(This article is written by Dr Inderjeet Tinani, my better half)

Fibonacci numbers are a sequence of numbers generated from two seed numbers 0 and 1.

The subsequent sequence of numbers is generated by adding the preceding two numbers to obtain the next number: for example, the 3rd number in this sequence would be

0 + 1 = 1, followed by

1 + 1 = 2

1 + 2 = 3

2 + 3 = 5

3 + 5 = 8

5 + 8 = 13

8 + 13 = 21

13 + 21 = 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, …, and so on and so forth.

What is so amazing about the Fibonacci sequence is not the numbers themselves but the mathematical relationships between these numbers when expressed in terms of ratios – for example, the ratio between any number in this sequence and its immediate predecessor always turns out to be what is called the Golden Ratio, a special number in mathematics, approximately equal to 1.618 (except for the few initial smaller terms, where it is slightly off the mark). And that between a number and its immediate successor is always 0.618 (inverse of the Golden Ratio). For instance:

2584/1597 = 1.618, 1597/2584 = 0.618

1597/987 = 1.618, 987/1587 = 0.618

987/610 = 1.618, 610/987 = 0.618

610/377 = 1.618, 377/610 = 0.618

377/233 = 1.618, 233/377 = 0.618

and so on and so forth. The ratios 1.618 and 0.618 correspond to percentage levels of 161.8% and 61.8%, respectively, translating to a corresponding increase of 61.8% and a decrease of 38.2%. Thus, each successive term in the sequence is 1.618 times, or 61.8% greater than, its immediate predecessor.

And each term is 0.618 times or 61.8% of its immediate successor, in other words 38.2% smaller than it.

Another fascinating thing about the Fibonacci sequence is that such unique mathematical relationships between the numbers in this sequence are not restricted just to adjacent terms (nearest neighbours) but even to distant neighbours.

For example, the ratio between alternate terms (2nd nearest neighbours) in the sequence is always 2.618 (or its reverse 0.382):

2584/987 = 2.618, 987/2584 = 0.382

1597/610 = 2.618, 610/1597 = 0.382

987/377 = 2.618, 377/987 = 0.382

610/233 = 2.618, 233/610 = 0.382

377/144 = 2.618, 144/377 = 0.382

and so on and so forth.

The ratios 2.618 and 0.382 correspond to percentage levels of 261.8% and 38.2%, respectively, translating to a corresponding increase of 161.8% and a decrease of 61.8%. Thus, each successive term in the sequence is 2.618 times, or 161.8% greater than, its 2nd predecessor. And each term is 0.382 times or 38.2% of its 2nd successor, in other words 61.8% smaller than it.

Extending the same logic to 3rd nearest neighbours yields ratios of 4.236 and 0.236, corresponding to percentage levels of 423.6% and 23.6%, or 323.6% increase (note that this double of 161.8%!) and 76.4% (note that this is double of 38.2%!) decrease, respectively.

**How Are Fibonacci Numbers Used in Technical Analysis**

**For calculating retracements and bounces**

In stock market technical analysis the most commonly used Fibonacci percentage levels are: 23.6%, 38.2% and 68.2%.

The reference points plotted on charts are 50% and 100%.

So let us take the example of Nifty Cash price today:

It opened around 11003 and hit a high of 11076, a rise of 73 points. Then it started reacting.

Now, a Fibonacci chartist will make the following assumptions:

The first fall will stop at 23.6% of rise (73 points) = 17 points

If the fall continues, the can be expected to stop at 38.2% of rise (73 points) = 28 points

If it continues falling, they will expect the fall to be 61.8% (of 73 points) = 45 points.

These chartisist will then draw the points to predict levels, and use these levels to snake in and out of their positions.

Likewise, if there is a fall, and if the market turns, the chartists will assume that the resistance points will be 23.6%, 38.2% and 61.8% of the fall.

This is by and large how the ratio concept works.

As I’m not a Fibonacci follower, this is the best I can explain.

**For Using Fibonacci Numbers in Moving Averages and Other indicators**

Traders use the Fibonacci numbers 3, 5, 8, 13, 21, 34, 55, 89, 144 and 233 to plot moving averages and crossovers.

For example, some day traders buy the stock when 3 MA crosses 5 MA or when 5 MA crosses 8 MA, or when price crosses above either MAs, and vice versa for shorts.

BTST traders typically use higher averages such as 34/55 or 34/89, etc.

I’m sure you get the picture.

**BACK TO MATHEMATICS**

**This is a technical discussion and only for those who are interested to know where the GOLDEN RATIO originates from:**

Mathematically speaking, Golden Ratio is defined as follows:

Two quantities are said to be in Golden Proportion if the ratio between them (larger to the smaller) is the same as the ratio between their sum and the larger one. Consider for example, a line divided into two parts a and b (a > b), as shown below:

So according to the definition of the Golden Ratio,

(a + b) / a = a / b = 𝝋

where the small Greek letter 𝝋 symbolizes the Golden Ratio.

This equation can be rewritten as

1 + (b / a) = a / b = 𝝋

or

1 + (1 / 𝝋) = 𝝋 [which also implies, (1 / 𝝋) = 𝝋 – 1)]

This relationship yields a simple school-level quadratic equation

𝝋2– 𝝋 – 1 = 0

which even a school student studying algebra can show has two solutions

𝝋 = (1 +5) / 2 = 1.618 and 𝝋 = (1 – 5) / 2 = – 0.618

corresponding to 𝝋 = 1.618 (Golden Ratio) and 1 / 𝝋 = 1.618 – 1 = 0.618, which is the same as the magnitude of the 2nd solution, i.e., without the minus sign). The inverse of the Golden Ratio (1 / 𝝋) is also called the Golden Ratio Conjugate, and is often represented by capital phi 𝜱. Thus, 𝜱 = 𝝋 – 1).

The above mathematical discussion shows where from does this amazing ratio originate. Have you by an chance noticed that5 equals 2.236, a number that also appeared while discussing raios between 3rd nearest neighbours.

**But why use at all Fibonacci ratios in technical analysis of financial markets?**

The Golden Ratio, also known by other names like the Golden Mean, the Divine Proportion, and represented by the Greek letter phi (𝝋).

It seems to have a ubiquitous presence in all natural patterns occurring in the Universe – from the patterns seen in incomprehensibly small entities (like the fundamental building blocks, the atoms) to the patterns found in unfathomably gigantic celestial bodies; from the arrangement of petals in a flower to the arrangement of seeds in a flower head; from the arrangement of seed pods in a pine cone to the arrangement of leaves on a stem; in spirals of galaxies, snail shells and hurricanes; in most beautiful faces; in eye-appealing man-made architectural designs; in family trees of honey bees; in DNA molecules; in body proportions of humans as well as animals; in fact, everywhere in Nature.

It is as if Nature finds the most perfect equilibrium in this fundamental ratio.

The question that tickles the mind of stock market analysts, therefore, is whether the financial markets, which to a large extent are based on collective human behaviour, too somehow conform to this Golden Ratio. After all, collective human behaviour may also be considered as a natural phenomenon and should probably follow the same mathematical behaviour.

Why 68.2% sir, is it 61.8%??

Very worthwhile reading the write up. Thanks.

Regards.

Sir,

Convey our 🙏 🙏 to madam… to present the concept in such a simple way…. if any reader of fiction can find it in the novel DaVinci Code too. Mathematics is Nature. It’s fun.

Am a teacher. I love mathematics