Phi number

Phi (Greek letter φ) – also known as the golden number or the golden ratio –  is an irrational number, approximately equal to 1.61803399, that can be used to predict market moves, as it is an indispensable element of such tools as Fibonacci retracement levels or Elliott wave theory. It's useful also for gold and silver investors.

Phi and Silver

The chart below (courtesy of presents an example of the use of Fibonacci retracement levels to find support and resistance levels on the silver market.

silver price and phi number

As you can see on the above chart, thanks to applying the Phi number through the Fibonacci retracements, we can identify the likely target prices during declines. The same can work for the gold market and also in case of retracements after downswings. In other words, the Phi number can help to determine the local tops and bottoms on the precious metals market as well as on other markets.

Phi and Gold

gold price and phi number

Similarly, in case of gold, we were able to use the Phi number through the Fibonacci ratios and create a reliable support for the gold price. As you can see on the above chart, gold stopped after moving (approximately) to the first Fibonacci (phi-based) retracement level.

Derivation of Phi

Phi can be obtained in various ways. It was first derived geometrically through the so-called golden ratio or golden section: a line segment is divided into two segments in such a way that the ratio of the longer one to the shorter one is equal to the ratio of the whole segment to the longer one obtained after the division. Below we can see this idea illustrated graphically:

golden section

The equality of ratios says that:

golden ratio

Another way of deriving phi is through solving a quadratic equation:

phi number quadratic equation

The solutions are:

phi number quadratic equation solutions

Yet another way of obtaining phi is through the Fibonacci sequence which is the following sequence of natural numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, …

In other words, each term (starting from the third) is a sum of the two preceding ones: 1+1=2, 1+2=3, 2+3=5, 3+5=8 and so on.

Phi is the limit of the ratio of an element of the sequence to its predecessor. This means that it approaches phi (approx. 1.618) as we proceed to infinity (the convergence begins to take shape after a couple of elements: 1/1=1, 2/1=2, 3/2=1.5, 5/3=1.667, 8/5=1.6, 13/8=1.625, 21/13=1,615, 34/21=1.619, 55/34=1.618, and so on).

Properties of Phi

Phi has myriad interesting mathematical properties. The most popular are presented below:

  1. Right from the equation phi number quadratic equation (whose solution – as has already been stated above – is phi) we get that phi^2 = phi + 1 (phi = 1.618, phi^2 =2.618)
  2. 1/phi = phi - 1
  3. phi number property: phi^(n+2) = phi^(n+1) + phi^n
  4. Phi and its inverse have exactly the same decimal part: φ=1.618033988749894848204586834365.... 1⁄φ=0.618033988749894848204586834365....
  5. Moreover, if n is odd then phi^n and 1/phi^n have exactly the same decimal parts. The table below provides an illustration of this fact for n from 1 to 7:
  • n
  • phi n
  • 1/ phi n
  • 1
  • phi 1 =   1.61803398875
  • 1/phi 1 = 0.61803398875
  • 2
  • phi 2 =   2.61803398875
  • 1/phi 2 = 0.38196601125
  • 3
  • phi 3 =   4.23606797750
  • 1/phi 3 = 0.23606797750
  • 4
  • phi 4 =   6.85410196625
  • 1/phi 4 = 0.14589803375
  • 5
  • phi 5 = 11.09016994375
  • 1/phi 5 = 0.09016994375
  • 6
  • phi 6 = 17.94427191000
  • 1/phi 6 = 0.05572809000
  • 7
  • phi 7 = 29.03444185375
  • 1/phi 7 = 0.03444185375

The Occurrence of Phi in the World

The golden ratio is present in so many areas that it is hard to list them all. Below there are just some examples.

Phi is omnipresent in the proportions of human body, for instance the distance from the head to the fingertips is approximately the golden section of the height of the body, and the distance from the top of the head to the chin is approximately the golden section of the distance from the head to the chest. Many scientists believe that it plays an important role in the human perception of beauty.

In nature it can be seen in the shape of some seashells and the arrangement of leafs of some plants.

It also appears in the human art – the most famous example is Leonardo da Vinci’s Mona Lisa whose face is a perfect golden rectangle, that is the ratio of her face’s height to its width is 1.618. But other renowned painters such as Salvador Dali (e.g. the dimensions of his painting – Sacrament of the Last Supper – are in a golden ratio to each other) also used the golden section to achieve geometrical beauty.

Golden section can be found in architecture as well. Examples include the proportions of such marvelous buildings as the Parthenon or the Chartres and the Notre Dame cathedrals. It continues to show up in the contemporary architecture, for instance in the proportions of the United Nations building or the CN Tower in Toronto.

Applications of Phi in the Financial Markets

Below there are two examples of the use of phi in the financial markets:

  1. Fibonacci retracement levels - the inverse of phi (approx. 0.618 or 61.8%) and its square (approx. 0.382 or 38.2%) are used as important retracement levels, i.e. levels that serve as support or resistance as the price retraces some of its previous movement in a trend. For instance, in an uptrend, after reaching a local top, price is likely to retrace between 38.2% and 61.8% of the distance from the last bottom to this top. The name corresponds to the way of obtaining phi by means of the Fibonacci sequence.
  2. Elliott wave theory - Phi is used in Elliot wave theory to set price targets for particular parts of a trend. For instance, it is assumed that an uptrend is formed of five “waves”: waves number 1, 3 and 5 represent moves up, while waves number 2 and 4 represent corrections in the main trend. To find the price target for the third wave one multiplies the length of the first wave by phi (approx. 1.618) and adds that to the bottom of wave 2. The price target for the fifth wave can be obtained by multiplying the length of the first wave by 2 times phi (approx. 2*1.618 = 3.236) and adding it to either the top or the bottom of the first wave to obtain maximum and minimum targets.


Phi and the idea behind it (the golden section) is an ubiquitous phenomenon in the world around us and in the society, even though we may not realize its existence. It is then no surprise that it also appears in the financial markets. It should be noted though, that in the technical analysis, various transformations are used much more frequently that the phi number (1.618) itself, for example its inverse (0.618) and its inverse square (0.382) to construct Fibonacci retracement levels.