**Fibonacci Numbers**

Fibonacci numbers are named after Leonardo Fibonacci, a twelfth century Italian mathematician, who discovered the unique properties of a particular number sequence; apparently from studying the dimensions of the Great Pyramid at Gizeh in Egypt.

*Fibonacci numbers*

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, etc.

Each number in the sequence is the sum of the previous two numbers:

1 + 1 = 2,

1 + 2 = 3,

2 + 3 = 5,

and so on….

**The Golden Ratio**

As we progress along the sequence, the ratio of each number to its preceding number approaches closer and closer to the golden ratio: approximately 1.618. The golden ratio, often represented by the Greek letter Φ (Phi), is calculated as:

**( 1 + √ 5 ) / 2**

Each number is also approximately 0.618 of its successor. This reciprocal number, known as φ (phi), is calculated as:

**( √ 5 – 1 ) / 2**

Where √ 5 is the square root of 5.

Fibonacci Golden Ratio and its Reciprocal |

Each number divided by its predecessor approaches 1.618 |
Each number divided by its successor approaches 0.618 |

1/1 |
1.0 |
1/1 |
1.0 |

2/1 |
2.0 |
1/2 |
0.5 |

3/2 |
1.5 |
2/3 |
0.666… |

5/3 |
1.666… |
3/5 |
0.6 |

8/5 |
1.6 |
5/8 |
0.625 |

13/8 |
1.625 |
8/13 |
0.61538… |

21/13 |
1.61538… |
13/21 |
0.61905… |

34/21 |
1.61905… |
21/34 |
0.61765… |

55/34 |
1.61765… |
34/55 |
0.61818… |

89/55 |
1.61818… |
55/89 |
0.61798 |

**Fibonacci Numbers in Nature**

Fibonacci numbers occur throughout nature:

- the arrangement of petals in most flowers
- the arrangement of leaves on most plants
- sea-shell spirals
- the arrangement of seeds on sunflowers, pine cones and many other plant species.

While the Fibonacci number sequence may be prevalent in nature, it is not a universal law. There are many exceptions.

**Fibonacci Ratios**

Four ratios are normally plotted:

- 0.618 (or 61.8 per cent), the reciprocal of the golden ratio, is the most important;
- 0.50 (or 50 per cent) – the second number divided by the third (1 divided by 2);
- 0.382 (or 38.2 per cent) – the reciprocal of the golden ratio squared (i.e. 89 / 233);
- 0.236 (or 23.6 per cent) – the reciprocal of the golden ratio cubed (i.e. 55 / 233).

**Fibonacci and Stocks**

Fibonacci ratios regularly occur in stock market cycles and in the determination of support and resistance levels. Some traders attach almost mystical significance to them, but I have yet to find any statistical support for this.

The weakest of the Fibonacci ratios is 0.50. In fact some maintain that 0.50 is not really a Fibonacci ratio at all because it has no connection to the golden ratio. Nevertheless, it is probably the most prevalent: the first line of support in a rally is the previous peak — which often equates to a 50% retracement.

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