Homework answers / question archive / The research question: How Niels Henrik Abel proved the binomial theorem at the age of sixteen, and how can this theorem apply to combinatorics, algebra, calculus, and many other mathematics areas? Is this still the research question you are trying to answer in your paper? I will assume so

The research question: How Niels Henrik Abel proved the binomial theorem at the age of sixteen, and how can this theorem apply to combinatorics, algebra, calculus, and many other mathematics areas? Is this still the research question you are trying to answer in your paper? I will assume so

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The research question: How Niels Henrik Abel proved the binomial theorem at the age of sixteen, and how can this theorem apply to combinatorics, algebra, calculus, and many other mathematics areas?

Is this still the research question you are trying to answer in your paper? I will assume so. But much of what you have written in your paper is not about this question, so you can cut a lot of it. For instance, the stuff about the unsolvability of the quintic is interesting, but I do not understand what it has to do with the binomial theorem. It is not clear how you have answered your research question either. The stuff about the binomial theorem needs more detail, and needs to be better organized. What is the theorem? How does he extend it from what Newton and others had done? How does he use it in other areas? You say this, “Cauchy's Cours d'analyse of 1821,5 intrigued Abel's curiosity in the binomial theorem by constructing a theory of infinite number based on the current norm of rigor. Abel set out on a course of investigation based on his interpretation of Cauchy in generalizing Cauchy's proofs of the binomial theorem and included complex exponents.” — can you go into more detail about this also? Running head: NIELS HENRIK ABEL’s BINOMIAL THEOREM How Niels Henrik Abel proved the binomial theorem at the age of sixteen MATH388-01: History of Mathematics Heba Aljabrine Prof. Ely February 1, 2021 1 NIELS HENRIK ABEL’s BINOMIAL THEOREM 2 Niels Henrik Abel was a well-known Norwegian mathematician who lived from 1802 to 1829. Even though he died at the age of 26, he contributed significantly to a variety of fields. Abel proved the binomial theorem at the age of 16. He independently invented group theory three years later, proving that it is challenging to solve quantic equations. For more than 350 years, there had been an unsolved issue. Moreover, he also explored Abelian functions and experimented on elliptic equations. Abel grew up in poverty, with six brothers, a father who died when he was 18, an inability to find employment at a university, and several mathematicians dismissing his work at first. The Abel Prize, another of the highest honors in mathematics, is named after him presently. The question need to answer is: How Niels Henrik Abel proved the binomial theorem at the age of sixteen, and how can this theorem apply to combinatorics, algebra, calculus, and many other mathematics areas? This paper looks into how Niels Henrik Abel proved the binomial theorem at the age of sixteen. In particular, it also explains how this theorem can be used in combinatorial optimization, calculus, algebra, as well as many other areas of mathematics. To begin with, Abel’s mathematics history can be traced back to when Bernt Holmboe, Abel's instructor, found that he had a firm grasp of mathematics. Abel was eligible to the University of Christiania because his father died in 1821. Such would have only arisen as a result of Holboe's assistance in securing a scholarship. He graduated from university one year after beginning his research, but he had already achieved so much. Niels Henrik Abel made several significant contributions to mathematics. Even though Abel lived just 27 years, he made several scientific breakthroughs. At the age of 16, Abel extended Euler's theorem by proving the Binomial Theorem to all figures, not only Rationals. He demonstrated that there had been no algebraic expression for any general probability density function of a degree greater than four NIELS HENRIK ABEL’s BINOMIAL THEOREM 3 when he was 19 years old. To do the same, he developed Group Theory, a fundamental branch of mathematics that is useful in many fields of mathematics and physics. Abel published articles on combinatorial optimization and integrals when he was 21 years old in 1823. Abel provides the very first application to an integral equation in this paper. He subsequently proved the inability to solve the general equation of the fifth degree algebraically a year later. He self-published it in the hopes of gaining an appreciation for his work. This finally led to him receiving a Norwegian government scholarship to move to Germany and France. When criticizing Cauchy's theorem on the validity of sums of linear equations, Abel, a proponent of Cauchy's improved intellectual rigor, used the term "exceptions." On the other hand, exceptions tend to be both accurate and viable structures in the early nineteenth century when considered in the context (Cauchy & Bottazzini, 1821). Initially, Abel's use of the word "exception" in his binomial work is known, as is the position of the exception. Between the mideighteenth and the mid-nineteenth centuries, the style of mathematical