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----- Original Message -----
From: "walt" <wa1ter@xxxxxxxxxxx>
Newsgroups: misc.invest.futures
Sent: Tuesday, September 25, 2001 9:13 AM
Subject: Re: More On Fibonacci
: zendial@xxxxxxxxx (Art Zapper) writes:
:
: > Cycle and astro types often talk about the Fibonacci ratio. It is
: > based on the so-called Fibonacci number series, which is formed by
: > starting with 0 and 1 and then adding each number to the previous one
: > to get the next number in the series. Here it is:
: >
: > 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc
: >
: > One interesting thing about this series is that (after about the 10th
: > number), if you divide any number by the following number, the ratio
: > remains roughly constant at about 0.618. For example 34/55=0.6181818
: > and 89/144=0.618055. ...
:
: Another interesting property that I've never seen in print is that
: the series can also be extended backwards by working to the left,
: subtracting the numbers instead of adding them:
:
:
: ... -21, 13, -8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, ...etc.
:
: This produces what engineers call an oscillator, i.e. the numbers
: swing above and below zero instead of just getting bigger or smaller.
:
: This property demonstrates the relationship between the Fib series
: and the mathematical 'exponential' function which is typical of so
: many natural phenomena like population growth--i.e. the rate of
: growth of a population is proportional to the total size of the
: population--a country of a billion people produces about 1000 times
: more babies each year than a city of one million would produce.
:
: But populations don't swing from positive to negative like the numbers
: to the left of zero, so what is that all about? This leads to the dark
: side of mathematics known as 'complex number theory' where the magic
: "imaginary" number 'i' rears its ugly head.
:
: 'i' is defined as the square root of -1, i.e. i x i = -1. You can
: only imagine what number that might be, so 'complex number' theory
: was often called 'imaginary number' theory by its many critics.
:
: It proved to be very useful over time, though, allowing the accurate
: mathematical modeling of natural systems that oscillate--like a
: weight hanging on the end of a rubber band that will bob up and
: down when you pull and release it. Or the pendulum on a clock,
: or the electronic oscillator that drives your digital watch.
: Or....maybe a market that swings up and down...
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