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Niels Henrik Abel Biography and Achievements

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    On August 5th, 1802, Soren George Abel and Marie Simonsen gave birth to their son Niels Henrik Abel. Niels Able was born in Nedstrand, Norway (Encyclopedia Britannica). Niels grew up in a very religious family. His father, Soren had a degree in theology and philosophy. Soren was a pastor at Finney. As well as his father being a pastor, Niels’ grandfather, Hans Mathias Abel was a pastor at Gjerstad; where Niels’ father Soren was a chaplain during his childhood (Wolfram Research). When Hans passed away, Soren took over as pastor in Gjerstad.

    Niels’ mother, Marie was brought up in Risor, where her father Niels Henrik Saxlid Simonsen was considered the wealthiest man in Risor. In Gjerstad Marie was used to the luxurious life, thus she was in charge of throwing the balls and gatherings. (Encyclopedia Britannica) It was said that Marie was not very interested in the upbringing of their children but instead, spent a lot of time with alcohol (Wolfram Research). Being elected a representative of Storting, Soren created contact with the Cathedral School in Christiania or also known as Olso, where instead of sending his eldest son Hans, Soren sent Niels (Encyclopedia Britannica).

    In 1815, Niels entered the Cathedral School at the age of 13. In 1816, Niels’ older brother Hans joined him at Olso. Hans and Niels shared the same room as well as sharing classes and much more at Olso. Niels and Hans loved to share rooms and they didn’t mind sharing classes. Although Hans was always being the one to flaunt that he was receiving better grades in the classes they shared, there was a different take on which one of the two Able brothers was the smarter of the two. This fact was proven in 1818 when Olso hired Brent Michael Holmboe, a new Mathematics professor who had a different perspective about the brothers.

    Brent would send homework for the students to do and realized that Niels was very good at mathematics and he told Niels that he could just tell that Niels and math could have a great future if Niels spent more time on it. He thought so highly about Niels and mathematics that Brent started to give Niels private lessons in mathematics (Stubhaug). While away at school the Abel family went through a very rough stretch. In 1818 Niels father Soren had was engaged in a tense theological argument with Stener Johannes Stenersen.

    This theological argument included concerns and frustration about Stener’s catechism or religious thoughts about the New Testament (Encyclopedia Britannica). It was a very big deal at the time that it was being covered by the press. During this argument, Soren insulted the wrong people at the wrong time and almost left himself with nothing. Instantly returning from this argument with Stener, Soren started drinking and after two years of constant drinking, Soren passed away and left his wife and kids in some serious depression.

    Each one of the family members acted differently to the death of Soren. The two sons went two completely different ways. Hans was greatly depressed and decided he needed to return home and grieve at home, while Niels was given a scholarship from Brent Holmboe, his professor who thought very highly of him (Encyclopedia Britannica). Brent didn’t just stop at obtaining a scholarship to be able to stay at Olso; he also fundraised and found ways to get Niels money for him to attend the Royal Frederick University (Wolfram Research). In 1821, Niels attended Royal Frederick University.

    At this time in Niels’ life, being only 19 years of age, he was already one of the top mathematicians and was looked up to by many in the mathematical world. During his first year at Royal Frederick University, Niels started on his first piece of work, the quintic equation in radicals. He graduated with a superior status in math, but an average elsewhere in 1822. Before he graduated Niels thought he found the solution to a 250-year-old problem (Stubhaug). He was so sure about his work on the quintic equation in radicals. Many sought-after mathematicians such as Ferdinand Degen and Christopher Hansteen found no errors in Niel’s work.

    Although Degen was so overwhelmed by the idea of a 250-year-old problem finally being solved he asked Niels to show him another example in which this would work (Wolfram Research). When Niels went through his work with the new example with Degen, Niels found out that his solution was incorrect, and the 250-year-old problem was still unsolved. After graduating from Royal Frederick University, Niels Abel was struggling to find a new home. He had no place to live so his professors had helped him financially for a while so that he could find a place to live.

    However, his inability to be financially stable did not stop him from his mathematic ambition. He was still determined to work hard and continue his great progress towards mathematic work. After 1822 Niel’s life began to take off in many different ways. One of which in 1823 Niels published an article in “Magazin for Naturvidenskaberne” This specific magazine was Norway’s first scientific journal. After Niels published his first article, he went on to publish many more in the same magazine. After being told that the articles he was publishing needed to be for the common person he took his work and wrote an entire paper in French.

