Mathematical Complications: Leonhard Euler, Mathematicals, And Present Mathematical Notations

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Abstract: - Leonhard Euler is considered to be one of the greatest mathematicians of all time. He made incredible discoveries to many subject areas, but most were to the field of mathematics. In this paper we explore Euler’s mathematical contributions including formulas/identities, concepts and notations. We see how Euler’s formula came to be and the story behind it along with Euler’s identity. We then explore Euler’s approaches to the Basel and Seven Bridges of Königsberg problems and his solutions to these. Lastly, we see Euler’s influence in current mathematical notations.

Key-Words: - Euler, notations, formulas, Basel, Königsberg

1 Introduction
Leonhard Euler was born on April 15th, 1707 in Basel, Switzerland. His father, Paul Euler,
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2 Seven Bridges of Königsberg [9]

In a letter dated from 1736, Euler wrote to Carl Leonhard Gottlieb Ehler, mayor of Danzig regarding this problem Ehler had presented to Euler. In the letter, Euler stated that the solution to the problem was on reason alone and even though he found the problem to be quite trivial, he was still very much interested on it. That same year, Euler wrote a letter to Italian mathematician and engineer Giovanni Marinoni explaining that even though the problem was very ordinary, it deserved attention as it could not be solved by geometry, nor algebra nor even art. [9]
On August 26, 1735, a paper by Euler was presented where the solution to this problem was included. To find the answer, Euler started by exploring different approaches that could be taken in order to reach a solution. He mentioned that finding all possible paths could be done but would take a great amount of time to do so and plus, he wanted to find solution that would apply to all similar problems. Euler then went ahead a used a method where he used lowercase letters to demonstrate a crossing of a bridge and capital letters for the landmasses. By doing this, Euler was able to find patterns and relationships that led to his
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He noticed that for each landmass, not including the initial or last one, two different bridges much be taken, one for entering and one for leaving that specific landmass. So, each landmass should be an endpoint to a number of bridges that is twice the number of times it is passed during the walk. So, it can be stated that each landmass, besides the initial and last, is an endpoint to an even number of bridges. Since the Königsberg problem consisted of a landmass that was endpoint of five bridges and three landmasses that were endpoints of five bridges, Euler was then able to conclude that such a walk was impossible to create. Euler described his work for solving this problem as “geometry of position” and together with future works of his became the vital ideas for topology. His work and ideas presented here inspired the areas of graph theory and topology which are not very important areas of mathematics.

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