## Fibonacci and Technical Analysis

Fibonacci numbers and Fibonacci ratios are used by a wide range of traders to spot potential turning points in the market. As far back as 1930's Ralph Elliott, who developed the Elliott Wave Theory, noted that price movements have three basic characteristics: pattern, time, and ratio; all of which observe the Fibonacci summation series. Indeed, Fibonacci ratios play a large role in Elliott Wave Theory, but Fibonacci numbers and ratios are not only used by Elliott Wave theorists, or Elliotticians, they are used by other traders too, mainly to determine support and resistance levels and to identify potential turning points in the market.

### The Fibonacci Summation Series

The Fibonacci Summation series was introduced to the Western world by an Italian mathematician, Leonardo Pisano Bogollo who was also known as Fibonacci (son of Bonacci), in his 1202 manuscript *Liber Abaci*, although the series had been previously described in Indian mathematics. The series is derived by finding the sum of the of the two preceding numbers in the series, with 0 and 1 being the seed numbers, or starting point, in the series. With ) and 1 being the first two seed numbers, the third number in the series is the sum of 0 and 1 (0 + 1), which is 1; the fourth number in the series is the sum of the second and third numbers in the series, namely, 1 + 1, and so forth. The first few numbers in the series thus are:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ... ∞

### Fibonacci Ratios

As the numbers the series increase towards infinity, **mathematical relationships**, expressed as ratios, appear between the numbers. For example, the ratio between consecutive numbers tends to oscillate closer and closer towards **1.618034**, which is known as the **golden ratio** or the **golden section** or the **golden mean** and is denoted by the upper-case Greek letter Phi Φ in mathematics. This ratio is reached by dividing a number in the summation series by the number that precedes it: 233 ÷ 144 = 1.618055

The inverse of this ratio is **0.618034**, which is denoted by the lower-case Greek letter phi φ in mathematics. This ratio is reached by dividing a number in the summation series by the number that succeeds it: 144 ÷ 233 = 0.618025

The square of 0.618034 approximates the ratio of **0.381966** and when we divide a number in the summation series by the number that is two places after it we tend closer towards this ratio. We also reach this ratio by subtracting 0.618034 from 1: 0.618034 x 0.618034 = 0.381966 or 144 ÷ 377 = 0.381962 or 1 - 0.618034 = 0.381966

When we multiply the ratio 0.381966 by 0.618034, we reach the ratio of **0.236068** and when we divide a number in the summation series by the number that is three places after it we tend closer towards this ratio. We also reach this ratio when we subtract the ratio 0.381966 from 0.618034: 0.381966 x 0.618034 = 0.236068 or 144 ÷ 610 = 0.236065 or 0.31966 - 0.618034 = 0.236068

The square root of 0.618034 gives us the approximate ratio of **0.786152**: √0.618034 = 0.786152

The square root of 1.618034 gives us the approximate ratio of **1.272020**: √1.618034 = 1.272020

The square of 1.618034 approximates the ratio of **2.168034** and when we divide a number in the summation series by the number that is two places before it we tend closer towards this ratio: 1.618034 x 1.618034 = 2.618034 or 144 ÷ 55 = 2.618181

When we multiply 2.618034 by 1.618034 we reach the ratio of **4.236068** and when we divide a number in the summation series by the number that is three places before it we tend closer towards this ratio: 2.618034 x 1.618034 = 4.236068 or 144 ÷ 34 = 4.235294

These ratios, along with the ratios of 50.0%, 100%, form the key Fibonacci levels that are used in technical chart studies, namely: **23.6%**, **38.2%**, **50,0%**, **61.8%**, **78.6%**, **100.0%**, **127.2%**, **161.8%**, **216.8%**, **423.6%**. Of these levels 38.2%, 61.8% and 161.8% are the most significant.

## The Golden Ratio

The Lucas number series developed by the French mathematician François Édouard Anatole Lucas is another sequence of numbers that adhere to the Golden Ratio. It is similar to the Fibonacci summation series with the key difference being that the seed numbers (the starting point) for Lucas numbers are 2 and 1. The numbers in this series are:

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, etc.

Like the Fibonacci sequence, the ratio between two consecutive Lucas numbers also converges closer to the golden ratio of 1.618034 as the numbers reach closer to infinity. In fact, any numbers series that is derived from the sum of the two previous numbers converge closer towards the golden ratio.