Carl Friedrich Gauss' Life

Carl Friedrich Gauss Sania Ali Carl Friedrich Gauss' life Carl Fridrich Gauss was born in Brunswick, Germany to a working class family on the 30th of April 1777. His father made a living by laying bricks and gardening, hence it was not likely that Gauss would pursue an education. His father also hoped for his son to follow in his footsteps and work as a labourer. His father was also of the mindset that education was not very helpful and would not make a difference to a labourer. With his own father discouraging him to get an education and his family's financial situation, Gauss did not look like he would end up in school anytime soon. However, his mother and uncle played a huge role in getting him there when they realized in his younger years how bright he was. When he was only five years old, he managed to find an error in his fatherr's payroll. His potential was even discovered later on by his teachers while he was still in grade school. Gauss' class was challenged by a mathematics teacher to find the sum of all the integers numbers from one to 101. Astonishingly, Gauss was the first to finish, while the rest of his class took understandably much longer. When his teacher checked the work, the vast majority of the children's solutions were incorrect, but Gauss's had the correct number, 5050, written as his answer. The child prodigy had recognized that if he paired off the integers,for example 1+100=101, 2+99=101, 3+98=101, etc he would have 50 pairs of integers, and because those 50 pairs all added up to 101, he multiplied 101 by 50 and got a total of 5050. As if this was not amazing enough, Gauss only continued to excell in the field of mathematics. With the help of Bartels and Bttne Gauss began his education at Gymnasium (a senior secondary education institute in Germany as well as other portions of mainland Europe) at the age of twelve in 1788 where he further learned more mathematics as well High German and Latin and began to question concepts such as Euclid's geometry. After ingesting many nontrivial concepts so quickly, Gauss caught the attention of Karl Wilhelm Ferdinand, the Duke of Brunswick, whom agreed to provide a stipend to help the prodigy learn more so that he could himself contribute to the field. That was how Friedrich Gauss found himself at the Brunswick Collegium Carolinum in 1972 where he independently rediscovered the binomial theorem, Bode's Law, the arithmetic- geometric mean, and the law of quadratic reciprocity. Towards the end of his stay at the Collegium, Gauss managed to discover a method to create a regular polygon with 17 sides, known as a heptadecagon, with the use of a straight edge and a compass. Many mathematicians believed such a feat was impossible and Guass was so confident in the importance of this discovery that he not only insisted that it be engraved on his tombstone, but it also inspired him to continue studying in the field of mathematics. During his stay in college, Gauss was very keen to learn and study languages, but he now knew that he would rather study and discover more in mathematics instead. Duke Ferdinand continued to support Gauss financially so that he may continue to discover more. Gauss then found himself attending the University of Göttingen in 1795. Here he befriended Farkas Bolyai whom he met in 1799 and continued to correspond with many years later. Gauss' next discovery was in an astronomical application. In 1801, some astronomers had discovered what they thought to be a planet and named it Ceres. Later on in the year, they lost track of where it was. However, Gauss had been in communication with them about the matter, and he then managed to pinpoint its exact location, therefore rediscovering it. He used this new means of calculations to create a method to determine the orbits of certain new asteroids. In the year 1818, Gauss applied his practical uses of mathematics to help him conduct a geodesic survey of the Kingdom of Hanover and used previous Danish surveys for further data. For the purpose of helping him with this survey, Gauss invented the heliotrope which was an instrument that used a mirror to reflect sunlight over vast distances to measure position. Such a project piked Gauss' interest in topology and diffrential geometry which are fields of mathematics that deal with curved surfaces. This led to the establishment of Gauss' Theorema Egregium (which is Latin for The Remarkable Theorem) in 1828 which was an incredibly important document that dealt with the notion of curvature. The theorem states that the curvature of an object can be determined by measuring only angles and distances on the surface, and therefore curvature is independent of whether or not the surface is embedded in a 2-dimensional or 3-dimensional space. This coined the phrase of a Gaussian curvature or a Gauss curvature of a surface in differential geometry. These such discoveries lead Gauss to his appointment as a professor of mathematics and the director of the observatory at the Gottingen University which is where he worked in this official position until his death on the 23rd of February 1855. Carl Friedrich Gauss' mathematical and other works Using Gauss' Theorema Egregium (translates from Latin into The Remarkable Theorem), the curvature of a surface such as Gaussian curvature seen in differential geometry can be calculated using K = k1 ∗ k2 where k1 and k2 are the principal curvatures. The following equation gives us the total curvature of a surface in an integral form: 3 X ZZ θi = π + KdA (1) i=1 T where the total curvature of a geodesic triangle is given by the deviation of the sum of its angles from π. The sum of the angles of a geodesic triangle's curvature that is positive will be greater than π, while the sum of the angles of a geodesic triangle's curvature that is negative will be less than π. This implies that for a surface with zero curvature, such as the Euclidian plane, the sum of angles will add up to exactly π radians, which is indeed the case. Friedrich Gauss was not only a renown mathematician, but he also made some very useful and important contributions to the field of physics. One of the most well known laws discovered by Guass would be his equation of Gauss's Law for electric fields, an equation that contributes greatly to the Maxwell's equations of electricity and mangetism. These diffrentiable equations lay the foundation towards many electrodynamic and classical circuit problems. The following is Gauss' Law as expressed with the electric field E. Q ΦE = (2) "o Where ΦE is the electric flux through an enclosed surface, Q is the total enclosed charge by the body, and "o is the electric constant. This equation can also be written in its integral form as follows: I ΦE = EdA (3) Where ΦE once again represents the electric flux through an enclosed surface, E is the electric field, and dA is the infinitesimal area that is thought of as a planar. Besides this, Gauss also found the Fundemental Theorem of Algebra which was the proof for how every algebraic equation has at least one root or solution. This theorem shook the basis of algebra and impacted 2 mathematics greatly for centuries afterwards. Other mathematicians had also attempted to prove the same theory such as Jean le Rond d'Alembert who had produced a false proof before him. In Gauss' dissertation he critiqued the mathematician's work although his own proof was questionable nowadays due to the use of the Jordan curve method. However, by 1849 he had rigorously stayed at improving the proof and managed to produce three more, and in the process substantially cleared up the use of complex numbers in the proof. In his book Disquisitiones Arithmeticae (Latin for Arithmetical Investigations), he also created the sign for congruency which he used to present clear modular arithmetics as well as contributed greatly towards number theory. Gauss's symbol for congruency can be seen as =∼ Despite the seemingly random occurence of prime numbers, Gauss attempted to find some sort of relation or pattern to their order. He approached the problem from a different angle such that he graphed the incedence of primes as the numbers increased. He noticed something that resembled a trend which was that as the primes increased by ten, the the probability of a prime number appearing reduced approximately by a factor of two. Although this was progress, Gauss was aware that it was not significant enough proof as it only managed to give an approximation of whether a prime number would appear or not. Hence he kept this discovery secret until much later. Gauss was a person that only published the work he was confident in and the work he thought would be both important and have an impact on the world. Therefore, whenever he wrote proofs he was not satisfied with or had any ideas that he had difficulty expanding, he never published them and kept them secret in his own journals and diaries which were discovered often after another mathematician had proved a certain relationship that Gauss had theorized about before as is supposedly the case here. In 1840, Guass published his paper titled Diotrische Untersuchungen which is German for Dioptric Investi- gations that explored the idea of Gaussian optics. The paper expanded upon this technique that would be applicable to geometrical optics that describe the behaviour of light rays in optical systems using Gauss' parax- ial approximation.

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