Fibonacci Numbers

Although most writers keep it at the .618, a few round to .62. For the 1.272, though, almost all references just go with 1.27. Hence when I derived this 1.13 number (which, in keeping with how all these numbers cross reference, is also the square root of the 1.27), I decided to just round it to two decimal places and go with the 1.13. To the best of my knowledge, this is the first time this number has been quantified.

So why was this so fascinating? Well, many, many times I saw the ‘double tops’ and ‘double bottoms’ slightly exceeded, and then tum and rocket off in the opposite direction. This was the so-called ‘shake out’, false breakout, gunning for the stops or whatever one wants to call it. It’s also seen in many popular false breakout strategies. But now I am quantifying it. I found a great deal of these moves that just exceeded the double tops and double bottoms were stopping very near the 1.13.

This allowed me to see where a shake out may go. Furthermore, it allowed me to make a judgment as to when I felt that a breakout was a real breakout, or was just a head fake. If I was finding the .886 was the most significant number I was working with, I would have expected the 1.13 to be very significant, too. In practice, I found that the assumption was valid. See figures 6.11 and 6.12.

So at this point we have added the following numbers to our basic numbers:

.236, .486, .886, l.13 and 2.058.

There is one more number I like to use that I haven’t seen in any trading references, although I suspect it has been used here and there. I noticed a fair amount of issues just shy of the 2.000/2.058 area. Again, does this happen for ‘harmonic’ reasons, or because many traders are watching the 2.000 area and are jumping in early, or none of the above? I don’t care. All I care about is trying to determine areas where I tend to see things happen that I feel might help my trading, things that might give me an edge. The reason behind it is of no consequence to my immediate trading. I only harp on this because I want to point out my emphasis is on the chart, not on trying to discover the fundamentals behind the chart. If you feel those fundamental reasons are important for you to know, by all means pursue that. I just prefer to trade off the chart for the most part.

I started playing around with different geometric figures that had Fibonacci lengths on them, to see if anything interesting popped out. The most obvious layout to me was a ‘golden’ rectangle, a rectangle whose longer side was 1.618 times the length of its shorter side. As you may recall from our earlier discussion, this is the rectangle that was universally found in studies to be the most pleasing to the greatest percentage of people. So I constructed and labelled a rectangle with the sides of 1.000 and 1.618 and I drew in a diagonal. See figure 6.13.

I immediately wanted to know what the diagonal length was, so I did some quick high school math and found it was 1.902. Now here was a number that was just shy of the 2.000/2.058 area, and I stumbled upon it quite quickly. I started looking at charts and found this did look like an area I wanted to watch. See figures 6.14 and 6.15.

In doing the high school math to calculate the diagonal, I noticed a few numbers I’d seen before. Here’s the Pythagorean theorem applied to the triangle in figure 6.13 (sorry to all those out there that hated high school math).

 Not only did I get the 1.902 I was looking for, I noticed 2.618 and 3.618.

The 2.618 is a fairly commonly used external retracement. As can be seen from the above derivation, it's just Phi (1.618) squared. I also noticed in my own studies that there may be an area around three and a half or so, and that might be it. These are pretty extreme external retracements, but you do see some extreme moves that come up occasionally. See figures 6.16 and 6.17.

Let’s add in a few more extreme external retracements while we are in this area, and we’ll sum up what we have and call it quits. I think we have plenty to work with. Notice, up until now, whenever I decided to add a number into the group I also liked to add its reciprocal. This only makes sense: if a number is valid, the number ‘on the other side of 1.000’ should be, too.

We haven’t determined what the reciprocal of the .382 is yet, so let’s do that one:

As you guessed, this isn’t a coincidence. We’ve seen that the numbers are frequently derived in multiple ways. So at this point I’m sure I want 2.618 in the group. I also haven’t flipped over the .236:

Here’s a number you will see in many references, usually as the 4.24. It is also generally the most extreme external retracement I will look at. See figure 6.18.

Now we have all the Fibonacci numbers we need for our group, and all the numbers are matched with their reciprocals. I’m sure you are glad to be done with all that, but I’ll have to disappoint you temporarily, as I want to add three more numbers into the group. First, I want to add in a non-Fibonacci number, but one we know is related to Fibonacci, as we discussed earlier. I want to add in pi (π): 3.14. I’ll just use two decimal places, since that’s how it is referred to most often. I first learned about this one from Scott Carmey. He’s used it quite a bit in his book, and especially on his website with some of his new patterns. See figures 6.19 and 6.20.

Second (do you believe me this time?), I want to add in 2.236, frequently rounded to 2.24. This number is the square root of 5. Recall how, way back, we started out with Phi, the golden ratio, and the formula was the square root of 5 plus 1, all divided by 2. I’ll leave it as an exercise for the reader to see if he or she can show whether the 2.236 is a ‘true’ Fibonacci number, or not. Lastly, let’s flip this one over to get .447, and stop there. If I don’t stop now,

I may just keep adding them on and lose the rest of my readers. See figures 6.21 and 6.22.

Let’s sum up our rather extensive list of numbers:

0.236, .382, .447, .486, .500, .618, .786, .886, 1.000, 1.13, 1.272,

1.618, 1.902, 2.000, 2.058, 2.236, 2.618, 3.14, 3.618, and 4.236.

I would like to add one final comment before we proceed to the next chapter. The use of the square root of2 (which equals approximately 1.414), and its reciprocal (approximately .707), have become increasingly popular to use in the same manner as the Fibonaccis. I do see a lot of moves that turn right on these numbers. Now, again, I don’t know if this is due to the increasing popularity of these numbers, or the fact that they lie right in between the .618/.786 and 1.272/1.618 areas, and are logical spots for overshoots. Or is it for another reason? I can’t say. I only know they are numbers that seem to be points at which a move may tum.

For now, I won’t add these in, or we will have so many numbers they will begin to lose any sense. I merely wanted to point them out so that the reader will be aware of them, and can do further research if so inclined. I will say that I have found two Fibonacci derived numbers that are very close to these two ‘square root of two’ numbers. I may present the numbers in future material, but the dilemma is certainly arising about how many numbers are practical to work with.