**Things to know about Fibonacci and his Numbers ***-(by request)*

Leonardo Pisano Bigollo *(known as Fibonacci, and also Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci)* —was an Italian mathematician, considered by some “the most talented Western* mathematician of the Middle Ages.”*

Fibonacci is best known for the spreading of the Hindu–Arabic numeral system which we use today in modern times - In his *Liber Abaci* (1202), Fibonacci introduced the *modus Indorum* (meaning method of the Indians), today known as Arabic numerals - which include the numbers 0 - 9 and was one of the earliest numerical systems to use zero as a place holder. The book also advocated place value in early hindu-arabic numerals.

^ modern Arabic numerals

**The Fibonacci sequence**

The Fibonacci numbers were introduced in his Liber Abaci which posed, and solved a problem involving the growth of a population of rabbits based on idealized assumptions.

The solution, generation by generation, was a sequence of numbers later known as Fibonacci numbers. The number sequence was known to Indian mathematicians as early as the 6th century, but it was Fibonacci’s Liber Abaci that introduced it to the West.

the Fibonacci sequence is widely known for its interesting properties. the one you may be most familiar with is that every term is the addition of the previous two terms:

for example, the Fibonacci sequence is represented here in this famous pattern.

Your sequence begins with a square with side length of 1. Imagine this is one rabbit - if you pair one rabbit with no rabbits you will have no offspring. we then add a partner rabbit, so you have 1 and 1 paired together.

the number of offspring they produce is the sum of the previous two generation’s population, in this case because we start with only 1 and 1 rabbits we get 2 in the next generation.

at this point your sequence looks like *1,1,2,*

your next population of offspring continues the same rule - the sum of the previous two populations of the rabbit generation. So in this case where X is our fourth population in the next generation (1,1,2,X). X is the sum of 1 and 2 - the previous two populations.

The Rule is Xn = Xn-1 + Xn-2

so we now have the sequence **1,1,2,3**

and the **1,1,2,3,5**

and the sequence can carry on to infinity:

*1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584…*

The characteristics of the Fibonacci sequence is commonly found in sunflower seeds and seashells as well as many other forms of nature, Art and Architecture.

**The Golden Ratio**

The Fibonacci numbers were first expressed in terms of the Golden ratio by *Daniel Bernoulli* in 1724.

The Golden ratio is one of the few Famous Mathematical constants along with e, √2, and π. It is an Irrational number.

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.

If we continue divide a term in the Fibonacci sequence by its preceding term we eventually approach the Golden ratio designated by the Greek numeral ϕ (phi, lowercase *φ*):

1/1= 1.000

2/1= 2.000

3/2= 1.500

5/3= 1.333

…..

55/34= 1.617

89/55= 1.618

etc.

the Golden ratio is approximated to the decimal **1.618033988**

with this we can show that each Fibonacci number can be written in terms of Phi.

^ The golden ratio fits coherently with the Fibonacci pattern (where the curve is the Golden ratio and the squares are the Fibonacci numbers.)

^ Fibonacci numbers can be found in many other mathematical discoveries, as it is the one of the most naturally occurring sequences in Mathematics. Fibonacci numbers can be found in the Pascal triangle when you add the numbers diagonally.

**Finding the Nth term in the Fibonacci sequence**

The Formula to find the Nth term in the Fibonacci sequence can be calculated with the Golden ratio:

sources - [1] [2]