study in analysis shifted from a formula-centered framework exemplified by L. Euler (1707–1783) to a concept-centered methodology presented by works by G. P. L. Dirichlet (1805–1859) and G. F. B (Sorensen, 2005). Multiple facets of the mathematical organization, such as notations, queries, results, processes, and techniques, were affected by the transformation. It was felt by active and innovative nineteenth-century statisticians who noticed a disparity between the computational mechanisms connected with formula-based computations and formula-based computations. During this period of transformation, the Norwegian N. H. Abel (1802–1829) burst onto the global mathematical landscape in the 1820s, raising a slew of new questions and producing jaw-dropping new results. Even though Abel's mathematical corpus was primarily concerned with algebraic issues, especially those involving elliptic and more excellent transcendence NIELS HENRIK ABEL’s BINOMIAL THEOREM 4 features, he also provided a new proof of the binomial theorem (Cauchy & Bottazzini, 1821). Abel offered further evidence of the binomial theorem—a hypothesis fundamental to efforts by Euler, J. L. Lagrange (1736–1813), and A.-L. Cauchy (1789–1857) establishes more robust frameworks for analysis—even though his computational framework predominantly dealt with algorithmic problems, especially regarding elliptic and more excellent transcendental functions (Sorensen, 2005). Cauchy's Cours d'analyse of 1821,5 intrigued Abel's curiosity in the binomial theorem by constructing a theory of infinite number based on the current norm of rigor. Abel set out on a course of investigation based on his interpretation of Cauchy in generalizing Cauchy's proofs of the binomial theorem and included complex exponents. Nevertheless, Abel noticed that one of Cauchy's core theorems (on the consistency of any probabilistic sum of linear equations) "insinuated exceptions" during the analysis, and this discovery, as well as the events that took place in the bridge between the two various computational types, is the focus of this paper. According to T. S. Kuhn's theory of science, the compilation of facts (assumptions and experimentation) contradicting the dominant paradigm, according to T. S. Kuhn's view of science, plays a significant role triggering conflicts that are eventually resolved by revolutions (Kuhn, 2012). Mathematics, according to others, varies from the sciences because mathematical arguments are considered to be right or wrong based on the time-independent validity of their ideas. More lately, for instance, though I. Lakatos, mathematics has been viewed as a human creation that has evolved into a discourse that enables theorems to be misrepresented. The current paper provides a diachronic interpretation of a significant primary source from the 1890s, without strict adherence to any concepts connected with Kuhn or Lakatos. NIELS HENRIK ABEL’s BINOMIAL THEOREM 5 Mathematical developments cover a broad range of topics and theories that were significant in the early 1800s. Abel's considerable contributions are widely regarded as someone in the view of algorithmic solubility of formulas, the digitization of sequence analysis, and the study of elliptic structures and higher transcendental (Sorensen, 2010). Many works (compiled or unpublished) do, nevertheless, have such a place in the collection. Such include: • The solution of specific forms of difference calculus. • The development of fractional calculus. • The concept of the numerical solution. • The analysis of generating functions. This subject, the theory of equations, was Abel's first and enduring engagement with mathematics; his first autonomous moves out from the influence of the mentors were unsuccessful ones once he presumed to have acquired a specific antidote formula for the quintic formula in 1821(Sorensen, 2010). After being persuaded to clarify his point, he discovered he was mistaken, and by 1824, he had provided evidence that no such approach equation existed. The mathematical science of evaluation seems to have been capable of creating a rising number of functions since the invention of calculus at the end of the 17th century. EULER raised the principle of feature to the core object of study in his 1748 textbook Introduction in analysis Infinitum (Sorensen, 2010). The brilliant cal was used to study structural components and their numerical solution expansions. Since it could only handle a restricted number of elementary transcendental functions, mathematicians felt and spoke of an unsatisfying limitation of the study. Accepting new roles into research entails gaining information about them that allows them to be used as explanations. If the contemporary definition of a function is simply a projection NIELS HENRIK ABEL’s BINOMIAL THEOREM 6 from one set to another, then series extensions and other representations, discrete and integral ties, functional relations, and so much more. When ABEL chose elliptic trigonometric functions as his essential research subject, there was already much information about these artifacts. GAUSS' analysis of the dividing question for the ring (building of normal n-gons) throughout the Disquisitiones arithmeticae had a significant influence on ABEL. GAUSS had implied that his method could be extended to the lemniscate integral, a straightforward case of elliptic integrals, and