Fibonacci Numbers


In the 1202 AD, Leonardo Fibonacci wrote in his book “Liber Abaci” of a simple numerical sequence that is the foundation for an incredible mathematical relationship behind Phi.  This sequence was known as early as the 6th century AD by Indian mathematicians, but it was Fibonacci who introduced it to the west after his travels throughout the Mediterranean world and North Africa. He is also known as Leonardo Bonacci, as his name is derived in Italian from words meaning “son of (the) Bonacci”.

Starting with 0 and 1, each new number in the sequence is simply the sum of the two before it.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,  233, 377 . . .

Fibonacci sequence in liber abaci

This sequence is shown in the right margin of a page in Liber Abaci, where a copy of the book is held by the Biblioteca Nazionale di Firenze.

The relationship of the Fibonacci sequence to the golden ratio is this: The ratio of each successive pair of numbers in the sequence approximates Phi (1.618. . .) , as 5 divided by 3 is 1.666…, and 8 divided by 5 is 1.60. This relationship wasn’t discovered though until about 1600, when Johannes Kepler and others began to write of it.

The ratio of successive Fibonacci numbers converges on phi table shows how the ratios of the successive numbers in the Fibonacci sequence quickly converge on Phi.  After the 40th number in the sequence, the ratio is accurate to 15 decimal places. 1

1.618033988749895 . . .


Leonardo Pisano is better known by his nickname Fibonacci. He was the son of Guilielmo and a member of the Bonacci family. Fibonacci himself sometimes used the name Bigollo, which may mean good-for-nothing or a traveller.

Fibonacci was born in Italy but was educated in North Africa where his father, Guilielmo, held a diplomatic post. His father’s job was to represent the merchants of the Republic of Pisa who were trading in Bugia, later called Bougie and now called Bejaia. Bejaia is a Mediterranean port in northeastern Algeria. The town lies at the mouth of the Wadi Soummam near Mount Gouraya and Cape Carbon. Fibonacci was taught mathematics in Bugia and travelled widely with his father and recognised the enormous advantages of the mathematical systems used in the countries they visited. Fibonacci writes in his famous book Liber abaci (1202).

When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians’ nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it, for whatever was studied by the art in Egypt, Syria, Greece, Sicily and Provence, in all its various forms.


Fibonacci ended his travels around the year 1200 and at that time he returned to Pisa. There he wrote a number of important texts which played an important role in reviving ancient mathematical skills and he made significant contributions of his own. Fibonacci lived in the days before printing, so his books were hand written and the only way to have a copy of one of his books was to have another hand-written copy made. 2



In the book, Leonardo pondered the question: Given ideal conditions, how many pairs of rabbits could be produced from a single pair of rabbits in one year? The answer, it turns out, is 144 ¬— and the relationship used to obtain that answer is, you guessed it, the Fibonacci sequence. This thought experiment artificially dictates that the female rabbits always give birth to pairs consisting of one male and one female.

¬At the start, two newborn rabbits are placed in a fenced-in yard and left to breed. After the first month only the original pair remains since rabbits can’t reproduce until they a¬re at least one month old. By the end of the second month, the first pair give birth, now leaving two pairs of rabbits. In month three the original pair of rabbits produce another pair of newborns while their earlier offspring grow to adulthood. This leaves three pairs of rabbits, two of which will give birth to two more pairs the following month.

The total number of rabbits follows the Fibonnaci sequence. After 12 months there will be 144 pairs of rabbits. After two years, the number would jump to 46,368 pairs!


There is a special relationship between the Fibonacci numbers and the Golden Ratio, a ration that describes when a line is divided into two parts and the longer part (a) divided by the smaller part (b) is equal to the sum of (a) + (b) divided by (a), which both equal 1.618. This is represented by the Greek letter (φ). The ratio of any two successive Fibonacci Numbers approximates the Golden Ratio value ( φ = 1.6180339887…). The bigger the pair of Fibonacci numbers, the closer the approximation. From there, mathematicians can calculate what’s called the golden spiral, or a logarithmic spiral whose growth factor equals the golden ratio.

Using the values of the sequence as the edge length of squares arranged as below, a spiral is generated.

In Nature

There are many examples of Fibonacci numbers (numbers that appear in the sequence) appearing in the natural world. However, just because a series of numbers can be applied to an object, that doesn’t imply there’s a correlation between the math and reality.

Fibonacci numbers do appear in nature often enough to prove they reflect some naturally occurring patterns. You can commonly spot these by studying the manner in which various plants grow.

