Leonardo P. Fibonacci

 

Quick Info

Born: 1170, Pisa, Republic of Pisa.

Died:  1250 (aged 79-80), Pisa, Republic of Pisa.

Introduction

Leonardo Fibonacci, also known as Leonardo of Pisa, was an Italian mathematician born around 1170 and died 1250. He is best known for introducing the Hindu-Arabic numeral system to Europe through his book “Liber Abaci“ (The book of calculation), which was published in 1202. Here are some key points about Leonardo Fibonacci and his contributions:

Introduction of Hindu-Arabic Numerals :

Fibonacci most significant contribution was the introduction of the Hindu-Arabic numeral system with ten digits including a zero and positional notation to the Western world. Before this, Roman numerals were predominatly used in Europe. Fibonacci’s book “Liber Abaci“ explained the advantages of the decimal system with its place value notation, which greatly simplified arithmetic operations.

Fibonacci Sequence:

Fibonacci is also associated with the Fibonacci sequence is the sum of the previous two numbers. Fibonacci omitted the “0“ and the first “1“ include todayand began the sequence with 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,...

The Fibonacci numbers may be defined by the recurrence relation:

\begin{equation} F_{n}=F_{n-1}+F_{n-2} \end{equation}

\begin{equation} with\ F_{0}=0\ and\ F_{1}=F_{2}=1. \end{equation}

Fibonacci Numbers in Nature:

The Fibonacci sequence and the related golden ratio (the limit oft he ratios of consecutive Fibonacci numbers). This ratio is found in various aspects of art, architecture and nature, such as in the arrangement of leaves on a stem, the spirals of a pinecone, or the seeds in a sunflower.

The value of the golden ratio is

\begin{equation} \varphi =\frac{1+\sqrt{5}}{2}=1.61803398874... \end{equation}

Binet's Formula for the Fibonacci numbers

\begin{equation} F_{n}=\frac{\left ( \frac{1+\sqrt{5}}{2} \right )^{n}-\left ( \frac{1-\sqrt{5}}{2} \right )^{n}}{\sqrt{5}} \end{equation}

Generating Function

\begin{equation} \sum_{n=0}^{\infty }F_{n}x^{n}=\frac{x}{1-x-x^{2}} \end{equation}

Leonhard Euler

Quick Info

Born: 15 April 1707, Basel, Switzerland.

Died:  18 September 1783, St Petersburg, Russia.

Introduction

Leonhard Euler (1707-1783) was a Swiss mathematician and physicist who made significant contributions to various branches of mathematics and physics. Euler is widely regarder as one of the greatest mathematicians of all time.

Here are some key aspects of Euler’s work:

• Inventing functions for sine, cosine and tangent.

• Introducing and conceptualizing functions for mathematical and complex analysis.

• Discovering the relationships  between trigonometric functions and exponential functions.

• Creating an entirely new branch of mathematics, analytic Number Theory.

• Establishing and proving a product (known as Euler's product) that is the Riemann Zeta Function.

• First formulating the law of quadratic reciprocity in number theory.

• And more...

Constants and Theorems named after Euler

Constants

• Euler’s Number: 

\begin{equation}  e=2.718281828459... \end{equation}

• Euler-Mascheroni Constant:

\begin{equation}  \gamma =0.577215664901... \end{equation}

• Euler-Gompertz constant:

 \begin{equation} \delta =0.596347362323... \end{equation}

Theorems

• Euler’s theorem : If a and n are coprime positive integers, then

\begin{equation} a^{\phi \left ( n \right )}\equiv 1 \left ( modn \right ) \end{equation}

• Euler's partition theorem, used in combinatorics and numbers theory, states that the amount of partitions of an integer n into odd parts is equal to the numbers  of partitions of n  into distinct parts.

• The Euclid-Euler theorem, which connects Mersenne primes to perfect numbers, states that an even number is perfect if can be written as:

\begin{equation} 2^{p-1}\left ( 2^{p}-1 \right ) \end{equation}

Formulas and identities

• Euler's Formula:

\begin{equation} e^{i\theta }=cos\left ( \theta  \right )+isin\left ( \theta  \right ) \end{equation}

With a special case known as Euler's Identity:

\begin{equation} e^{i\pi  }+1=0 \end{equation}

• Euler’s reflection formula:

 \begin{equation} \Gamma \left ( z \right )\Gamma \left ( 1-z \right )=\frac{\pi }{sin\left ( \pi z \right )} \end{equation}

• Euler’s polyhedron formula:

\begin{equation} V-E+F=2 \end{equation}

• Euler's Totient functions Phi is a method for determining the amount of relatively prime numbers.