551. back to text

Disclaimer : The posts on this site are my own and doesn’t necessarily represent any organization’s positions, strategies or opinions.

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