Monday, March 08, 2010

Math is not hard, but roots of polynomials are always complex


From the latest entry -- Finding Your Roots -- in Steven Strogatz's series on Math: From Basic to Baffling:

Complex numbers are magnificent, the pinnacle of number systems. They enjoy all the same properties as real numbers — you can add and subtract them, multiply and divide them — but they are better than real numbers because they always have roots. You can take the square root or cube root or any root of a complex number and the result will still be a complex number.

Better yet, a grand statement called The Fundamental Theorem of Algebra says that the roots of any polynomial are always complex numbers. In that sense they’re the end of the quest, the holy grail. They are the culmination of the journey that began with 1.

A little later, he describes a numerical experiment -- with Newton's method of root finding -- that leads to some pretty stunning fractal patterns. You've got to see the accompanying video that keeps zooming into the pattern; it's absolutely mesmerizing!

Strogatz calls this work by his Cornell colleague John Hubbard "an early foray into what’s now called 'complex dynamics,' a vibrant blend of chaos theory, complex analysis and fractal geometry." He then brings that should perk up those with an interest in ancient Indian contributions to mathematics:

In a way it brought geometry back to its roots. In 600 B.C. a manual written in Sanskrit for temple builders in India gave detailed geometric instructions for computing square roots, needed in the design of ritual altars. More than 2,500 years later, mathematicians were still searching for roots, but now the instructions were written in binary code.

As a reference to the stuff about the ancient Indian "geometric instructions for computing square roots," Strogatz cites Experiencing Geometry on Plane and Sphere (Prentice Hall, 1996) by D. W. Henderson.

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2 Comments:

  1. gaddeswarup said...

    I think that the methods at the end are described in Kim Plofker's article in
    http://press.princeton.edu/TOCs/c8583.html
    (The Mathematics of Egypt, Mesopotamia, China, India, and Islam:A Sourcebook Edited by Victor J. Katz) sections of which are available in google books. More recently Kim Plofker has written a much longer book ( the earlier article is over 100 pages which read long ago, but I have not read the book) "Mathematics in India: 500 BCE-1800 CE". If the article is any indication, this book will be a definitive one for some time.

  2. அகிலன்(Akilan) said...

    Thanks for the feed link.