Fibonacci Numbers
Fibonacci numbers are a sequence of numbers where every number is the sum of the preceding two numbers. It starts from 0 and 1 as the first two numbers. This sequence is one of the famous sequences in mathematics. You can find Fibonacci numbers in plant and animal structures. These numbers are also called nature's universal rule or nature's secret code.
Fibonacci numbers were first discovered by an Italian mathematician called Leonardo Fibonacci in the 13^{th} century. The sequence begins with 0 and 1, and each subsequent number is the sum of the two preceding numbers. So the first few numbers in the sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on. Let us learn more about Fib numbers.
What are Fibonacci Numbers?
Fibonacci numbers are a sequence of whole numbers arranged as 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,... , where every number is the sum of the preceding two numbers. 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, Fib numbers can be represented as a spiral, if we make squares with those lengths. 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. To know how this Fibonacci spiral is formed, click here.
Fibonacci Meaning
In mathematics, Fibonacci is a concept that can be represented as numbers, sequences, or series such that each term is the sum of the two terms preceding it and the first two terms are 0 and 1. A Fibonacci number is generally denoted by F_{n}, where n is a whole number. The Fibonacci sequence is given by, F_{0} = 0, F_{1} = 1, F_{2} = 1, F_{3} = 2, F_{4} = 3, F_{5} = 5, and so 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 Fib 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 sequence 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 to Find 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, and so on. 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 5^{th} column is the summation of the values in the 3^{rd} column and 4^{th} column which represent the two preceding numbers.
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.
Fibonacci Numbers List  

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 sequence 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 this:
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.
n^{th} Fibonacci Number and the 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 = 1.615.
 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.
 Even the seeds of sunflowers are said to follow a Fibonacci pattern.
 The spirals that are found on the pinecone are equal to Fibonacci numbers.
 The seashells, and 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.
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 to 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.
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Fibonacci Numbers Examples

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.
Answer: ∴ 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 9^{th} and 10^{th} 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
Answer: ∴ 12^{th} and the 13^{th} Fibonacci numbers are 89 and 144 respectively.

Example 3: Given that F_{14} = 377 calculate the next Fibonacci number.
Solution:
We know that F_{n + 1} = F_{n} × Golden ratio
Here, golden ratio = 1.61803398875.
Thus, F_{15} = F_{14} × 1.615 = 377 × 1.61803398875 ≈ 610
Answer: F_{15} = 610
FAQs on Fibonacci Numbers
What are Fibonacci Numbers?
Fibonacci numbers form a sequence of numbers where every number is the sum of the preceding two numbers. It starts from 0 and 1 as the first two numbers.
Why are the Fibonacci Sequence 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.
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 0 a Fibonacci Number?
Yes, 0 is a Fibonacci number, and it's the first Fibonacci number. It is denoted by F_{1}.
What are the Applications of Fibonacci Sequence 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 nature can be studied using the Fibonacci numbers.
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 sequence numbers 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 Difference Between Fibonacci Numbers and Lucas Numbers?
The formation of both Fibonacci and Lucas numbers is the same (where every number is obtained by additing the previous two numbers). The only difference is, the Fibonacci numbers start from 0 and 1 whereas Lucas numbers start from 2 and 1. i.e.,
 Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, …
 Lucas numbers are 2, 1, 3, 4, 7, 11, 18, …
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 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 is Fibonacci Numbers List?
The Fibonacci numbers list is infinite but some 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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