How Fibonacci Numbers Work in Math and Everyday Life
Leonardo Pisano, also called Fibonacci (Fibonacci stands for filius Bonacii) was born in Pisa during 1170. His father, Guglielmo dei Bonacci, a rich Pisan merchant and a representative of the merchants of the Republic of Pisa present in the area of Bugia in Cabilia (in modern north-eastern Algeria), after 1192, he took his son with him, because he wanted Leonardo to become a merchant like him.
So the solution to this problem is the famous “Fibonacci sequence” which is: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89… a sequence of numbers where each of them is the sum of the previous two numbers.
What is Fibonacci Number?
Let’s know what a fibonacci number is. A Fibonacci number is known to be a series of numbers where each of the Fibonacci numbers is found by adding the two preceding numbers. It also means that the next number in the series is the addition of the two previous numbers. Let us take the first two numbers in the series as 0 and 1. So by adding 0 and 1, we will get the third number as 1, and by adding the second and the third number which is 1 and 1, we get the fourth number to be 2, and likely, the process goes on and on.
So, we get the Fibonacci series as 0, 1, 1, 2, 3, 5, 8, ……. Therefore, the obtained series is called to be the Fibonacci number series. Now we know what the fibonacci number is.
What is the Fibonacci Series?
We will discuss what is the Fibonacci series. The list of the numbers of Fibonacci Sequence is given below. This list is created by using the Fibonacci formula, which is also mentioned in the above definition.
The Fibonacci sequence is a set of the numbers that starts with a one or a zero, which are followed by a one, and then proceeds based on the rule that each of the numbers (called a Fibonacci number) equals to the sum of the preceding two numbers. If the Fibonacci sequence is put up as F (n), where n is the first term in the sequence, so the following equation obtains for n = 0, where the first two terms are put up as 0 and 1 by the convention:
Fibonacci Series Formula
F (0) equals 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ...
In some of the texts, it is mandatory to use n = 1. So, the first two terms are defined as 1 and 1 by default, and we see:
F (1) equals 1, 1, 2, 3, 5, 8, 13, 21, 34 …
Fibonacci Numbers, Fibonacci Formula
The sequence of the Fibonacci numbers can be written as:
Fn = Fn-1 + Fn-2
Where Fn is the nth term or the number
Fn-1 is the (n-1)th term
Fn-2 is the (n-2)th term
From the equation above, we can also write up the definition as the next number in the sequence, and is the sum of the previous two numbers which are present in the sequence, starting from 0 and 1. So, let's make a table to find the next term of the Fibonacci sequence, using the above Fibonacci formula.
Fibonacci Series List
In the above table, we can see that the numbers in each of the columns are relational, and also diagonally the numbers are the same in all three columns.
Fibonacci Number Properties
The following points below are the properties for the Fibonacci numbers:
In the Fibonacci series, let us take any of the three consecutive numbers and add those numbers. When we divide the result by 2, we will get the three numbers. Example, let’s take 3 consecutive numbers like 1, 2, 3. when we add these numbers that are 1+ 2+ 3 = 6. When 6 is divided by 2, the result will be 3.
Take four of the consecutive numbers other than “0” in the Fibonacci series. Then multiply the outer number and also then multiply the inner number. When we subtract these numbers, we will get the difference “1”. For example, if we take 4 consecutive numbers like 2, 3, 5, 8. Multiply the outer numbers (i.e) 2(8) and then multiply the inner number which is 3(5). Now we subtract these two numbers 16-15 =1. So, the difference is 1.
Applications for the Fibonacci numbers will also include the computer algorithms like the Fibonacci search technique and then the Fibonacci heap data structure, and the graphs called as Fibonacci cubes which are used for interconnecting the parallel and distributed systems.
Limitations of Using the Fibonacci Numbers and Levels
The usage of the Fibonacci studies is very subjective since the trader has to use the highs and the lows of their choice. Whichever highs and lows that are chosen will also affect the results the trader gets.
