fibonacci numbers

Fibonacci Numbers

Fibonacci Numbers are a very popular tool among technical traders and are based on a sequence of integers identified by mathematician Leonardo Fibonacci in the 13th century. However, Fibonacci’s sequence is not as important as the mathematical relationships expressed as ratios between the numbers in the series.


Fibonacci Numbers are used in technical analysis by taking two extreme points (usually a major peak and trough) on a stock chart and dividing the vertical distance by the key Fibonacci ratios of 23.6% and 38.2%, 50%, 61.8% and 100%. Once these levels are identified, horizontal lines are drawn and used to identify possible support and resistance levels. Before we can understand why these ratios were chosen, we need to better understand the Fibonacci number series.

Sequence of numbers

The Fibonacci sequence of numbers is as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. Each term in this sequence is simply the sum of the two preceding terms, and on it goes until the sequence reaches infinity. One of the remarkable characteristics of this numerical sequence is that each number is approximately 1.618 times greater than the preceding number. This common relationship between every number in the series is the foundation of the common ratios used in retracement studies.

The Golden Ratio

The key Fibonacci ratio of 61.8% – also referred to as “the golden ratio” or “the golden mean” – is found by dividing one number in the series by its number. For example: 8/13 = .6153, and 55/89 = .6179. The 38.2% ratio is found by dividing one number in the series by the number found two places to the right. For example: 55/144 = .3819. The 23.6% ratio is found by dividing one number in the series by the number that is three places to the right. For example: 8/34 = .2352.

For unclear reasons, these ratios seem to play an essential role in the financial market, just as they do in nature and can be used to determine critical points that cause an asset’s price to reverse. The direction of the prior trend is likely to continue once the asset price has retraced to one of the ratios listed above.

Fibonacci and the Golden Ratio 

A unique ratio can be used to describe the proportions of everything from nature’s smallest building blocks, such as atoms, to the most advanced patterns in the universe, such as unimaginably large celestial bodies. Nature relies on this innate proportion to maintain balance, but the financial markets seem to conform to this ‘golden ratio’. Here we take a look at some technical analysis tools developed to take advantage of it.

The Mathematics 

Mathematicians, scientists, and naturalists have known this ratio for years. It’s derived from the Fibonacci sequence, named after its Italian founder, Leonardo Fibonacci (whose birth is assumed to be around 1175 AD and death around 1250 AD). Each term in this sequence is simply the sum of the two preceding terms (1, 1, 2, 3, 5, 8, 13, etc.).

But this sequence is not all that important; instead, the quotient of the adjacent terms possesses a significant proportion, roughly 1.618, or its inverse 0.618. This proportion is known by many names: the golden ratio, the golden mean, PHI and the divine proportion, among others. So, why is this number so important? Almost everything has dimensional properties that adhere to the ratio of 1.618, so it seems to have a fundamental function for the building blocks of nature.

Fibonacci numbers
Fibonacci numbers
Prove It! 

Don’t believe it? Take honeybees, for example. If you divide the female bees by the male bees in any given hive, you will get 1.618. Sunflowers, which have opposing spirals of seeds, have a 1.618 ratio between the diameters of each rotation. This same ratio can be seen in relationships between different components throughout nature.

Still don’t believe it? Need something that’s easily measured? Try measuring from your shoulder to your fingertips, and then divide this number by the length from your elbow to your fingertips. Or try measuring from your head to your feet, and divide that by the length from your belly button to your feet. Are the results the same? Somewhere in the area of 1.618? The golden ratio is seemingly unavoidable.

But that doesn’t mean that it works in finance, does it? Actually, the markets have the very same mathematical base as these natural phenomena. Below we will examine how this ratio can be applied to finance, and we’ll show you some charts to prove it!

The Fibonacci Studies and Finance 

When used in technical analysis, the golden ratio is typically translated into three percentages: – 38.2%, 50% and 61.8%. However, more multiples can be used when needed, such as 23.6%, 161.8%, 423%, etc. There are four primary methods for applying the Fibonacci sequence to finance: retracements, arcs, fans and time zones.

Fibonacci Retracements 

Fibonacci retracements use horizontal lines to indicate areas of support or resistance. They are calculated by first locating the high and low of the chart. Then five lines are drawn: the first at 100% (the high on the chart), the second at 61.8%, the third at 50%, the fourth at 38.2%, and the last at 0% (the low on the chart). After a significant price movement up or down, the new support and resistance levels are often at or near these lines.

Fibonacci studies are not intended to provide the primary indications for timing the entry and exit of stock; however, they help estimate support and resistance areas. Many people use combinations of Fibonacci studies to obtain a more accurate forecast. For example, a trader may observe the intersecting points in various Fibonacci arcs and resistances.

Many more use the Fibonacci studies in conjunction with other forms of technical analysis. For example, the Fibonacci studies are often used with Elliott Waves to predict the extent of the retracements after different waves. Hopefully, you can find your niche use for the Fibonacci studies and add it to your investment tools!

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