The Origin, Formulas, and Patterns of the Fibonacci Numbers

Fibonacci numbers

Fibonacci numbers are a sequence of figures where each number is the sum of the two previous ones, starting with 0 and 1. The series of numbers begins as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, and continues infinitely. Fibonacci numbers are not just an abstract mathematical concept, but rather a common pattern in many areas of life and manifestations of nature. The arrangement of leaves on a stem, the branching of trees, the spirals of shells, and even the breeding patterns of rabbits are only a small handful of examples of the Fibonacci sequence in the natural world.

The history of the Fibonacci sequence

The mathematical sequence known as the Fibonacci sequence is named after Leonardo of Pisa, who posthumously came to be known as Fibonacci. The sequence had long been known in India, but Fibonacci introduced it to the Western mathematical community in his book Liber Abaci (Book of Computations), published in 1202. Fibonacci was primarily concerned with the practical application of mathematics, so he introduced the sequence as a solution to a problem about the growth of a population of rabbits.

Fibonacci number templates and formulas

The Fibonacci sequence can be calculated by the formula:

Fn = Fn–2 + Fn–1

where F0=0, F1=1, and n is greater than or equal to 2 and is an integer.

This simple formula generates an endless series of numbers that show intriguing properties and patterns on closer inspection. For instance, the ratio of successive Fibonacci numbers tends to approach the golden ratio, approximately 1.618, as the sequence progresses.

Other patterns in the Fibonacci numbers

1. Every third Fibonacci number is a multiple of 2, every fourth is a multiple of 3, and every fifth is a multiple of 5.

2. Fibonacci numbers contain the pattern where, modulo 10, the last digits repeat every 60 numbers.

Golden Ratio / Phi (φ)

The golden ratio, often symbolized by the Greek letter φ (phi), is an important aspect of the Fibonacci sequence. It is an irrational number approximately equal to 1.618033988749895. Mathematically, the golden ratio is calculated as:

The link between the Fibonacci numbers and the golden ratio is that the ratio of consecutive Fibonacci numbers approaches φ as the numbers increase.

This ratio is often observed in nature, art, architecture, and even the movements of financial markets.

Using the Fibonacci numbers in stock trading

In stock trading, Fibonacci number patterns are often used to predict future price movements based on an asset’s previous performance. Traders apply the Fibonacci system to identify potential reversal or retracement levels (where the price may pull back before continuing in the original direction). Based on key Fibonacci ratios derived from the sequence, these retracement levels may be deviated from to some degree, depending on a trader’s experience.

What are Fibonacci ratios?

Fibonacci ratios, also known as Fibonacci retracement levels, are deduced from the Fibonacci sequence. The most popular Fibonacci ratios used in trading are 23.6%, 38.2%, 50%, and 61.8%. The 50% level is not derived from the Fibonacci sequence, but it is often used due to its significance in trading history.

These ratios help traders identify levels of support or resistance where price corrections may occur during a trend. When the market is trending upward, Fibonacci ratios can be used to identify support levels where the price may retrace before continuing higher. Conversely, in a downtrend, these ratios help pinpoint resistance levels.

The Fibonacci retracement level formula

Fibonacci retracement levels in trading are calculated by taking two extreme points (usually a high and a low in the price) and dividing the vertical distance by the key Fibonacci ratios in percentages of 23.6%, 38.2%, 50%, 61.8%, and 100%.

To calculate a Fibonacci retracement level, you need two key points on a price chart: the high point (HHH) and the low point (LLL) of a price movement. The Fibonacci retracement levels are then calculated as follows:

Where:

• H = The high point of the price movement.

• L = The low point of the price movement.

• F = The Fibonacci level (commonly 0.236, 0.382, 0.500, 0.618, 0.764).

How to apply Fibonacci retracement levels on charts

To use Fibonacci retracement levels on a chart, identify a significant price movement — either up or down — and plot the key Fibonacci levels over the price action. Draw a line from the swing high to the swing low for a downtrend, or from the swing low to the swing high for an uptrend. The resulting Fibo lines on the chart represent potential levels of support or resistance.

How to draw Fibonacci retracement

To draw Fibonacci retracement levels:

1. Identify the high and low points of a recent price move on the chart.

2. Draw a trendline from the high point to the low point in a downtrend, or from the low point to the high point in an uptrend.

3. Important: you must use the shadow of the candle, not the body.

4. The key Fibonacci levels (23.6%, 38.2%, 50%, 61.8%, 100%) will automatically be plotted on the chart.

5. Observe these levels as potential reversal points for price action.

How to trade using Fibonacci retracement

To use Fibonacci retracement levels correctly, traders look for potential reversal zones. For example:

• In an uptrend: Traders may look for buying opportunities near the 38.2% or 61.8% retracement levels.

• In a downtrend: Traders might consider shorting the market after rebounding off the retracement level.

• There is a strong Fibonacci zone at 61.8% and 65.0%, called the golden ratio. The market often reacts to this significant zone.

The strategy usually involves entering a trade when the price shows signs of reversаl near one of these levels, and plаcing stop-losses just beyond the next retrаcement level to minimize risk. However, it is better to use Fibonacci levels as a strategy in conjunction with other indicators or technical analysis metrics.

Placement of stop-loss and take-profit

When trading with Fibonacci retracement, stop losses are usually placed just beyond the next Fibonacci level to limit potential losses. For instance, if entering a trade at the 61.8% retracement level, a stop-loss could be placed just below the 76.4% level or the previous high/low.

Take-profit levels can be set at the previous high or low (depending on the trend direction) or at key Fibonacci extension levels such as 127.2% or 161.8% beyond the initial price move, etc.

Combining Fibonacci retracement with other tools

Fibonacci retracement is often used in conjunction with other technical analysis tools like moving averages, trendlines, candlestick patterns and oscillators like RSI (Relative Strength Index), Momentum, MACD, etc. Combining these tools can increase the accuracy of trade setups and help confirm potential reversal points more precisely.

For example, if a Fibonаcci retrаcement level coincides with a moving average or a trendline, it strengthens the likelihood that the level will hold and act as support or resistаnce.

Limitations of using Fibonacci retracement levels

While Fibonacci retracement levels are popular with traders and can be effective, they are not foolproof. The levels should not be relied on alone. Market conditions and price action can sometimes overshoot or fail to reach these levels.

Fibonacci retracement levels do not predict future movement or the length of time a retracement will last. It is crucial to understand the broader market context and potential pivot points, so you really should use these levels in conjunction with other indicators and tools.

Summary

Fibonacci numbers and their calculated retracement levels are powerful tools in both mathematics and trading. However, like any tool, they should be used as part of a comprehensive strategy in general view, rather than in isolation. Understanding the underlying patterns and formulas of the Fibonacci sequence can enhance your trading approach and provide a deeper insight into market dynamics.