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Fibonacci theory

4-minute read

The principles of Fibonacci theory provide the basis for multiple different technical analysis tools, indicators, and strategies. In this lesson, we're going to run through Fibonacci ratios, retracements, and more.

A bit of history of Fibonacci

Before we get in too much about what Fibonacci is, let’s first answer the question “who is Fibonacci?” Leonardo Pisano, or Leonardo Fibonacci as he is most widely known, was a European mathematician in the Middle Ages who wrote Liber Abaci (Book of Calculation) in 1202 AD. In this book he discussed a variety of topics including how to convert currencies and measurements for commerce, calculations of profit and interest, and a number of mathematical and geometric equations. However, there are two things that jump to the forefront of our discussion in today’s world. First, in the beginning portions of Liber Abaci he discussed the benefits of using the Arabic numeral system. At the time, the influence of the defunct Roman Empire was still strong, and the preference of most European citizens was to use Roman numerals. However, in Liber Abaci, Fibonacci provided a very powerful, influential, and easy-to-understand argument for using the Arabic numeral system. From that point on, the Arabic numeral system got a strong foothold in the European community and soon became the dominant method of mathematics in the region and eventually throughout the world. It was so strong that we still use the Arabic numeral system to this day.

The second important section of Liber Abaci that we use today is the Fibonacci sequence.

What is the Fibonacci sequence?

The Fibonacci sequence is a series of whole numbers where each figure is the sum of the two before it. It starts with zero and one, which are known as the 'seed numbers'. The next number is (0 + 1) one, followed by (1 + 1) two and so on.

Here's what the beginning of the sequence looks like:

Fibonacci sequence

That might look a bit confusing at first, but looks much clearer when you include the math behind each figure:

Maths behind Fibonacci sequence

But the Fibonacci sequence on its own isn't hugely important to traders. Instead, it supplies the numbers for Fibonacci ratios.

What are Fibonacci ratios?

Fibonacci ratios are a series of percentages calculated by dividing figures along the Fibonacci sequence. There are quite a few different ratios, but the key ones are 23.6%, 38.2%, 61.8%, 78.6% and 161.8%.

To see how they work, let's take a closer look at the math behind the 61.8% ratio.

To find the 61.8% ratio, all you have to do is divide each number in the Fib sequence by the one that follows it. Do this along the chain, and you'll quickly spot that it comes out at roughly 0.618 each time – particularly from 21 ÷ 34 onwards.

0 ÷ 1 = 0
1 ÷ 1 = 1
1 ÷ 2 = 0.5
2 ÷ 3 = 0.67
3 ÷ 5 = 0.6
5 ÷ 8 = 0.625
8 ÷ 13 = 0.615
13 ÷ 21 = 0.619
21 ÷ 34 = 0.618
34 ÷ 55 = 0.618
55 ÷ 89 = 0.618

If we then convert 0.618 into a percentage, we get 61.8%.

To find 161.8%, meanwhile, you divide each number by the one that precedes it.

1 ÷ 0 = 0
1 ÷ 1 = 1
2 ÷ 1 = 2
3 ÷ 2 = 1.5
5 ÷ 3 = 1.67
8 ÷ 5 = 1.6
13 ÷ 8 = 1.625
21 ÷ 13 = 1.615
34 ÷ 21 = 1.619
55 ÷ 34 = 1.618
89 ÷ 55 = 1.618
144 ÷ 89 = 1.618

61.8% and 161.8% might be the most important Fibonacci ratios of them all. Also known as the golden ratios, they appear frequently across maths, geometry, architecture, art and more.

