Fibonacci sequence: interpretation, examples, and how to use in trading

The Fibonacci sequence is a fascinating numerical pattern that has intrigued mathematicians, traders, and analysts for centuries. Derived from a simple set of rules, this sequence holds a deep connection to various aspects of life and is found to have remarkable applications in financial markets.

In this article, we will explore the significance of Fibonacci numbers in finance, uncovering how they are used to identify key levels, predict potential price movements, and analyse market trends through tools like Fibonacci retracements, extensions, arcs, fans, and time zones.

History of Leonardo Fibonacci of Pisa

Leonardo Fibonacci of Pisa, better known by his nickname Fibonacci (which means son of Bonacci), was born in the Italian city of Pisa and lived from 1170 to 1250.

He called himself “Bigollo” which means “good for nothing”. Leonardo starts to shape his identity as a merchant and mathematician in the city of Bugia, now known as Bejaia, a port in the northeast of Algeria.

Very little is known about his life. However, in the preface of one of his most significant books, the Liber Abaci, Leonardo remarks that it was his father who taught him arithmetic and encouraged him to pursue the study of mathematics.

In Bugia, Leonardo received such teachings from Arab instructors, which undoubtedly was the most advantageous experience for a young medieval Italian eager to grasp mathematics. He specialised in arithmetic and the various numeral systems in use during that era.

It wasn't long before he became convinced of the superiority of the Indo-Arabic system over those employed in the different countries he had visited. He decided to introduce this system to Italy, and ideally, to the entirety of Europe, where Roman numerals and the abacus still prevailed.

The study of mathematics and its practical applications as an indispensable tool for trade development consumed a significant portion of his life.


Leonardo returned to Pisa around 1200, where he authored numerous books and texts on mathematics. During his time, the printing press had not yet been invented, so his works were handwritten, and the circulated copies were also painstakingly produced by hand.

It is easy to envision the limited number of copies that could circulate during that era. However, despite seeming implausible, several books have managed to survive to this day, including Liber Abaci, written in 1202; Practica Geometriae, written in 1220; Flos, written in 1225; and Liber Quadratorum, written in 1227. Sadly, many more works were lost over the course of history.

In 1200, Fibonacci published his renowned Liber Abaci, which introduced one of the most significant mathematical tools in history—the decimal numeral system—to Europe. This book introduced fundamental arithmetic operations in the form of whole numbers and fractions. It also delved into the advanced trigonometry and algebra of the Arab scholars—a breakthrough so momentous in mathematics that Fibonacci's solutions and discoveries remained benchmarks throughout the Western world for the next three centuries.

Leonardo's reputation continued to soar, and by 1225, he was widely recognised as one of the best mathematicians. Courts and businesses sought his advice.

Fibonacci sequence

The Fibonacci sequence, featured in the Liber Abaci, presents the challenge of calculating the growth of a rabbit population from its initial state. To tackle this, Fibonacci puts forward the following solution.

You commence with two rabbits (couple A) who reproduce and give birth to a male-female pair (referred to as couple B) during the second month. Subsequently, we have two pairs of rabbits.

Moreover, in the second month, the original couple reproduces once more and produces a new pair of offspring (couple C) who are born in the third month. Additionally, during the third month, couple B enters the reproductive phase and gives birth to a pair of rabbits (couple D) in the fourth month.

Similarly, couple A has a third litter (couple E). It is always assumed that when rabbits reproduce, they produce a pair and become fertile at the conclusion of their first month of life.

Under these assumptions, Leonardo solves the problem by introducing a recurrence relation that was later named the Fibonacci Sequence by the 19th-century French mathematician Édouard Lucas in his honour. It is presented as a solution to a mathematical problem concerning the rate of rabbit reproduction under specific circumstances.

Hence, the Fibonacci sequence or series is calculated by adding the current number to the previous number, such that: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…

Properties of the Fibonacci sequence

  • The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones.
  • The ratio between a number and its predecessor (tn/tn-1) approaches the golden ratio, Phi, which is approximately equal to (1 + √5) / 2 ≈ 1.618. Similarly, the ratio between a number and the subsequent one (tn-1/tn) tends to approach the inverse of Phi, which is approximately 0.618. These relationships hold true for most numbers in the Fibonacci sequence, except for the first three numbers.
  • Another interesting property of the Fibonacci sequence is that the sum of the first ten terms is equal to eleven times its seventh term. This property holds true for any sequence constructed following the assumptions of the Fibonacci sequence, regardless of the choice of the initial two numbers.
  • Furthermore, if any number from the sequence is taken starting from the 3rd term, multiplied by 4, and then added to the number corresponding to the three previous terms, the result is the number of the sequence located three positions ahead. This can be expressed algebraically as tn*4 + tn-3 = tn+3. For example, if we consider the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, taking the number 21, multiplying it by 4, and adding the number 5 (which is three positions before 21), we obtain the number 89.

Fibonacci in technical analysis

The Fibonacci sequence is widely used in technical analysis. Several tools are derived from this sequence. These tools use the ratio between the different digits of the series, mainly between a number and the previous one. The higher these ratios, the closer the result approaches the golden ratio (1.618). The equation of the golden ratio is:

Based on this golden ratio, a set of proportions emerges that exhibit a direct mathematical relationship with this value, and these proportions can be employed in technical analysis when analysing graphs.

