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Trading using The Elliott Wave III: Applying Fibonacci ratios and the square root of two to Elliott waves




Traders use the Elliott Wave mostly as a continuously developing price map, on which they try to guess the most probable future path. Sometimes the trader waits for some unfinished wave to end, to pull the trigger or take profits. When this occurs, he or she commonly uses Fibonacci ratios in trading to forecast a price level for that event.

Ian Copsey on his book The Case for Modification of R.N. Elliott’s Impulsive Wave Structure, says he has found, that harmonic ratios derived from the square root of two are also a very helpful tool.

My personal belief is that those ratios are really artefacts, a product of the random nature of the trading activity. In the age of computers and Big Data, a statistical study on the retracements ratios might reveal much more precise information about those proportions. Even better, a computer study might be developed to show the most likely retracement levels as a function of the latest N-retracements, taking account of the recent volatility changes.

Nonetheless, Fibonacci ratios and Sqrt-2 ratios may serve as an approximation to forecast retracements or extensions when a better information tool is unavailable.


Leonardo Pisano Bigollo (1170-1250), known as Leonardo Fibonacci, was an Italian mathematician and traveller, who studied and brought the Indo-Arabic numerical system to Europe. This revolutionary numerical system on which the absolute value of a digit is established by its position within the number made possible the mathematical and scientific revolution in Europe.

In his Liber Abaci book of 1202, Fibonacci introduced the arithmetic systems he learned from the merchants working on the Mediterranean coast that he called modus Indorum (The way of the Indians). The book made a case for a 0-9 digits and place value, as well as examples of how to use it in business to calculate interest rates, money-changing, and other applications.

The Fibonacci sequence

The book also discusses irrational numbers,  prime numbers, and the Fibonacci series, as a solution of the problem of the growth of a population of rabbits.

The Fibonacci sequence starts with two ones: 1,1. The following numbers in the series are calculated as the sum of the preceding two numbers. He carried the calculation up to 377, but he didn’t discuss the golden ratio as the limit ratio of consecutive numbers in the sequence.

Below, Table 1 shows in yellow the first 27 Fibonacci numbers. The other columns, from 1 to 6 shows the results of the n-following divisions, as a percentage. That is the result of dividing the Fibonacci number by next one, two apart, three apart etc.

Fibonacci ratios in trading

The last column shows the stabilized Fibonacci ratios generated:

elliott wave fibonacci ratios


Table 2 shows the Fibonacci ratios of the n-preceding division as a percentage.


As with the preceding table, the last column shows the stabilized Fibonacci ratios generated:

Fibonacci ratios generated

Two more sets of fibonacci multiplying ratios are obtained by multiplication and division of the N-following ratios:




As we can observe, except for ratios smaller than 5% and the 100% ratio, all of them are already present in the original series.

The Square root of two

Well, the square root of two is the first known irrational number, and the one that raised heated passions in ancient Greece, that ended with a crime. At a date around 520 BC, a man called Hippasus of Metapontum was dropped from a boat into open waters to die.  The man’s crime was revealing to the world a “dirty” mathematical secret. The secret of the relation between the sides of the square triangle of length 1, and its hypotenuse.

According to the well known Pythagoras theorem, the sum of the squares of the sides a rectangle is equal to the hypotenuse squared.

Therefore, for unity sides: 12+12 = 2, therefore the hypotenuse length is the square root of 2.

Pythagoras Theorem - Forex Academy

The square root of two including its four decimal places is 1.4242

This article is not focused on the proof that the square root of two is not rational, so I’d recommend the curious reader the following page:


Ian Copsey explains that he was told about this ratio applied to the markets by some acquaintance, who stated that it was commonly occurring between musical notes.

After studying it, Mr Copsey began to find out that two derivations of this ratio usually happened: 41.4% and it complementary 58.6%, being 100-41.4%.

Usual wave relationships

Ian Copsey says in his book that, after many research hours into normal relationships between waves, he has found the most usual to be:

Fibonacci: 5.6%, 9%, 14.6%, 23.6%, 33.3%, 38.2%, 50%, 61.8%, 66.6%, 76.4%, 85.4%, 91%, and 94.4%

To this list, you can add those derived from the square root of 2: 41.4% and 58.6%.

And, specifically on Wave (iii), it’s possible to take those ratios and add 100%, 200% and on occasions 300% and 400%.

The most common extensions he has found were: 138.2%, 176.4%, 223.6%, 261.8%, 276.4%, and 285.4%. Additionally, but less frequently, he found 158.6%, 166.7%, 238.2%, and 361.8% and occasionally 423.6% and also 461.8%

It’s important to observe the underlying ratios of a particular market trend. It’s better to stick with the ratios that often show in the most recent retracements of the same kind.