Another argument against the Fibonacci number trading methods is that there are also so many of these levels that the market has to bounce or to change the direction near one of them, making the indicator look significantly in hindsight. The problem is that it is very difficult to know which of the numbers or the level will be important in real-time or in the future.
FAQs on Fibonacci Numbers: Meaning, Series & Applications
1. What is the Fibonacci sequence and how does it start?
The Fibonacci sequence is a unique series of numbers where each subsequent number is the sum of the two preceding ones. The sequence most commonly starts with 0 and 1. For example, the beginning of the sequence is 0, 1, 1 (0+1), 2 (1+1), 3 (1+2), 5 (2+3), and so on, creating a progression based on addition.
2. What is the formula used to find the nth term of the Fibonacci sequence?
There are two primary ways to find a term in the Fibonacci sequence:
- Recursive Formula: The most common representation is F(n) = F(n-1) + F(n-2), where you must know the two previous terms to find the current one.
- Binet's Formula: This is an explicit formula that calculates the nth term directly: F(n) = [φ^n - (1-φ)^n] / √5, where φ (phi) is the Golden Ratio, approximately 1.618. This is useful for finding large Fibonacci numbers without calculating all the steps before them.
3. Can you list the first 20 Fibonacci numbers?
Yes, starting from 0, the first 20 numbers in the Fibonacci sequence are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, and 4181. Each number from the third one onwards is the sum of the previous two.
4. What makes the Fibonacci sequence so special in mathematics?
The Fibonacci sequence is special due to its surprising and frequent appearance in various fields and its connection to the Golden Ratio (φ). As you go further into the sequence, the ratio of two consecutive Fibonacci numbers (e.g., 89/55 or 144/89) gets progressively closer to approximately 1.618. This unique mathematical property links it to concepts of natural growth, geometry, and aesthetics.
5. How can you test if any given number is part of the Fibonacci sequence?
A simple and definitive test exists to check if a number 'N' is a Fibonacci number. A positive integer 'N' belongs to the Fibonacci sequence if and only if either (5 * N² + 4) or (5 * N² - 4) is a perfect square. For example, to test if 8 is a Fibonacci number, we calculate (5 * 8² + 4) = 324, which is the perfect square of 18. Therefore, 8 is a Fibonacci number.
6. What are some common examples of Fibonacci numbers found in nature?
The Fibonacci sequence is famously observed in natural patterns and growth. Key examples include:
- The number of petals on flowers like lilies (3 petals), buttercups (5 petals), and daisies (often 34 or 55 petals).
- The spiral arrangements of seeds in a sunflower head or the scales on a pinecone.
- The branching of trees and the arrangement of leaves on a stem (phyllotaxis).
- The shape of a nautilus shell, which grows in a logarithmic spiral closely related to the Golden Ratio.
7. How does the Fibonacci sequence differ from an arithmetic or geometric sequence?
The main difference lies in how each sequence progresses:
- An Arithmetic Sequence grows by adding a constant difference (e.g., 3, 6, 9, 12...).
- A Geometric Sequence grows by multiplying by a constant ratio (e.g., 3, 9, 27, 81...).
- The Fibonacci Sequence grows by a rule of addition, but the number added is not constant; it is the preceding term in the sequence itself. This makes its growth pattern exponential but distinct from a simple geometric progression.
8. What are the important applications of Fibonacci numbers outside of nature?
Beyond its natural occurrences, the Fibonacci sequence has several important practical applications in various fields:
- Computer Science: It is used to design algorithms, such as the Fibonacci search technique and data structures like Fibonacci heaps.
- Financial Markets: Traders use Fibonacci retracement levels (23.6%, 38.2%, 61.8%) to analyse stock charts and predict potential areas of price support and resistance.
- Art and Design: The related Golden Ratio is a fundamental principle in art, architecture, and design, used to create compositions that are considered aesthetically pleasing.
