Birthday Special: Here's How Leonhard Euler Changed The Course Of Mathematics.

I know, most of you would be under the impression of mathematics being as a boring and incompetent subject, Right? But the reality is not what it seems, as studying mathematics is a fascination in it's own. From number theory to applied mathematics, it never fails to lure the curiosity of a seeker. While we may be busy praising mathematicians such as Ramanujan, Carl Friedrich Gauss, Pythagoras etc, Leonhard Euler has acquired himself a place in the hearts of all mathematicians with his innumerable contributions to the field of Mathematics and changing its face as we all know today.

Euler

This day marks the 312th birth anniversary of Leonhard Euler. Born on 15th April 1707, this Swiss prodigy is known for his work in physics, astronomy, engineering, etc and above all, being an eminent mathematician in his century. To highlight upon his work, here are his five major contributions to Mathematics.

Mathematical Notations

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Euler introduced and popularized many of the mathematical terms that we frequently use today. So it would not be over-exaggerating to say that he contributed towards giving the language of mathematics. He gave us the notation f(x) to describe a mathematical function of x. He was the first one to use the letter e for the base of the natural logarithm. Later, this symbol e came to be known as Euler's number. He was also the first to use summation notation using the Greek letter ∑ and introduced the standard notation for trigonometric functions. The use of the Greek letter pi (π) to denote the ratio of a circle's circumference to its diameter was also popularized by him. Along with these contributions, he is also credited for inventing the notation i ( pronounced as iota ) to represent √-1.

The Number Theory

The number theory was one of Euler's most favorite parts to ponder upon. The interest can be traced back to the influence of his friend Christian Goldbach. His curiosity guided him to discover the proofs of Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two squares, and not to forgot about the discovery of a connection between the prime numbers and Reimann-Zeta function, better known as "Euler product formula for the Riemann zeta function". He invented the Totient Function, contributed to the theory of perfect numbers and even discovered the Euclid-Euler Theorem. His works were so influential that they acted as the stepping stones for the world-famous mathematician Carl Friedrich Gauss.

Complex Analysis

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Euler not only gave the notation i to represent the complex number √-1, but also gave one of the most remarkable formula in mathematics, the Euler Formula. According to this formula - " e^(iα) = Cos(α) + iSin(α) " Euler's formula basically represents a complex exponential function that satisfies any real number α. Euler's identity is a special case of Euler's formula and it states that - " e^(iπ) + 1 = 0 ". This identity is one of the most remarkable identity ever known in mathematics for involving e, π, i, 1, and 0, which are arguably the five most important constants in mathematics.


Euler's formula illustrated in the complex plane.

Applied Mathematics

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Some of Euler's greatest successes were achieved in the field of applied mathematics and applying analytic methods to real-world problems. His works briefly describe applications of Bernoulli's numbers, Fourier series, Venn diagrams, Euler numbers, and uses of constants e and π. Not only that, he even worked upon improving Calculus we know of today. This eventually leaded him to discover what is known as "Euler Approximations of Integrals".

Graph Theory

In 1735, Euler presented a solution to the problem known as the Seven Bridges of Königsberg. He also discovered the formula V - E + F = 2, where V is the number of vertices, E is the number of edges and F is the number of faces of a plane graph. The study and characterization of this formula is at the heart of topology.

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