Main Proportions

Fibonacci tools technical analysis

Fibonacci time zones, also known as Fibonacci temporal zones, are a technical analysis tool used to identify potential reversal or continuation points based on the Fibonacci sequence.

They are typically applied by counting forward from a significant high or low point on a chart to the next significant high or low point. The lines marked on the chart are interpreted as potential inflexion points in the future.

On a daily chart, Fibonacci trading days are counted using numbers from the Fibonacci sequence, such as 5, 8, 13, 21, 34, and so on. However, many charting platforms have built-in tools that automatically display the Fibonacci time zones, eliminating the need for manual counting. It's worth noting that the effectiveness of Fibonacci time zones may vary on shorter time charts.

Fibonacci fans are another tool used in technical analysis. They help identify potential levels of support and resistance and provide insights into the timing and depth of corrections following an impulse move. To use Fibonacci fans, a line is drawn from the low to the high of the impulse move.

The tool then generates a series of lines with different angles of inclination based on important Fibonacci ratios. If the main impulse is bullish, the Fibonacci fans are drawn from the low to the high. Conversely, if the main impulse is bearish, they are drawn from the high to the low.

Fibonacci Expansion (or Extension) is a tool used in technical analysis to project potential price targets or levels of extension in an ongoing trend. Unlike other Fibonacci tools, it requires both an initial impulse and a corrective movement. The purpose of Fibonacci expansion is to estimate how far the impulse wave may continue.

To plot a Fibonacci expansion, the cursor is placed at the base of the impulse wave and then moved to the peak of the first internal sub-wave within the corrective phase. Next, the cursor is brought back to the base of the second internal sub-wave. The tool then generates different Fibonacci proportion lines that indicate potential levels to which the impulse wave could extend.

These Fibonacci proportion lines provide traders with potential price targets or areas of interest where the trend may experience significant resistance or support. By projecting these levels, traders can gauge the potential strength and duration of the ongoing trend.

Fibonacci retracements are one of the most widely used tools derived from the Fibonacci sequence in technical analysis. This tool is employed to identify potential areas where a corrective movement may find support or resistance levels. To use Fibonacci retracements effectively, it is necessary to have a completed previous impulse movement. Without a prior impulse, the tool may not provide useful information.

To apply Fibonacci retracements correctly, we need to identify the lowest and highest points of the completed impulse and draw the retracement levels based on these points. Typically, the default levels marked on the chart are 23.6%, 38.2%, 50%, and 61.8%, with the latter being equivalent to the golden ratio.

According to the theory, the most probable zone for a corrective process to end is between the 50% and 61.8% retracement levels. These levels are often considered as areas where price may find support or resistance before resuming the main trend. In the event of an upward correction following a bearish impulse, for example, the price may find a potential reversal point around the 50% retracement level and continue its bearish trend.

Check out our technical analysis guide to find out more about the most important tools and theories used by traders.

Interpretation of numbers and Fibonacci lines in trading

Some traders consider Fibonacci numbers to be of vital importance. The Fibonacci numerical sequence can be used to create proportions or percentages commonly used by traders, such as 23.6%, 38.2%, 50%, 61.8%, 78.6%, 100%, 161.8%, 261.8%, and 423.6%.

These percentages are used in various techniques, such as:

  • Fibonacci Retracements: horizontal lines on a chart indicating areas of support and resistance.
  • Fibonacci Extensions: horizontal lines on a chart indicating price levels where a strong wave or trend could potentially reach.
  • Fibonacci Arcs: curved lines derived from a significant high or low point that represent areas of potential support and resistance.
  • Fibonacci Fans: diagonal lines generated using a significant high or low point to identify areas of potential support and resistance.
  • Fibonacci Time Zones: vertical lines extending into the future on a chart, intended to predict potential time periods for significant price movements.

Among these techniques, Fibonacci retracements are the most commonly used form of technical analysis based on the Fibonacci sequence. Traders utilise this tool to determine potential levels of retracement or correction within the context of the current trend.

To conclude, Fibonacci numbers are employed to establish proportions and percentages derived from the Fibonacci sequence. Traders and analysts have adapted these Fibonacci methods to financial markets, utilising them as tools for technical analysis. It's important to note that Fibonacci methods were not originally created for trading purposes but have been incorporated and applied by traders and analysts in the financial markets.

Learn more about trading and technical analysis:


How is the Fibonacci sequence used in technical analysis?

Fibonacci numbers have gained significance in financial markets due to their ability to identify key levels of support and resistance, predict potential price targets and extensions, and analyse market trends. Traders and analysts utilise Fibonacci tools such as retracements, extensions, arcs, fans, and time zones to enhance their technical analysis and make informed trading decisions.

How are Fibonacci retracements used in technical analysis?

Fibonacci retracements are horizontal lines drawn on a chart to identify potential areas of support and resistance. Traders use these levels to gauge potential price retracements or corrections within an ongoing trend.

Can Fibonacci tools accurately predict market movements?

While Fibonacci tools provide valuable insights, it's important to note that they are not infallible predictors of market movements. They should be used in conjunction with other technical analysis tools.

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