A BEL decided to back up his argument (Sorensen, 2010). A BEL reversed the analysis of elliptic integrals into the study of elliptical components by a new theory that would soon be celebrated as one of the best in assessment. Rather than just seeing an integral's value as a component of its theoretical threshold, he viewed the upper limit as a function of the integral's value. ABEL derived elliptic functions of a differential equation using proper adjustments and some addition formulae. ABEL put the whole theory of elliptic integrals on a new footing by inverting the emphasis. The position and significance of ABEL's efforts in the area of algebra The theory of equations became a mathematical discipline in the nineteenth century, with its collection of questions, approaches, and justifications. ABEL served an essential role in the implementation. His dissertation on the algebraic insolubility of the specific quintic equation, as well as his in-depth analyses of the so-called Abelian equations, are among the first findings to emerge from this fledgling discipline. While ABEL's inquiries posed new questions and provided answers to some of them, his methodology and technique were profoundly associated with the work of past generations of mathematicians. A BEL relied on the algebraic re-searches by EULER (1707–1783) in general. Such problems have been fundamental to mathematical progress since the Renaissance. However, they gave birth to a solid mathematical model in the NIELS HENRIK ABEL’s BINOMIAL THEOREM 7 second half of the 18th century; after describing the theory-building method, the focus shifts to ABEL's contribution to algebraic problems. The quintic equation experiments of ABEL show how a change in the way questions are asked can lead to unexpected results. Therefore, because of the similarities in methods and sources of inspiration, ABEL's questions are addressed to show how an algebraic subject arose from an otherwise non-algebraic domain. The quest for generalizability to quin-tic formulas has already been on since procedures for algebraic expressions computing the origins of cubic and bi-quadratic formulae were invented in the mid - sixteenth century, after R. DUP. D ESCARTES' (1596–1650) new notational framework converted the question into strictly algebraic symbol manipulation techniques, assuming that such a generalization could spread. For example, EULER was confident enough in the generalized algebraic solubility of equations that he used to prove another seemingly self-evident result: the fundamental theorem of algebra. As a result, ABEL emphasizes algebraic connections relating to and current among such objects in the middle of a spatial and temporal map acquired by extremely transcendental entities. ABEL's algebraic fixation influenced subsequent advances during the second quarter of the nineteenth century when the higher metaphysical theory was in a transitional phase. The way ABEL asked questions played a significant role in changing people's minds about the quintic's solubility. ABEL stressed in one of his notebooks that every mathematical dilemma is decidable when adequately implemented, whether affirmatively or negatively. As a result, ABEL proposed that goals that had been unachievable for years. How ABEL asked questions played a significant role in changing people's minds about the quintic's solubility. ABEL stressed in a passage in another one of his journals that every mathematical dilemma is NIELS HENRIK ABEL’s BINOMIAL THEOREM 8 decidable when adequately implemented, whether affirmatively or negatively. ABEL proposed a reformulation of the issue with targets that had been impossible to achieve for decades. As JACOBI soon learned, a shift in attitude toward mathematical goals signals a shift toward more general and abstract mathematics. ABEL utilized explicit quantification of total potential answers to the dilemma to address concerns about the probability of life. His methodology was focused on the classification and normalization of such objects, which were then examined as items belonging to a collection defined by definition rather than as individuals. ABEL brought up the question of deciding if a given equation is fixable or not in the concept of equations after proving the validity of both algebraic expressions solvable and unsolvable illustrations. ABEL sought to answer such a problem in a notebook document. He demonstrated effectiveness with some forms of equations; nevertheless, it was left to G ALOIS to formulate a theory based on his findings. In the Direction of Unsolvable Equations The theory of equations addressed a wide range of questions by the dawn of the nineteenth century. The critical issue is algebraic solvency for the time being, but in the 18th century, it was mixed with a slew of other concerns about the presence and interpretation of roots. The characterization of the foundation of the theorem changed a little bit for subsequent generations of mathematics. They interacted with the essence of the supposed origins where DESCARTES had not. The dilemma of proving that all (envisioned) roots of a branch are. The hypotheses of the basic theorem of mathematics were often existence proofs with no mention of the computational part. Such nonconstructive outcomes were also sought. A significant subfield of equation theory was developed in order to define