Origin of the Fibonacci Number Sequence

  Fibonacci numbers are based upon the Fibonacci sequence discovered by Leonardo de Fibonacci de Pisa (b.1170-d.1240). His most famous work, the Liber Abaci (Book of the Abacus), was one of the earliest Latin accounts of the Hindu-Arabic number system. In this work, he developed the Fibonacci number sequence, which is historically the earliest recursive series known to date.
  The series was devised as the solution to a problem about rabbits. The problem is: If a newborn pair of rabbits requires one month to mature and at the end of the second month and every month thereafter reproduce itself, how many pairs will one have at the end of n months? 

The answer is: un. This answer is based upon the equation: un+1 = un+un-1

  Although this equation might seem complex, it is actually quite simple. The sequence of the Fibonacci numbers is as follows:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144, 233, 377 ...... up to infinity 

  Starting with zero and adding one begins the series. The calculation takes the sum of the two numbers and adds it to the second number in the addition. The sequence requires a minimum of eight calculations.

(0+1=1)...(1=1=2)...(1+2=3)...(2+3=5)...(3+5=8)... 5+8=13)…(8+13=21)…(13+21=34) 

  After the eighth sequence of calculations, there are constant relationships that can be derived from the series. For example, if you divide the former number by the latter, it yields 0.618.

34/55 = 0.618181 ~ .618 
55/89 = 0.618181 ~ 0.618 
And, if you divide the latter number by the former, it yields 1.618.
144/89 = 1.617977 ~ 1.618 
233/144 = 1.618055 ~ 1.618 

Not only do these constant numeric relationships occur in the Fibonacci series, there are also universal examples that exhibit this phenomenon. For example, Venus takes 225 days to complete a revolution around the sun. As we all know, the Earth requires 365 days to complete one revolution. If you divide 225 by 365, the result is approximately .618 of a year. (225/365 = .6164 ~ .618) That's amazing!

Fibonacci Ratios -> Harmonic Ratios

  The 0.618 and the 1.618 are the two numeric results derived by the Fibonacci Sequence comprising the Primary Ratios that are employed in Harmonic Trading measurements.  These Fibonacci-related numbers are utilized to validate price action as harmonic price action. 

From these  two Harmonic Trading Primary Ratios, other numbers can be derived from these  in the same manner as they were from the Fibonacci series.  Specifically, the 0.786 and 1.27, are the square root of the 0.618 and the 1.618, respectively.  The square root of the 0.786 and 1.27 yield another pair of ratios of primary importance. These ratios - although derived from the 0.618 and 1.618 of the Fibonacci Sequence - serve as an effective means to decipher´price action via the measurement of harmonic ratios.  Other harmonic ratios such as 38.2%, 50% and more that exist within the Harmonic Trading approach are derived indirectly from the primary ratios of the Fibonacci Sequence, as well.  This derivation is important as an effective strategy in the financial markets.  These harmonic ratios define specific and pertinent trading behavior  of harmonic price action that identifies extremely accurate natural opportunities.