You can find other Fibonacci ratios with other dividing patterns. Here are a few common variations:

Divide by second following Divide by second preceding Divide by third following Divide by third preceding

0 ÷ 1 = 0

1 ÷ 0 = 0

0 ÷ 2 = 0

2 ÷ 0 = 0

1 ÷ 2 = 0.5

2 ÷ 1 = 2

1 ÷ 3 = 0.333

3 ÷ 1 = 3

1 ÷ 3 = 0.333

3 ÷ 1 = 3

1 ÷ 5 = 0.2

5 ÷ 1 = 5

2 ÷ 5 = 0.4

5 ÷ 2 = 2.5

2 ÷ 8 = 0.25

8 ÷ 2 = 4

3 ÷ 8 = 0.375

8 ÷ 3 = 2.666

3 ÷ 13 = 0.231

13 ÷ 3 = 4.333

5 ÷ 13 = 0.385

13 ÷ 5 = 2.6

5 ÷ 21 = 0.238

21 ÷ 5 = 4.2

8 ÷ 21 = 0.381

21 ÷ 8 = 2.652

8 ÷ 34 = 0.235

34 ÷ 8 = 4.25

13 ÷ 34 = 0.382

34 ÷ 13 = 2.615

13 ÷ 55 = 0.236

55 ÷ 13 = 4.231

21 ÷ 55 = 0.382

55 ÷ 21 = 2.619

21 ÷ 89 = 0.236

89 ÷ 21 = 4.231

34 ÷ 89 = 0.382

89 ÷ 34 = 2.618

34 ÷ 144 = 0.236

144 ÷ 34 = 4.235

55 ÷ 144 = 0.382

144 ÷ 55 = 2.618

55 ÷ 233 = 0.236

233 ÷ 55 = 4.236

89 ÷ 233 = 0.382

233 ÷ 89 = 2.618

89 ÷ 377 = 0.236

377 ÷ 89 = 4.236

However, there is one other way to get Fib ratios: by finding the square root of an existing one. The square root of 0.618, for example, is 0.786. Convert that into a percentage and you get 78.6% – one of our key ratios.

Here are some other common ratios calculated using square roots:

Fibonacci ratio Operation Result

0.382

Square root of 0.382

0.618

0.618

Square root of 0.618

0.786

1.618

Square root of 1.618

1.272

2.618

Square root of 2.618

1.618

Gann ratio

50%

There is another ratio that is commonly used in Fibonacci analysis, but isn’t technically a Fibonacci ratio: 50%. It doesn’t appear in the sequence, but like the key ratios, it arises often in the markets.

Some argue that the 50% ratio is a ‘Gann ratio’, created by W.D Gann in the early 1900s. Whatever the source, the 50% ratio seems to be a rather important and relevant level when trading, so it is often included in technical analysis as if it were a Fibonacci ratio.

Just like the Fibonacci ratios, many people will either take the inverse or square root of the “sacred ratios” to form more values. Some examples can be found in the table below.

SACRED RATIO OPERATION RESULT INVERSE OF SACRED RATIO

1

Square root of 1 1 1
2 Square root of 2 1.414 0.5
3 Square root of 3 1.732 0.333
4 Square root of 4 2 2.236
5 Square root of 5 0.25 0.2

Whatever the source, the 50% ratio seems to be a rather important and relevant level when trading, so often times it is included in Fibonacci analysis as if it were a Fibonacci ratio. Some of the other numbers included in the table have been mistaken as Fibonacci ratios as well, but obviously are not.

Fibonacci retracements

But how can you use Fibonacci theory in your trading? The most common way is through Fibonacci retracements, which traders use to predict support and resistance levels when a market retraces after a significant move.

Say, for instance, that Brent crude tumbles 150 points as part of a bear trend. You expect a countertrend to form as buyers briefly arrest crude’s fall. According to Fibonacci theory, that countertrend may find support or resistance at a Fibonacci ratio of the initial move: often 23.6%, 38.2%, 61.8% or 78.6%.

Example of Fibonacci theory

You can add these ratios to any FOREX.com trading chart using the Fibonacci retracement drawing tool. This automatically adds lines at key Fibonacci ratios (and 50%) on your chart, so you can plot where a reversal may arise in an upcoming countertrend and project potential support and resistance levels in advance.

Fibonacci retracement factsheet

Type: Drawing tool
Used in: Retracements
Used for: Finding support and resistance levels
Markets: Any
Timeframes: Any

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