Introduction to Fibonacci Mathematics

Fibonacci mathematics can help traders to reveal the hidden proportionality of market behavior. Fibonacci extension analysis studies the extends of prime trends and countertrends in order to identify key reversal zones, or else levels where a trending market may lose momentum and reverse.

Calculating the Basic Ratios using the Fibonacci Sequence

The Fibonacci sequence of numbers begins as follows: 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 and etc.

The above sequence can then be broken down into ratios. The Fibo ratios can be found by dividing the Fibonacci numbers into each other.

If we do the math by excluding the first numbers, we realize that:

• Every number is approximately 1.618 times the preceding number
• Every number is 0.618 of the number to the right of it

Note that 1.618 is the golden ratio, and its inverse is 0.618.

In financial trading, the key ratios are 0.236, 0.382, 0.618, 1.618, 2.618, and 4.236. Many traders also use 0.5, 1.0.

Table: Key Fibonacci ratios for Financial trading

 0.236 1 0.382 1.618 0.5 2.618 0.618 3.618 0.786 4.236

Fibonacci Extensions

The Fibonacci sequence of numbers produces useful trading tools such as the Fibonacci retracement and the Fibonacci extensions.

Fibonacci extensions can identify the extent of prime trends and countertrends to spot potential price reversal zones. Let's start by calculating the Fibo extensions above 100.

eBook: Trading World Markets Using Phi and the Fibonacci Numbers

The Complete Guide to Fibonacci Trading and Phi by George M. Protonotarios

The complete guide to Fibonacci trading and Phi with reference to Elliott Waves, Dow Theory, Gann Numbers, and Harmonic Patterns, for trading successfully the Global Financial Markets (Forex currencies, Stocks, Indices, Metals, and Energies).

This Book covers an enormous range of trading theories and methodologies involving the Fibonacci numbers and their products. You will find all the basic Fibonacci trading practices and tools based on Fibonacci ratios in it. Phi and the Fibonacci numbers do not form just another tool of technical analysis. Phi proportions are everywhere: in arts, architecture, our DNA’s helix spiral, and even in our nature’s plant formations.

The first chapter begins with the mathematical properties of Phi and several of its applications outside the financial markets. In the next few chapters, you will find information about the Dow Theory, the Elliott Wave Theory, and the Gann numbers. At the end of each chapter, you will be able to detect the correlation of each theory with the Fibonacci numbers. You will learn also how you to use the Fibonacci numbers in order to create a trading system based on Fibonacci Moving Averages (MAs). In addition, you will find information about some popular Fibonacci trading tools such as the Fibonacci Retracement, the Fibonacci Extension, and the Fibonacci Fan. There are two chapters dedicated to Harmonic Trading and Harmonic Patterns. Harmonic trading is one of the most sophisticated trading practices and it is entirely based on Fibonacci proportions. Six basic harmonic patterns are presented with their properties and charts with examples. The last chapter is dedicated to money management and the effect of the irrational brain in our everyday decision-making process.

INTRODUCTION

• CHAPTERS:
1. THE ORIGINS OF FIBONACCI NUMBERS AND PHI (Φ)
2. THE DOW THEORY AND FIBONACCI NUMBERS
3. THE ELLIOTT WAVE THEORY AND FIBONACCI NUMBERS
4. W. D. GANN AND FIBONACCI NUMBERS
5. FIBONACCI NUMBERS AND MOVING AVERAGES
6. FIBONACCI RETRACEMENT AND OTHER TOOLS
8. SIX HARMONIC PATTERNS
9. THE IRRATIONAL BRAIN & MONEY MANAGEMENT
• APPENDIX
1. Dow Jones Industrial Record Highs
2. Heikin-Ashi Charts

By combining the information and tools presented in all chapters you have the chance to build the foundations of a trading system out of chaos. A trading system that can make you less emotional when trading the global markets and significantly improve your odds of winning

Trading World Markets Using Phi and the Fibonacci Numbers

Fibonacci Sequence and Phi (Φ)

The Fibonacci sequence was originally discovered by the Italian mathematician Leonardo de Fibonacci de Pisa (1170–1240). The basic concept of the Fibonacci sequence is that each number equals the sum of the two previous numbers.

