“Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.”
— Leonhard Euler
Euler, who ???
Euler was a Swiss mathematician, physicist, astronomer, geographer, logician, and engineer who made important and influential discoveries in many branches of mathematics.
But what makes him great in the first place?
He authored over 800 papers, his collected works fill 92 volumes, more than anyone else in the field and that too after going blind.
Much of the notation used by mathematicians today — including e, i, f(x), ∑, and the use of a, b and c as constants and x, y and z as unknowns — was either created, popularized or standardized by Euler. His efforts to standardize these and other symbols (including π and the trigonometric functions) helped to internationalize mathematics and to encourage collaboration on problems.
“Approximately one-third of the entire corpus of research on mathematics and mathematical physics and engineering mechanics published in the last three-quarters of the eighteenth century is by him.”
- Clifford Truesdell
Let’s look at some of the famous work contributed by Euler
Euler’s identity is amazing because it is simple to look at and yet incredibly profound. It has all five of the most important constant of mathematics 0, 1, e, i and π combined with three basic operations addition, multiplication, and exponentiation.
This is what famous 20th-century physicist Richard Feynman said about it.
“Our jewel … one of the most remarkable, almost astounding, formulas in all of mathematics.”
– Richard Feynman
The Basel Problem
Take infinite natural numbers 1, 2, 3, 4, 5, 6, ……
Square them 1,4, 3, 16, 25, 36, ..
Now, find the sum of their reciprocals ?? Is it even possible to find some of such infinite series?
If you look closely you will notice that as the series go on, the total sum is increased by a very small proportion because the denominator is increasing exponentially(infinite at some point).
This is the result calculated by Euler. Again in the form of π.
Seven Bridges of Königsberg
Königsberg (Kaliningrad) is divided by a river (Pregel), which contained two islands with seven bridges linking the various landmasses. The puzzle was to find a walk through the city that crossed every bridge exactly once.
Euler observation: If a path through this network is going to cross every link exactly once, then each node within the path must have an even number of links attached to it.
That’s because whenever you enter the node by one link, you need to leave it by another, so the node needs two links if you visit it once, four if you visit it twice, and so on. The only nodes that can have an odd number of links attached to them are the nodes where the walk starts and ends (if they are distinct).
In the third representation of bridges in the graphical form, it is visible that there are no such nodes. So, it is not possible.
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n.
So what it has to do with Euler?
By 1772, Euler had proved that 2³¹ − 1 = 2,147,483,647 is a Mersenne prime. It remained the largest known prime until 1867.
The number 2,147,483,647 (or hexadecimal 7FFF,FFFF16) is the maximum positive value for a 32-bit signed binary integer in computing. It is, therefore, the maximum value for variables declared as integers (e.g., as
int) in many programming languages, and the maximum possible score, money, etc. for many video games.
In December 2014, PSY’s music video “Gangnam Style” had exceeded the 32-bit integer limit for YouTube view count, necessitating YouTube to upgrade the counter to a 64-bit integer.
Euler’s Polyhedron Formula
Euler gave the relationship between the number of faces(F), edges (E), and vertices (V)of the Polyhedron.
V — E + F = 2
Euler’s laws of motion
In classical mechanics, Euler’s laws of motion are equations of motion which extend Newton’s laws of motion for point particle to rigid body motion.
They were formulated by Leonhard Euler about 50 years after Isaac Newton formulated his laws.