The Golden Ratio

Hi. This page will be about the golden ratio based on the Fibonacci sequence.

Recall that the Fibonacci sequence of numbers is an infinite (never-ending) sequence of numbers. Here are some of the starting terms of the Fibonacci sequence (beginning with 1 and 1).

    \[1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...\]


The Road To The Golden Ratio

The golden ratio can be achieved from the Fibonacci sequence of numbers by obtaining ratios of two successive numbers.

The ratio of the first two terms is \dfrac{1}{1} = 1. The ratio of the second number and the third number with the more recent number as the numerator is \dfrac{2}{1} = 2. The ratio of the fourth number over the third number is \dfrac{3}{2} = 1.5. We continue the pattern and notice that we eventually reach a number. Refer to the chart below.

Source: http://quicklatex.com/cache3/73/ql_8a001be5a3130da8206217270700d673_l3.png

 

 

We could continue the chart but since the Fibonacci sequence of numbers is never-ending (infinite) our chart would be never ending as well. We notice that the ratio reaches the number of 1.618 (to 3 decimal places). This number is referred to as the golden ratio.

The golden ratio is an irrational number as it is not a whole number and it has a never ending amount of decimal places. The irrational number Pi also has a never ending amount of decimals places with \pi \approx 3.14.

From the golden ratio, we can achieve the golden mean which is the reciprocal of the golden ratio. The reciprocal of 1.618 is \approx 0.6180 = 61.8\%

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