Fibonacci Numbers
Fibonacci numbers are special kinds of numbers that form a special sequence. This sequence is one of the famous formulas in mathematics. You can find Fibonacci numbers in plant and animal structures. These numbers are also called nature's universal rule, and nature's secret code.
In this article, let's learn about Fibonacci numbers, their sequence with rules, and solved examples.
What are Fibonacci Numbers?
Fibonacci numbers are a sequence of whole numbers arranged as 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,... Here are some interesting facts about the Fibonacci numbers:
 This sequence is called the Fibonacci sequence and it's an infinite sequence.
 Each number in the Fibonacci series or sequence is represented as \(F_n\).
As shown below, Fibonacci numbers can be represented as a spiral, if we make squares with those widths. In the given figure, we can see how the squares fit neatly together. For instance, 5 and 8 add up to 13, 8 and 13 add up to 21, and it goes on.
Fibonacci Formula
Fibonacci numbers follow a specific pattern. To find the Fibonacci numbers in the sequence, we can apply the Fibonacci formula. The relationship between the successive number and the two preceding numbers can be used in the formula to calculate any particular Fibonacci number in the series, given its position.
Formula to Find Fibonacci Numbers
The formula to calculate the (n + 1)^{th} number in the sequence of Fibonacci numbers can be given as,
\(F_n = F_{n1} + F_{n2}\)
where,
n > 1
F\(_{n1}\) n^{th }Fibonacci number
F\(_{n2}\) (n  1)^{th} Fibonacci number
Rules for Fibonacci Numbers
The rules for the Fibonacci numbers are given as :
 The first number in the list of Fibonacci numbers is expressed as \(F_0\) = 0 and the second number in the list of Fibonacci numbers is expressed as \(F_1\) = 1
 Fibonacci numbers follow a rule according to which, \(F_n = F_{n1} + F_{n2}\), where n > 1
 The third fibonacci number is given as \(F_2 = F_{1} + F_{0}\). As we know, \(F_0\) = 0 and \(F_1\) = 1, the value of \(F_2\) = 0 + 1 = 1
 The sequence of fibonacci numbers goes like 0, 1, 1, 2 and so on.
The rule for Fibonacci numbers, if explained in simple terms, says that every number in the sequence is the sum of two numbers preceding it in the sequence.
How Can We Calculate Fibonacci Numbers?
Let's calculate the Fibonacci numbers using the rule from the above section. The sequence is given as 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. Let's see how the first ten terms come about in the sequence. If we tabulate the calculation, we get:
n  Term  \(F_{n1}\)  \(F_{n2}\)  \(F_n = F_{n1} + F_{n2}\)_{,} ( for n >1) 

0  First      \(F_0\) = 0 
1  Second  \(F_{0}\) = 0    \(F_1\)= 1 
2  Third  \(F_1\) = 1  \(F_0\) = 0  \(F_2\) = 0 + 1 = 1 
3  Fourth  \(F_2\) = 1  \(F_1\) = 1 
\(F_3\) = 1 + 1 = 2 
4  Fifth  \(F_3\) = 2  \(F_2\) = 1  \(F_4\) = 2 + 1 = 3 
5  Sixth  \(F_4\) = 3  \(F_3\) = 2  \(F_5\) = 3 + 2 = 5 
6  Seventh  \(F_5\) = 5  \(F_4\) = 3  \(F_6\) = 5 + 3 = 8 
7  Eighth  \(F_6\) = 8  \(F_5\) = 5  \(F_7\) = 8 + 5 = 13 
8  Ninth  \(F_7\) = 13  \(F_6\) = 8  \(F_8\) = 13 + 8 = 21 
9  Tenth  \(F_8\) = 21  \(F_7\) = 13  \(F_9\) = 21 + 13 = 34 
From the above table, we can infer that:
 In the sequence formed by the Fibonacci numbers, the first term is always 0, and the second term is always 1.
 The output obtained in the 4^{th} column is the summation of the values in the 2^{nd} column and 3^{rd} column which represent the two preceding numbers.
