Fibonacci Numbers
The Fibonacci sequence mentioned in part 3 has many interesting properties. If we divide the golden ratio formula Φ2 = Φ + 1 by Φ, we get Φ = 1 + 1/Φ. Since the right side of the equation is equal to Φ, we can substitute it for Φ in the right side of the equation. That gives us Φ = 1 + 1/(1 + 1/Φ). This process could go on indefinitely, giving us the continued fraction expansion of (1 + √5)/2. The successive approximations to Φ, obtained by truncating the expansion are always the ratio of two successive Fibonacci numbers.
When the successive terms of the Fibonacci series are written as follows
.0112358
.00000013
.000000021
.0000000034
.00000000055
.000000000089
.00000000000144
etc.,
and the whole table is added, the total is the repeating decimal
.01123595505617977528089887640449438202247191,011235955056179...
which is equal to 1/89. In a sense, the Fibonacci number 89 is composed of the entire Fibonacci series! When we consider that the successive repeats of the decimal number are generated by entirely different and ever larger terms of the series, it is apparent that this cannot be a coincidence. Certain properties of Fibonacci numbers appear to occur only in bases 2,3,8, & 10.
Part 5 will be Calculations of pi
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Number Theory & Mysticism Part 4
