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GAUSS VS EULER

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Leonhard Euler (Basel, Switzerland, 15 April St. Petersburg, Russia, 18 September 1783) He was a Swiss mathematician and physicist. This is the main eighteenth century mathematician and one of the largest and most prolific of all time. He lived in Russia and Germany most of his life. At the age of 13 he enrolled at the University of Basel. Because of the friendship between the two families, Euler received private lessons from Johann Bernoulli, who quickly discovered the incredible talent of his new pupil for mathematics.

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**Contribution to mathematics and other scientific areas**

Mathematical Notation Analysis The Number and Number Theory Graph Theory And Geometry Applied Mathematics Physics and Astronomy Logic Architecture and Engineering

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**Mathematical Notation**

Euler introduced several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function and was the first to write f (x) to denote the function f applied to the argument x. He also introduced: The modern notation for the trigonometric functions. The letter e for the base of the natural logarithm (now also known as Euler's number). The Greek letter Σ for summations and the letter i to denote the imaginary unit. The use of the Greek letter π to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it did not originate with him.

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Analysis Euler is well known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms. He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric functions. For any real number φ, Euler's formula states that the complex exponential function satisfies

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**Geometric Interpretation of Euler’s Formula**

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Number Theory Euler linked the nature of prime distribution with ideas in analysis. He proved that the sum of the reciprocals of the primes diverges. In doing so, he discovered the connection between the Riemann zeta function and the prime numbers: the Euler product formula for the Riemann zeta function. Euler proved Newton's identities, Fermat's theorem on sums of two squares, and he made distinct contributions to Lagrange's four-square theorem. He contributed significantly to the theory of perfect numbers, which had fascinated mathematicians since Euclid.

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Graph theory Euler also discovered the formula V − E + F = 2 relating the number of vertices, edges, and faces of a convexpolyhedron, and hence of a planar graph. In analytic geometry also found that three of the significant points of a triangle, centroid, orthocenter and circumcenter-could obey the same equation, to the same line. In the line containing the centroid, orthocenter and circumcenter is called "Euler line" in his honor.

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**Euler solved the problem known as THE SEVEN BRIDGES OF KÖNIGSBERG**

Euler solved the problem known as THE SEVEN BRIDGES OF KÖNIGSBERG. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not possible: there is no Eulerian circuit.

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Applied mathematics Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of the Bernoulli numbers, Fourier series, Venn diagrams, Euler numbers, the constants e and π, continued fractions and integrals.

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Physics and astronomy Euler development the equation of the elastic curve, which became the cornerstone of engineering. Aside from successfully applying his analytic tools to problems in classical mechanics, Euler also applied to the problems of celestial movements of the stars. In the field of mechanics Euler explicitly introduced the concepts of particle mass and punctual and vector notation to represent the velocity and acceleration. In studied hydrodynamic flow of an ideal fluid incompressible Euler equations detailing of hydrodynamics. Over a hundred years ahead Maxwell predicted the phenomenon of radiation pressure, fundamental unified theory of electromagnetism.

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Logic Euler is also credited with using closed curves to illustrate syllogistic reasoning (1768). These diagrams have become known as Euler diagrams. An Euler diagram shows that the set of "four-legged animals" is a subset of "animals", but all the "mineral" is disjoint (no common members) with "animals"

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Carl Friedrich Gauss (April 30, 1777, Brunswick - February 23, 1855, Göttingen) was a mathematician, astronomer, geodesist, and German physicist who contributed significantly to many fields, including number theory, mathematical analysis, differential geometry, statistics, algebra, geodesy, magnetism and optics. Considered "the prince of mathematics" and "the greatest mathematician since antiquity“. Along with Archimedes and Newton, Gauss is undoubtedly one of the three geniuses in the history of mathematics.

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The Polygon The first contribution of Gauss, with 17 years, mathematics was the construction of the regular polygon of 17 sides. Gauss not only managed the construction of the 17-sided polygon, also found the condition to be met polygons can be constructed by this method. Gauss proved this theorem combining with other geometric algebraic reasoning. The technique used for the show, has become one of the most used in math: move a problem from an initial domain (geometry in this case) to another (algebra) and solve the latter.

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The Disquisitions In 1801, when he was 24 years, Gauss published his first major work "Disquisitiones Arithmeticae". Besides organizing what already exists on the integers, Gauss contributed ideas. He based his theory from a congruent number arithmetic used in the proof of important theorems, perhaps the most famous of all and is the favorite of Gauss quadratic reciprocity law, which called Gauss theorem aureus. It is also noted the contribution of Gauss's theory of complex numbers. Also developed a method to decompose the product of prime numbers in complex numbers.

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A New Planet The discovery of the "new planet", called Ceres by Giuseppe Piazzi. It was necessary to accurately determine the orbit of Ceres to put it back to reach the telescopes, Gauss accept this challenge and Ceres was rediscovered a year later, in the place that had predicted with detailed calculations. His technique was to demonstrate how variations in experimental source data could be represented by a bell-shaped curve (now known as the bell curve). He also used the least squares method. Had similar success in the determination of the orbit of the asteroid Pallas.

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Geodesy By 1820 Gauss began working in geodesy (determination of the shape and size of the earth). In 1821 he was commissioned by the governments of Denmark and Hannover, Hannover geodetic study. To this end Gauss invented the heliotrope, an instrument that reflects sunlight in the specified direction and can reach a distance of 100 km and enabling alignment of surveying instruments. Working with data from observations developed a theory on curved surfaces, the surface characteristics can be known by measuring the length of curves contained therein.

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Magnetism From 1831 he began working in the theoretical and experimental investigation of magnetism. Gauss was able to prove the origin of the field was in the interior of the earth. Gauss also worked with the possibilities of the telegraph, his was probably the first to run in a practical way in seven years ahead of the Morse patent. The main feature of the work of Gauss, especially in pure mathematics is particularly so have reasoned like general. After his death it emerged that Gauss had found the double periodicity of elliptic functions.

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**THANK’S FOR ATTETION ANY QUESTIONS? Carlos Fernández Martín**

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