 ### Fibonacci Retracements

The use of Fibonacci levels in trading is based on the principle that the ratios of the Fibonacci sequence tend to coincide with key support and resistance zones, often signaling key pivot areas of price movement. ### FIBONACCI EXTENSIONS

While Fibonacci retracements examine price action following a breakdown from the pivot cycle highs, Fibonacci extensions establish target levels following a breakout from pivot cycle highs. ### FIBONACCI TIME ZONES & RATIOS

A similar tool, Fibonacci Time Ratios, applies the same technique across the x-axis to measure activity across time, but instead of producing bands related to a fixed period, a Fibonacci Time Ratio produces vertical bands corresponding to Fibonacci Ratios. ### FIBONACCI FANS & ARCS

While Fibonacci retracements and extensions measure changes in price (y-axis, horizontal bands), and Fibonacci time zones and ratios measure changes in time (x-axis, vertical bands), Fibonacci fans measure changes in both time and price.

### What is Fibonacci?

Fibonacci refers to the sequence of numbers made famous by thirteenth-century mathematician Leonardo Pisano, who presented and explained the solution to an algebraic math problem in his book Liber Abaci (1228). The Fibonacci sequence and the ratios of its sequential numbers have been discovered to be pervasive throughout nature, art, music, biology, and other disciplines, and they form the foundation for Fibonacci trading tools. Traders apply these Fibonacci levels to help interpret market behavior and to isolate higher probability setups and market pivots.

The sequence begins with 0 and 1 and is comprised of subsequent numbers in which the nth number is the sum of the two previous numbers. The equation for finding a Fibonacci number can be written like this:

Fn = F(n-1) + F(n-2). The starting points are F1 = 1 and F2 = 1.

Each number in the Fibonacci sequence is identified with a subscript 1, 2, 3, 4 …… to indicate which term of the sequence we are talking about. Thus F16 refers to the sixteenth Fibonacci number.

### What is the Golden Ratio?

Related to the Fibonacci sequence is another famous mathematical term: the Golden Ratio. When a number in the Fibonacci series is divided by the number preceding it, the quotients themselves become a series that follows a fascinating pattern: 1/1 = 1,   2/1 = 2,   3/2 = 1.5,   5/3 = 1.666…,   8/5 = 1.6,   13/8 = 1.625,   21/13 = 1.61538,   34/21 = 1.619,   55/34 = 1.6176…, and   89/55 = 1.618… The first ten ratios approach the numerical value 1.618034… which is called the “Golden Ratio” or the “Golden Number,” represented by the Greek letter Phi (Φ, φ). After these first ten ratios, the quotients draw ever closer to Phi and appear to converge upon it, but never quite reach it because it is an irrational number. Phi (Φ), 1.61803 39887…, is also the number derived when you divide a line in mean and extreme ratio, then divide the whole line by the largest mean section; its inverse is phi (φ), 0.61803 39887…, obtained when dividing the extreme (smaller) portion of a line by the (larger) mean. In the image below, the ratio of the smaller part of a line (CB), to the larger part (AC) – i.e. CB/AC – is the same as the ratio of the larger part, AC, to the whole line AB. Therefore, CB/AC = AC/AB. Phi and phi are also known as the Golden Number and the Golden Section. The formula for Golden Ratio is: F(n) = (x^n – (1-x)^n)/(x – (1-x)) where x = (1+sqrt 5)/2 ~ 1.618 The Golden Ratio represents a fundamental mathematical structure which appears prevalent – some say ubiquitous – throughout nature, and is used as the basis for Fibonacci tools in trading.