As per some older definitions, the value \(F_{0}\) = 0 is omitted, so that the list of Fibonacci numbers starts with \(F_{1}\) = \(F_{2}\) = 1. \(F_{n}\) = \(F_{n1}\) + \(F_{n2}\) is valid for n>2 . But as per the original definition, the Fibonacci numbers start with \(F_{1}\) = 1, and \(F_{2}\) = 2.
Fibonacci Numbers List
Using the Fibonacci numbers formula and the method to find the successive terms in the sequence formed by Fibonacci numbers, explained in the previous section, we can form the Fibonacci numbers list as shown below,
\(F_0\) = 0  \(F_{10}\) = 55 
\(F_1\) = 1  \(F_{11}\) = 89 
\(F_2\) = 1  \(F_{12}\) = 144 
\(F_3\) = 2  \(F_{13}\) = 233 
\(F_4\) = 3  \(F_{14}\) = 377 
\(F_5\) = 5  \(F_{15}\) = 610 
\(F_6\) = 8  \(F_{16}\) = 987 
\(F_7\) = 13  \(F_{17}\) = 1597 
\(F_8\) = 21  \(F_{18}\) = 2584 
\(F_9\) = 34  \(F_{19}\) = 4181 
Properties of Fibonacci Numbers
Fibonacci numbers are used in many computer algorithms such as Fibonacci cubes, Fibonacci heap data structure, and the Fibonacci search technique. Let's see the different properties of the Fibonacci numbers based on the number's position above and below zero.
The first 10 Fibonacci numbers in the sequence can be shown as:
\(F_0\)  \(F_1\)  \(F_2\)  \(F_3\)  \(F_4\)  \(F_5\)  \(F_6\)  \(F_7\)  \(F_8\)  \(F_9\) 
0  1  1  2  3  5  8  13  21  34 
 The sequence of Fibonacci numbers can be extended to negative index n also by rearranging the recurrence relation \(F_{n2}\) = \(F_{n}\)  \(F_{n1}\)
 This yields the sequence of negafibonacci numbers which has the relation \(F_{n}\) =( 1)^{n+1} × \(F_{n}\)
Thus, for fibonacci numbers, the bidirectional sequence looks like:
\(F_{5}\)  \(F_{4}\)  \(F_{3}\)  \(F_{2}\)  \(F_{1}\)  \(F_{0}\)  \(F_{1}\)  \(F_{2}\)  \(F_{3}\)  \(F_{4}\)  \(F_{5}\) 
5  3  2  1  1  0  1  1  2  3  5 
We can observe from the above table that the Fibonacci numbers below zero are the same as the Fibonacci numbers above zero, with the only difference that they follow the +  + pattern. It is interesting to note that Fibonacci numbers are used in planning poker games.
Relation of Fibonacci Numbers to Golden Ratio
When any two consecutive Fibonacci numbers are taken, their ratio is very close to 1.618034. Let's take a random example of two consecutive numbers:
 Let A = 13, B = 21 and, let's divide B by A. We get 21 ÷ 13 = 1.625.
 This ratio of successive Fibonacci numbers is known as the Golden Ratio.
We can calculate any Fibonacci number using this Golden Ratio as per this formula: \(F_{n}\) = ((ɸ)^{n}−(1−ɸ)^{n}) ÷ √5. Here, ɸ = 1.618034.
Let's calculate \(F_{6}\) = ((1.618034)^{6}−(1− 1.618034 )^{6}) ÷ √5. When this calculation is done using a calculator, we get the value of \(F_{6}\) as 8.00000033, which when rounded off to the nearest whole number becomes 8.
Fibonacci Numbers in Nature
We can find Fibonacci numbers everywhere in nature. Here are some of the most common patterns and sequences of Fibonacci numbers in nature:
 The petals in certain plants such as sunflowers, lilies, roses, and buttercups follow the Fibonacci pattern, and these flowers are called Fibonacci flowers.
 The spirals that are found on the pinecone are equal to Fibonacci numbers.
 Even the seeds of sunflowers are said to follow a Fibonacci pattern.
 The seashells, starfish that we find on the seashores, follow the pattern of Fibonacci numbers.
One of the practical applications of the concept of Fibonacci numbers is that it was applied in the construction of the Great Pyramid at Giza.