    The paper was said to “generalize a representation of the possibility to integrate differential formulas” (Encyclopedia Britannica). After all his publishing, Niels was given a chance to go to Copenhagen and work on Fermats Last Theorem. While in Copenhagen, Niels was able to visit his uncle Peder Tuxen. On his visit, Niels met his soon-to-be fiancee Christine Kemp. During all these quests, Niels published one of his first notable pieces of work, “Memoir on Algebraic Equations” (Encyclopedia Britannica). Later on, in 1823 Niels finally proved his quintic equations to radicals. It was later on named “Abel-Ruffini Theorem”.

    While studying these two languages, Abel published his first notable work in 1824, Memoire Sur Les equations algebriques ou on demonstrating impossibility de la resolution de equation general du cinquieme degre, Memoir on algebraic equations, in which the impossibility of solving the general equation of the fifth degree is proven (Wikipedia). He was also receiving scholarship money because he was not able to receive any more money from his professors. Abel continued to be anxious about the publication of his large memoir on integration, for which he originally had requested university support.

    The manuscript circulated with the documents concerning his fellowship until it disappeared without a trace in some public archive. The formula he indicated is not quite intelligible as it stands, but there can be no doubt that it contains the idea for the result which later was called the great Abelian theorem, his momentum is perennials, a moment more lasting than bronze (Ore). In September 1825 he was granted permission to travel abroad. He traveled again to Berlin and the Alps with four friends to study geology. He was to visit Carl Gauss by traveling to Gottingen then continuing to Paris.

    On his trip, he spent four months in Berlin, becoming close friends with August Crelle. At this time, August Crelle was about to publish his mathematical journal, Journal fur die Reine und Angewandte Mathematik. In this journal, Abel contributed to seven articles in the first year. After Berlin, Abel traveled to Leipzig, Freiberg, Dresden, Prague, Vienna, Trieste, Venice, Verona, Bolzano, Innsbruck, Luzern, and Basel. In Freiberg, Abel made great work and research in the theory of functions, specifically elliptic, hyperelliptic, and a new function now known as abelian functions.

    He finally traveled to his last destination, Paris, in July of 1826. Abel sent a part of his work to Berlin to be published in Crelle’s journal and saved the most important work for the French Academy of Sciences. He was going to show them his general theorem on integrals, which would be later known as the great Abelian theorem. His work was not known well in France and ended up being forgotten. His time on his study abroad trip was cut short due to financial problems and arrived back in Berlin in January of 1826. He was back in Norway by May 1827.

    His scholarship was not renewed because they felt that his travels were unsuccessful. He needed money so he reverted to tutoring and continued to work with Crelle’s journal. Arriving back in Norway, Niels Abel had contracted tuberculosis from his stay in Paris. On Christmas 1828, he traveled to visit his fiancee. His illness got worse during the travel. The couple was able to enjoy the holidays but Abel would eventually pass away. Abel died on April 6, 1829. After his death, a letter from Crelle arrived to announce to Abel that he had been offered a professorship at the University of Berlin.

    His death cut very short a career of brilliance and promise to mathematics. The French Academy published his memoir in 1841. Abel had entered the elite of the world’s mathematicians, and nothing he produced could from now on be neglected. After being granted many scholarships, Niels was able to study many different languages such as German and French and he also was permitted to travel abroad to learn much more about mathematics. Niels Henrik Abel was one of the most influential mathematicians to ever come out of Scandinavia.

    His work as a mathematician was very short-lived due to him only living 27 years as stated previously. Abel’s contributions to math started very early for most people when he gave his first contribution as a 16-year-old. At 16 years old, Abel gave proof of the Binomial Theorem valid for all numbers, not just rational numbers. This was originally proven but only for rational numbers by Leonhard Euler. Abel’s Binomial Theorem is as follows (Abel’s Binomial Theorem): At the age of 19, Abel invented a famous and important branch of mathematics with many applications in physics. He developed the theory which shows that there is no algebraic solution to find the roots of a quintic number, or any others greater than degree 4. During his time, degrees of three or less were able to be found but this solution that he found was the most sought after. For Abel to find this solution was very big for mathematics and for him to do it at such a young age was even more impressive. After Abel had proved that there were no such solutions for degrees of four or more, Abel gained a lot of ground in the mathematics world. He was then inspired to produce Abel’s Theorem of Convergence.

