Let’s start with a refresher on Fibonacci numbers. If we start at 0 and then go to the next whole integer number, which is 1, and add 0 to 1, that gives us the second 1. If we then take that number 1 and add it again to the previous number, which is of course 1, we have 1 plus 1 equals 2. If we add 2 to its previous number of 1, then 1 plus 2 gives us 3, and so on. 2 plus 3 gives us 5, and we can do this all the way to infinity. This series of numbers, and the way we arrive at these numbers, is called the Fibonacci sequence. We refer to a series of numbers derived this way as Fibonacci numbers.



We can go back to the beginning and divide one number by its adjacent number – so 1÷1 is 1.0, 1÷2 is .5, 2÷3 is .667, and so on. If we keep doing that all the way to infinity, that ratio approaches the number .618. This is called the Golden Ratio, represented by the Greek letter phi (pronounced “fie”). It is an irrational number, which means that it cannot be represented by a fraction of whole integers. The inverse of .618 is 1.618. So, in other words, if we carry the series forward and take the inverse of each of these numbers, that ratio also approaches 1.618. The Golden Ratio, .618, is the only number that will also be equal to its inverse when added to 1. So, in other words, 1 plus .618 is 1.618, and the inverse of .618 is also 1.618.

…Now, bear with me as I lay down some more necessary groundwork before we start looking at price charts in Chapter 3. I want to come back to this diagram of Fibonacci ratios again to point out another interesting Fibonacci relationship. Look at the series at the top. Instead of dividing adjacent numbers (1 by 2, 2 by 3, and 3 by 5 to infinity), if we divide alternate numbers (1÷3, 2÷5, and 5÷13 to infinity) we will get the ratio of .382. 1 minus .618 is also .382. The inverse of .382 is 2.618. If we look at the second alternate – meaning, if we divide 1 by 5 and 2 by 8, etc. – we will get .236 if we carry that to infinity. The inverse of .236 is 4.236.

These are all different Fibonacci ratios. I also want to point out two important numbers. We are going to use the number .5 – or 50% — a lot, since we will see it demonstrated in patterns in financial markets. Remember .5. It is not the Golden Ratio, but it is related to Fibonacci numbers. Another number we want to keep in mind is .786, which represents the square root of .618. These are all numbers that we will use to analyze wave patterns in various markets.