Mathematical Contributions of Leonhard Euler

Everyone knows that e is called Euler number in honor of Leonhard Euler. But the contribution of this great guy is far beyond that.

Euler Photo

Before we head to his contributions though, let me introduce Euler. He was born on April 15, 1707 in Basel, Switzerland. He got married twice. This great mathematician got blind due to cataract. At the age of 76, following the announcement of the discovery of planet Uranus and its orbit, he died because of Brain Hemorrhage.


* Discovered the base of natural logarithm e, an irrational constant equal to 2.7182818…

* Discovered the gamma constant also called the Euler-Mascheroni  Constant that is approximately equal to 0.57721. It is still not known if this constant is rational or irrational, algebraic or transcendental.

* First to introduced the concept of functions and used f(x) to denote a function.

* First used the symbol   \sum,  for summation, i for imaginary and  \pi  for the ratio of circumference to its diameter.

* Advances infinitesimal calculus, he is fond of power series. One of this is power series is his own number.

e^x=\displaystyle\sum_{n=0}^{\infty} \displaystyle\frac{x^n}{n!}=\lim_{n\to \infty}(\displaystyle\frac{1}{0!}+ \displaystyle\frac{x}{1!}+ \displaystyle\frac{x^2}{2!}+\ldots +\displaystyle\frac{x^n}{n!})

* Derived his famous equation e^{i\varphi}=\cos\varphi+i\sin\varphi  in which if  \varphi=\pi   this equation reduces to   e^{\pi i}+1=0.   This equation is believed to have proved the existence of God. It is called the Euler’s Identity.

* Discovered the connection between the Riemann Zeta Function and prime numbers.

* He proved the newton’s identity and Fermat’s little theorem.

* He also invented the Totient function  \varphi n  which is defined as the number of positive integers less than or equal to the integer   n  that are coprime to n.

* Conjectured the law of quadratic reciprocity, the theorem in modular arithmetic that gives conditions on the solvability of quadratic equations modulo prime numbers.

* Discovered the formula relating the number of vertices, edge, and faces of a Convex Polyhedron. That is   V-E+F=2.

These are just some highlights of his contributions in Mathematics. He also contributed a lot in Astronomy, Mechanics, Philosophy, etc.

You are welcome to add more of his achievements by leaving a comment below.

If you are a Mathematician reading this post, you might already realize how wide his contributions in Mathematics that made our lives easier. Totally opposite if you will share this with your friends thinking that they should have not failed their Mathematics if Euler didn’t invent these stuff. 😀


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