### Who was Fibonacci?

During the Middle Ages, Europe emerged from the Dark Ages as a land that thirsted for knowledge. Goods, new ideas, and mathematics flowed from areas not ravaged by the Dark Ages, such as Arabia and India. Born into this flow of ideas, Leonardo Fibonacci da Pisa was born sometime between 1170 and 1180. He was the son of a Pisan merchant who also served as a customs officer in North Africa. Leonardo traveled widely in Algeria, and was later sent on business trips to Egypt, Syria, Greece, Sicily, and Provence.

In 1200, he returned to work in Pisa and used the knowledge he gained on his travels to write Liber Abaci (literally “book of calculations”) in which he introduced the Latin-speaking world to the decimal number system. The first chapter of part one starts, “These are the nine figures of the Indians: 9, 8, 7, 6, 5, 4, 3, 2, 1. With these 9 numbers and the sign “0”, which in Arabic is called zephirum, or sifr (zero), any number can be written. Fibonacci went on to change the world of mathematics in Italy and the World from the Roman numeral system, to a pure numeric system.

### The Fibonacci Sequence

If this feat was not noteworthy enough, perhaps the wisest human being in the Renaissance Period became famous for a simple series of numbers. The Fibonacci sequence as it was later named goes like this: 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, and on to infinity. After zero and one, the sequence is derived by adding the previous two numbers to get the next number. Fibonacci discovered the sequence of numbers in a national mathematics tournament held in Pisa, ordered by Emperor Frederick II.

The problem presented was this: beginning with a single pair of rabbits, if every month each productive pair bears a new pair, which becomes productive when they are one month old, how many rabbits would there be after n months. The Fibonacci sequence was the answer. From the sequence however, the phenomenon of the discovery was just the beginning. If you take a number in the sequence and divide it by the next number, the farther out (larger) in the sequence you go, the closer the division equates to the number .618. 3/5=.60, 5/8=.625, 8/13=.6154, 13/21=.619, 21/34=.6175, 34/55=.61818, and so forth.

It works in similar fashion the other way around; that is, dividing a number by the one smaller yields an answer closer to 1.618 as follows, 5/3=1.666, 8/5=1.6, 13/8= 1.625, 21/13=1.6154, 34/21=1.619, 55/34=1.676, and so on. From this Fibonacci defined the golden section, or phi (φ), as the limit of b/a, so 0 = 1/0.

### Golden Section

(a+b)/a+ a/b= φ

### The Golden Ratio

Phi is the only number that when added to 1 yields its inverse: .618 + 1 = 1 ÷ .618. This alliance of the additive and the multiplicative produces the following sequence of equations:

- .6182 = 1 - .618,
- .6183 = .618 - .6182,
- .6184 = .6182 - .6183,
- .6185 = .6183 - .6184, etc.

or alternatively,

- 1.6182 = 1 + 1.618,
- 1.6183 = 1.618 + 1.6182,
- 1.6184 = 1.6182 + 1.6183,
- 1.6185 = 1.6183 + 1.6184, etc.

Some statements of the interrelated properties of these four main ratios can be listed as follows:

1) 1.618 - .618 = 1,

2) 1.618 x .618 = 1,

3) 1 - .618 = .382,

4) .618 x .618 = .382, 5) 2.618 - 1.618 = 1,

6) 2.618 x .382 = 1,

7) 2.618 x .618 = 1.618,

8) 1.618 x 1.618 = 2.618.

Besides 1 and 2, any Fibonacci number multiplied by four, when added to a selected Fibonacci number, gives another Fibonacci number, so that:

- 3 x 4 = 12; + 1 = 13,
- 5 x 4 = 20; + 1 = 21,
- 8 x 4 = 32; + 2 = 34,
- 13 x 4 = 52; + 3 = 55

### Fibonacci and the Stock Market

Since Fibonacci mathematical properties show up in virtually all of nature we can assume that since the stock market unfolds in Fibonacci math, then the stock market acts like all of nature as well. And it does. The Fibonacci sequence governs the numbers of waves that form in the movement of both stock and commodity prices.

Consider again the Fibonacci Sequence: o,1,1,2,3,5,8,13,21,34,55,89….infinity.

Consider then the chart of simple Elliott Waves below:

Wave 1 exists when we add 0 to 1. 1 added to 1 gives us wave 2. Wave 2 plus the preceding number in the sequence (1) gives us wave 3. Wave 3 adding the 2 that preceded it gives is 5 waves (and a completed impulse wave). Wave 5 plus the 3 preceding it gives us 8 waves (a completed impulse and a 3 wave corrective wave).

Taking wave 8 and adding the 5 preceding it equals total 13 waves (an impulse, corrective, and another impulse). Now adding 13 to the Fibonacci number preceding it (8) gives us 21 total waves. This also is 3 impulse waves and two intervening corrective waves—a completed Elliot wave sequence.

If we took this out as far as we wanted, we would have the same thing,—the added Fibonacci sequence would continue to unfold in Elliott waves at many degrees of trend. This implies that the markets follow a natural pattern, like the rest of nature, which flies in the face of “cause and effect” that so many use to explain the value of stocks and/or commodities.

In fact it means that markets follow the same path of the rest of creation and for good reason. It’s part of our DNA when you think about it. Why wouldn’t our very actions ebb and flow like the rest of creation? And our actions, that’s mans’ collective actions of buying and selling assets in a public market in which every single transaction is recorded and can be plotted on a chart, shows that each wave up and down follows the same math that is inherent in sunflowers, pine cones, oceanic waves, our galaxy the milky way, and even fractal nature of our brains.

The prices of stock market indices continue in these “wave patterns” as discovered by R.N. Elliott in geometric fractals and unfold in exact Fibonacci numbers, even the number of waves themselves. The markets wax and wane to Fibonacci retracement areas as well. And these levels are also derived from the Fibonacci sequence. In the prior discussions you likely learned the phenomenon of Fibonacci proportions like .382 or 38.2% and .50 or 50% and .618 or 61.8%. These proportionate levels see many markets pull back and “correct” a previous impulse wave move before reversing again.

For example, in the following chart of the NASDAQ 100 beginning in 1997:

NASDAQ 100 WEEKLY

The NASDAQ 100 “crashed” beginning in 2000 and fell from 4,821.36 to 797.73. Notice that in late October 2007, as the Dow and S&P were making all-time highs, the NASDAQ 100 made it to almost its 38.2% Fibonacci retracement level before reversing. It then returned to dance around the 38.2% retracement area for most of 2011, and finally made it to the Fibonacci 50% (by dividing the Fibonacci number 2 by Fibonacci number 1).

Fibonacci retracements as well as projections are useful in stock market forecasting. In fact it’s much more common than most people realize. Virtually every technical analysis software product as well as every trade platform has conveniently inncluded Fibonacci retracement, projection, and even time calculators.

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