The Fibonacci Sequence of Numbers Explained

As it was mentioned, Fibonacci discovered a unique numerical sequence according to which each number equals the sum of the previous two numbers, as follows:

• 1+1=2

• 1+2=3

• 2+3=5

• 3+5=8

• 5+8=13

• 8+13=21

• 13+21=34

to infinity

or else:

F(n) = F(n-2) + F(n-1)

Table: The first 60 Fibonacci Numbers

The Fibonacci Ratios

If you start dividing the Fibonacci numbers the result is always close to three ratios: 1.618, 0.618, or 0.382. For example, let's take three Fibonacci numbers 144, 233, and 377:

• If you divide 233 by 144, the result is 1.618

• If you divide 144 by 233, the result is 0.618

• If you divide 144 by 377, the result is 0.382

The ratios 1.618, 0.618, and 0.382 are very important when trading stocks, Forex, or any other financial asset. The Fibonacci ratios create an extended series of trading tools that are available in all modern trading platforms.

Additionally, the Fibonacci sequence of numbers provides the background of various methodologies when trading the global markets. There are numerous theories and methods explaining the behavior of financial markets based solely on the Fibonacci numbers and their derived ratios. This sequence of numbers and their ratios can be found behind hundreds of trading systems worldwide.

Fibonacci Sequence of Numbers and Phi

As it was mentioned before, if we divide each Fibonacci number to the previous Fibonacci number the result is always close to 1.618. The number 1.618 is known as Phi (Φ).

Phi (Φ) = 1.618033… (until infinity)

The Ancient Origins of Phi

Originally, the Phi number was properly defined by the ancient mathematician Euclid (around 300-B.C.). Before Euclid, the ancient philosopher Plato (428-B.C. – 347-B.C.) considered the Golden Ratio to be the most universally binding of all mathematical relationships. Before Plato, the great mathematician Pythagoras (570-B.C. – 475-B.C.) also referred to the Golden Ratio. Actually, no one knows for sure who really discovered Phi. Some believe that ancient Egyptians initially discovered and applied it to the Pyramids, others believe that the ancient Sumerians discovered it.

Phi ‘Φ’ is the first letter of the name Phidias (Greek:Φειδίας), the ancient sculptor who created the statue of Zeus at Olympia, one of the seven wonders of the ancient world. He applied the golden ratio in the construction of the Athenian Parthenon.

Unique mathematical properties of Phi

It is impossible to pin the golden ratio down. The first 10 million digits were computed in 1996, but they were never repeated. Here are some basic mathematical properties of Phi.

Key Properties of Phi:

(i) If you square Phi (Phi²), you get the number Phi+1, or else 2.618…, and that means:

Φ² = Φ + 1 = 2.618

(ii) If you divide Phi into 1 (1/Phi) you get the number Phi-1, or else 0.618…, and that means:

1 / Φ = Φ – 1 = 0.618

0.618 is sometimes symbolized as a small ‘p’

Golden Spiral

A golden spiral in Geometry is a logarithmic spiral whose growth factor is Phi (Φ). Α golden spiral gets wider by a factor of Phi for every quarter turn it makes.

Image: Fibonacci Golden Spiral

More about the Golden Spiral: https://math.temple.edu/~reich/Fib/fibo.html

Applications of Phi in the Universe

Phi is found everywhere in our body, nature, and even in our universe:

• The diameters of Moon and Earth form a mathematical triangle with dimensions close to phi properties
• The distances between our solar system’s sun and its planets correlate closely to Phi exponential powers
• NASA, in 2003, released that the shape of our Universe (Milky Way) is a dodecahedron based on Phi. The Milky Way has a number of spiral arms, each of which has a logarithmic spiral of roughly 12 degrees. The shape of the spiral is identical to the Golden Spiral.

Image: Applying two Golden Spirals on the nearby spiral galaxy M74 (Qexpert.com, original image: NASA/ESA Hubble Space Telescope)

Fibonacci Sequence and Phi (Φ)

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Fibonacci mathematics aims to reveal the hidden proportionality of market behavior. Find Fibonacci trading tools and tutorials:

eBOOKTRADING WORLD MARKETS USING PHI AND THE FIBONACCI NUMBERS (2018)

Guide to Fibonacci Trading with Reference to Elliott Waves, Gann Numbers, and Harmonic Patterns