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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.

But before we get on to that, we need to cover 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 maths 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 maths 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 2nd following	Divide by 2nd preceding	Divide by 3rd following	Divide by 3rd 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

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.

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 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.