    This was one of Abel’s most famous works and it stated, “that if a real power series converges for some positive value of the argument, the domain of the uniform convergence extends at least up to an including this point. Furthermore, the continuity of the sum of the function extends at least up to and including this point (Wolfram Alpha).”

    At the age of 21, Abel had published numerous papers on functional equations and integrals. In one of those papers, Abel gave the first solution to an integral equation. One year after he solved an integral equation, Abel proved the impossibility of solving the general equation of the fifth degree algebraically. After Abel had accomplished this, he had become extremely ill and returned to Norway where he continued to work on equation theory and elliptic functions.

    His work on elliptic functions had major implications on the whole theory of elliptic functions. An important process in these discoveries was the studying on inverses of these functions. Abel realized that to determine the answer, he had to study the inverse of the elliptic function. Niels Henrik Abel was the center of a mathematical transformation. In the early nineteenth-century math analysis was mainly formula-centered and turned into a more concept-centered practice. The issues that were fixed when changing from one practice to the other were the subject matter, notions of rigor, methods, and objects (Sorensen page 40).

    The subject matter was categorized into independent disciplines and sub-disciplines. The theory of equations was separated into its category which was algebra. What was left of analysis became focused on the study of real or complex functions. Abel’s elliptic function would become categorized into this section. Since new complex methods were introduced into math there was a shift to move towards a more arithmetical base for analysis. This meant that functions were no longer explained by their formal relationships but by a numerical value.

    In order to move objects into the new practice, they needed to be redefined by class as well. Abel played a huge part in the transformation of analysis. Abel wrote a paper and sent it to many different mathematicians in the hope to be recognized for the type of work that he was doing. In his original draft he did not know that an Italian mathematician, Paolo Ruffini had published the same proof years earlier. Even in Abel’s early proof there were things in his paper that were not fitting to a proof. A mathematician, Ferdinand Degen, suggested he work in the field of elliptic integrals.

    Without this suggestion, Abel might not have made his discoveries. Abel traveled the world trying to get recognition for the work that he was doing but failed to do so. When he returned home he found he contracted tuberculosis and tried to feverishly get some recognition for what he was doing. Mathematicians slowly started to notice what Abel was doing and the type of impact it would have on the mathematical world. From an early age Abel started giving proofs for different equations. At age 16 he gave the proof for the binomial theorem.

    This theorem was important because it expanded on Euler’s idea that was only true for rational numbers. Abel’s theorem expanded Euler’s theorem because Abel’s theorem says that the binomial theorem is possible for all numbers. After the binomial theorem, Abel started his mathematical journey, at age 19, with trying to solve the quintic equation, which resembled a x5 + b x4 + c x3 + d x2 + e x + f = 0. Abel proved that there was no general solution for the algebraic solution of roots of a quintic equation greater than four. In order to do this Abel had to create a whole new area of maths called group theory.

    This area of math became vital for math and physics. Group theory is the study of groups. It is a method for analyzing abstract and physical systems which shows symmetry and the importance it has for physics. Abel began to work on elliptic functions and his theory of equations. Boneedthese needs to be described with very technical formulas and without this particular language, neither of these two formulas could be described without this language. Once calculus was discovered it was possible to find integrals of square roots and an expression of degrees of three and four.

    A problem that had simple calculations of the length of arcs for ellipse was called elliptic integrals. There was no expression yet, that was found, for these known functions (Ore pg 67). Abel came up with the idea that instead of investigating integrals he turned them into inverse functions. This method gave everything a different aspect. This theory of elliptic functions became the generalization of trigonometry. The properties became the laws for sine, cosine, and tangent. Another area of study for Abel was equation theory. Babylonians and Greeks could solve quadratic equations where they learned it from the Arabs.

    About 1500 A. D. Scipione del Ferro found the solution to the cubic equation. Ferro tried to keep it a secret because he knew how valuable this information was. The equation passed through many different mathematicians. Gerolamo Cardano publish the Ars Magna which contained not only the cubic equation but the solution for the equation to the fourth degree. Once the equation for the fourth degree was out the race to find the fifth and higher degrees were on. Abel attempted to show that higher degrees could be found but after some time he demonstrated that a solution did not exist.

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