As is usual, the help of visual channels, spotting important supports and resistances or pivot points may show which one of those ratios best fit the rest of the clues.

It is also noteworthy that the projections of the Wave (v) and also of the Wave (c) should match the end of higher order waves as well, so the most probable final ratio is the result of that confluence.

The data below was taken from Mr Copsey’s study, published in the referred book.

Wave (i)

There is no relationship to any previous wave as this is the start of a five wave sequence.

Wave (ii)

This is a corrective wave of Wave (i). This retracement is one of the most difficult to assess. According to Ian Copsey, it can go from 14.5% up to 100%. He also mentions that on a 5 min chart it’s complicated to observe the sub-waves composing Wave(ii), however on a daily chart it shows the typical A-B-C pattern or, even, more complex patterns.

Wave (iii)

Wave (iii) is an extension of Wave (i), projected from the end of wave (ii).


  • The most typical forecast are 176.4%, 185.4%, 190.02%, 223.6%, 276.4%, and 285.4% projections of Wave (i).
  • Less recurring projections are: 138.2%, 166.7%%, and 261.8%.
  • Sporadically it goes to: 123.6%, 238.2%, 361.8%, 423.6%, and 476.4%.

Wave (iv)

Wave (iv) is, of course, a retracement starting from the top of Wave (iii). At this stage, noting the implications of the alternation rule with Wave (ii) or Wave (b) there is a stronger basis to identify the end of the pullback.

Potential retracement percentages:

  • For small retracements: 14.6%, 23.6%, 33.3%, and 38.2%.
  • For mid retracements: 41.4% and 50%.
  • For intense retracements: 58.6% and less often 61.8% or 66.7%.

Wave (v)

Wave (v) is an extension of the total price move from the beginning of Wave (i) to the end of Wave (iii), projected from the end of Wave (iv).

Having identified Wave (iv) makes it much easier to build up a projection for Wave (v).


  • The bulk of projections go to 61.8%, 66.7%, and 76.4%.
  • In a truncated Wave (v), usual ratios are 58.6% and 50%.
  • In an extended Wave (v), the most usual projections are: 85.6%, 100%, 114.4%, and sometimes 123.6% and 138.2%.

Wave (A)

Wave (A) is similar to Wave (I) in its unpredictability. There is no reference to spot its end because there is no relation to other prior waves. The best method is to find a higher order price channel in which this wave might end, observe support/resistance levels or pivot points.

Another method is to go to a shorter time frame, watch the 5-wave pattern that constitutes the A wave and try to project Wave (v) by matching it with a previous Wave (B) of Wave (v) or the prior Wave (iv).

Wave (B)

Wave (B) is a retracement of Wave (A), but it’s a correction within a correction so that it can be really complicated and random. The retracement ratios range from 15% to 100%. The use of pivot, swing high and low, and support/resistance levels give more clues than simple mathematical ratios.

As stated in other articles, it doesn’t pay to trade Wave (B) or any other 3-wave corrective pattern for that matter, because of it’s poor reward-to-risk ratio.

Wave (C)

Wave (C) is an extension of Wave (A) projected from the end of Wave (B).


  • The most usual projections are: 100%, 105.6%, 109.2%, 114.4%, 138.2%, and 161.8%.
  • Less common are: 76.4%, 85.6%, 123.6%, and 176.4%.
  • Sporadic projections are: 123.6%, 223.6%, and 261.8%, but sometimes, as short as 61.8%.

It is important to note that Wave (C) is related to the next higher and lower degrees. Thus, its sub-wave (v) should, also, match Wave (A) extension and, if it’s part of a higher degree’s Wave (iii) or Wave (v), their possible projections.

Wave (x)

Wave (x) usually retrace similarly to Wave (b).


Wave a: retraces deeply. In a Wave (iv) this exceeds 50% of Wave (iii)

Wave b: commonly retraces beyond 76% of Wave a

Wave c: projects  66.7% to 76.4% of Wave a, from the end of Wave b

Wave d: 66.7% to 76.4% of Wave b from the end of Wave c

Wave e: a zigzag less than 66% of Wave d

Expanded Flat Corrections

Wave a: 50% of the preceding wave

Wave b: 15% to 38%, occasionally as low as 9% and rarely up to 41%

Wave c: Back to the end of Wave or beyond.

Final guidance

Those values are only a guide. Every market has its characteristics; therefore you should know them to trade efficiently. Additionally, every timeframe behaves differently.

In intra-day trading, you should add pivot points to this analysis as pivots are used heavily by professional traders.

Visual clues offer better information than numerical values. If the projection or retracement touches a trendline drawn on price channel or support/resistance area, then the chance of that projected value increases substantially.



The Case for Modification of R.N. Elliott’s Impulsive Wave Structure, Ian Copsey


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