Related Articles:
Check out the following pages related to the Fibonacci numbers
 Fibonacci Numbers Calculator
 Arithmetic Progression
 Sequence Calculator
 Arithmetic Sequence Calculator
Important Notes on Fibonacci Numbers:
Here is a list of a few points that should be remembered while studying the Fibonacci numbers
 The concept of Fibonacci numbers is only applicable for whole numbers and decimal numbers from a financial perspective.
 The sequence of Fibonacci numbers also applies to numbers below zero.
 The first Fibonacci number is always 0 and the second Fibonacci number is always 1.
Examples on Fibonacci Numbers

Example 1: Find the sum of the first ten Fibonacci numbers.
Solution:
The list of Fibonacci numbers is given as 0, 1, 1, 2, 3, 5, 8, 13, 21, 34
On summation of numbers in the sequence, we get
Sum = 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 = 88.
Thus, the sum of the first ten Fibonacci numbers is 88.

Example 2: Calculate the value of the 12^{th} and the 13^{th} Fibonacci numbers. The 9th and 10th terms in the sequence are 21 and 34.
Solution:
We can calculate the 12^{th} and the 13^{th} Fibonacci numbers in the following way:
11th term can be obtained by adding the 9^{th} and 10^{th} term as 21 + 34 = 55
12th term can be calculated as 34 + 55 = 89
13th term can be calculated as 55 + 89 = 144
Thus, the 12^{th} and the 13^{th} Fibonacci numbers are 89 and 144 respectively.
FAQs on Fibonacci Numbers
Why are the Fibonacci Numbers So Important?
The Fibonacci numbers have a lot of practical applications in computer technology, music, financial markets, and many other areas. Fibonacci numbers exist in nature in various forms and patterns.
Is 0 a Fibonacci Number?
Yes, 0 is a Fibonacci number, and it's the first Fibonacci number. It is denoted by F\(_0\).
Are the Fibonacci Numbers Infinite?
Yes, the Fibonacci list consists of infinite Fibonacci numbers where every number is calculated by simply adding the two numbers that are before it. Each number in the sequence of Fibonacci numbers is represented as F\(_n\).
Is There a Formula for Finding Fibonacci Numbers?
Yes, there is a formula for finding Fibonacci numbers. Fibonacci numbers follow this formula according to which, F\(_n\) = F\(_{(n1)}\) + F\(_{(n2)}\), where F\(_n\) is the (n + 1)^{th} term and n > 1. The first Fibonacci number is expressed as F\(_0\) = 0 and the second Fibonacci number is expressed as F\(_1\) = 1.
How to Calculate the Fibonacci Numbers?
Fibonacci series depending upon their position in the series can be calculated using the general formula for Fibonacci numbers given as, F\(_n\) = F\(_{(n1)}\) + F\(_{(n2)}\), where F\(_n\) is the (n + 1)^{th} term and n > 1.
What is the Formula to Find Fibonacci Numbers?
The formula to find the (n + 1)^{th} term in the sequence formed by Fibonacci numbers can be given as, F\(_n\) = F\(_{(n1)}\) + F\(_{(n2)}\), where n > 1.
What are the Applications of Fibonacci Numbers?
Fibonacci numbers have various applications in the field of mathematical and financial analysis. We use Fibonacci numbers in the computational runtime analysis of Euclid's algorithm to find HCF. Also, many patterns in the nature can be studied using the Fibonacci numbers.
What are the First 10 Fibonacci Numbers?
The First 10 Fibonacci numbers are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. Here, we can see that the first Fibonacci number is always 0 and the second Fibonacci number is always 1.
What are Fibonacci Numbers in Nature?
We can find Fibonacci numbers everywhere in nature. Here are some of the most common patterns and sequences of Fibonacci numbers in nature:
 The petals in certain plants such as sunflowers, lilies, roses, and buttercups follow the Fibonacci pattern, and these flowers are called Fibonacci flowers.
 The spirals that are found on the pinecone are equal to Fibonacci numbers.
 Even the seeds of sunflowers are said to follow a Fibonacci pattern.
 The seashells, starfish that we find on the seashores, follow the pattern of Fibonacci numbers.
What are the First 20 Fibonacci Numbers?
The First 10 Fibonacci numbers are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181.
Is 33 a Fibonacci Number?
No, 33 is not a Fibonacci number as it is not present among the first 10 Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.
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