While learning about current day techniques for root finding, we should realize the roots from which root finding was born; almost 4000 years ago. Some of the first approximations of roots came from Babylonian mathematics written in cuneiform script on clay tablets dating back to 1800 BC. These clay tablets include topics including fractions, algebra, quadratic, and even cubic equations. One of these tablets approximates sqrt(2) accurately to 5 decimal places. (1.414212963) (actual = 1.414213562).
The “Babylonian Method” of finding the square root of a number N goes as follows…
Pick a number somewhat close to sqrt(N). We’ll call this guess Xn The following iteration is the arithmetic mean of Xn and N/Xn. By following this procedure then the limit of Xn as n approaches infinite goes to sqrt(N).
A major limitation to the ancient mathematics was that it was done through geometrical means. All root finding today is done through pure arithmetic.
It is interesting to look back and find the similarities between the earliest root finding and our current methods to date.
- Both required initial approximations.
- Both made attempts to bound the root
- Both worked recursively
References
http://en.wikipedia.org/wiki/Babylonian_mathematics
http://mipagina.cantv.net/arithmetic/rmdef.htm#NewAlgorithms
http://en.wikipedia.org/wiki/Square_root_of_two
You say:
“A major limitation to the ancient mathematics was that it was done through geometrical means. ”
That’s true, all methods was based on geometry.
and you say:
“All root finding today is done through pure arithmetic.”
That’s not true. Newton’s, Halley’s, Householder’s methods are all based exclusively on geometry (analytic geometry and infinitesimals) as well as many other well-known modern methods.
There is only one general root-solving method based exclusively on Pure Arithmetic (No geometry, at all):
THE RATIONAL PROCESS shown at the webpage:
mipagina.cantv.net/arithmetic/roots.htm
And that means a lot, mainly, when considering the whole history of roots solving.
The ‘Rational Process’ have been recently developed by using the new and extremely simple and general ‘Rational Mean’ concept.
Greetings.
D. Gomez.