Many seed heads, pinecones, fruits and vegetables display spiral patterns that when counted express Fibonacci numbers. Look at spirals of seeds in the center of a sunflower and you’ll observe patterns curving left and right. If you count these spirals, your total will be a Fibonacci number. Divide the spirals into those pointed left and right and you’ll get two consecutive Fibonacci numbers. You can decipher spiral patterns in pinecones, pineapples and cauliflower that also reflect the Fibonacci sequence in this manner. 3

The Fibonacci sequence can be applied to finance by using four techniques including retracements, arcs, fans, and time zones.

Fibonacci retracements require two price points chosen on a chart, usually a swing high and a swing low. Once two points are chosen, the Fibonacci numbers and lines are drawn at percentages of that move. If a stock rises from $15 to $20, then the 23.6% level is $18.82, or $20 – ($5 x 0.236) = $18.82. The 50% level is $17.50, or $15 – ($5 x 0.5) = $17.50.

Image by Sabrina Jiang © Investopedia 2021

Fibonacci retracements are the most common form of technical analysis based on the Fibonacci sequence. During a trend, Fibonacci retracements can be used to determine how deep a pullback may be. Traders tend to watch the Fibonacci ratios between 23.6% and 78.6% during these times. If the price stalls near one of the Fibonacci levels and then start to move back in the trending direction, an investor may trade in the trending direction.

Arcs, fans, and time zones are similar concepts but are applied to charts in different ways. Each one shows potential areas of support or resistance, based on Fibonacci numbers applied to prior price moves. These supportive or resistance levels can be used to forecast where prices may fall or rise in the future. 4


“What Are The Real Life Applications Of Fibonacci Series?” 2022. Quora.

“What Are The Five Important Applications Of The Fibonacci Sequence?” 2022. Quora.

  1. The golden ratio is approximately 1.618. And the mile to kilometer ratio is 1.609 which is within 0.5% of the Golden ratio. So if we take 2 consecutive Fibonacci numbers, we get the approximate mile to kilometer conversion.
  2. Fibonacci numbers are example of complete sequence. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most. 6=1+5, 9=1+3+5, etc.
  3. The Fibonacci search technique. Fibonacci search technique
  4. The worst case input for Euclidean algorithm, which finds the greatest common factor of 2 numbers is a pair of consecutive Fibonacci numbers.
  5. Fibonacci numbers are used by some pseudorandom number generators.

“Does The Fibonacci Sequence Have Any Practical Uses?” 2022. Quora.

See Theoretical Knowledge Vs Practical Application.


Many of the References and Additional Reading websites and Videos will assist you with understating and applying Fibonacci numbers.

As some professors say: “It is intuitively obvious to even the most casual observer.


1 Meisner, Gary. 2012. “What Is The Fibonacci Sequence (Aka Fibonacci Series)? – The Golden Ratio: Phi, 1.618”. The Golden Ratio: Phi, 1.618.

2 “Fibonacci – Biography”. 2022. Maths History.

3 “The Fibonacci Sequence”. 2022.

4 Mitchell, Cory. “Fibonacci Sequence “. 2022. Investopedia.

Additional Reading

⭐ “Are Fibonacci Numbers And Golden Ratio Related To Nature And Arts? | Kinetrika Blog”. 2020. Kinetrika Blog.

“Fibonacci Number — From Wolfram Mathworld”. 2022.

“Fibonacci Number – Wikipedia”. 2022.

“Fibonacci Numbers – List, Meaning, Formula, Examples”. 2022. CUEMATH.

“Fibonacci Numbers and the Golden Section”. 2022.

“Fibonacci Sequence”. 2022.

“Fibonacci Sequence – Definition, List, Formulas And Examples”. 2022. BYJUS.

Flom, Peter. “Elementary Math, Beautiful Math: Introduction And A Property Of Fibonacci Numbers”. 2018. Medium.

Ghose, Tia. “What Is The Fibonacci Sequence?”. 2022.

Landau, Elizabeth. 2020. “The Fibonacci Sequence Is Everywhere—Even The Troubled Stock Market”. Smithsonian Magazine.

Sautoy, Marcus. 2022. “What Is The Fibonacci Sequence? | BBC Science Focus Magazine”.

Sheldon, Robert. “What Is The Fibonacci Sequence And How Does It Work?”. 2022.

“The Fibonacci Sequence – The Important Applications Of The Fibonacci Sequence –”. 2019.

“The Fibonacci Sequence and the Golden Ratio”. 2021.


Mathematics – Fibonacci Sequence and the Golden Ratio
Patterns – JW.ORG

The natural world is filled with intriguing patterns. Are they just random, or are they in fact evidence of careful design?

The magic of Fibonacci numbers | Arthur Benjamin
What is the Fibonacci Sequence and Why is it Important?

⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.

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