and explain attributes of the roots of a given equation based on a priori investigations of the equation and without the use of exponents. NIELS HENRIK ABEL’s BINOMIAL THEOREM 9 Other nonconstructive findings were also followed. A significant subfield of equation theory was created to define and explain the roots of a given equation based on priori evaluations of the formula without knowing the roots specifically. LAGRANGE'S research into the characteristics of the origins of specific equations grew out of his solutions to fix greater level formulas using algebraic expressions (Sorensen, 2010). LAGRANGE's focus on mathematical equations, or detailed equations with specific coefficient correlations, can be categorized into three parts: the nature and number of roots and limits for the values of the coefficients. Such equations defined the feature equation's essential symmetric connections between the roots and coefficients. When theories of such relations first appeared, they were based on formal manipulation techniques of the tacitly implemented variables and thus fit neatly into the existing algebraic design. Among the many possible questions about the unidentified roots, one is related to the theory of algebraically numerically solving. It started when mathematicians looked into the different ways the roots could be written, and it was the first step toward asking general solubility questions. The varied and essentially empirical attempts to provide substantial reductions were overtaken by conceptual and generalized investigations, primarily by LAGRANGE 1770–1771 in the last half of the 18th century. The tendency towards broad analysis was followed by the concept of investigating permutations in LAGRANGE's function (Sorensen, 2010). Both sections were critical in proving that the algebraic answer to the quintic equation, which had been tried for a long time, was impractical. ABEL'S classification of algebraic equations was hierarchical, and the two definitions of order and degree were his way to obtain framework. The order was implemented to capture the depth of the entangled root extraction method, while the degree imposed a more delicate structure to keep track of root extractions at the same stage. The order definition was introduced NIELS HENRIK ABEL’s BINOMIAL THEOREM 10 by ABEL, who defined rational expressions as being of order. ABEL defined a hierarchy governed by the principle of degree within each order. Abel's definition of the extent of an algebraic equation calculated the number of co-ordinate root extraction methods at the highest level. In contrast, the sequence acted to represent the number of nesting root extraction methods of the primary degree. Abel never contemplated whether his definitions of order and degree were complete, that is, if any algebraic expression might (distinctively) be assigned order and a degree; instead, he assumed that any such entity referred to a particular order and a specific degree in his analyses of algebraic expressions. These ideas clearly established a hierarchy within the algebraic exponents class. Abel proved a core theorem about these newly defined objects using his structure of arithmetic operations. Its aim was to provide a concrete standard form for arithmetic operators. He showed that all elements of a top-level arithmetic sequence solving a correctable equation became rational representations of the equation's basis in this way. Abel offered that this related to any component involved in the answer by examining any of these items and operating downward in the structure. Abel's initial analysis of algebraic expressions yielded two main proof findings. It had first developed an algebraic expression hierarchy based primarily on nesting of root extraction methods. Second, it had led to the auxiliary theorem mentioned earlier, which assured Abel that any concept he encountered in the hierarchy of a possibility to solve formula would have been rationally dependent on the specified equation's roots. Abel's analysis of combinations and proofs of the CAUCHY-RUFFINI theorem, which describes the possible combinations of quantities of linear equations under permutations of their hypotheses, formed the other conceptual pillar of his uncertainty proof. Abel outlined most of what CAUCHY had accomplished before providing his evidence of this central finding. NIELS HENRIK ABEL’s BINOMIAL THEOREM 11 However, whereas CAUCHY started separating substitutions from the concepts they operated, Abel carried on the L LAGRANGE practice. Even though he rarely mentioned the "substitution" as a separate object, all of his calculations were about their expression effects. To conclude Abel's statement, he presented a crucial connection between the two preliminary rounds mentioned above. He related the minimum degree of a polynomial equation with v as a root and symmetric formulas as coefficients to the number of values taken by a function v under all permutations of its hypotheses (Schwartz, 2006). This formula became later discovered to be the indefinable equation relating to v. This is the irreducible equation that corresponds to v, and it would later play a key role in his general theory of solvation. Abel connected the relatively recent principle of distribution of elements under permutations to the older principle of possible values of representations of the type v y in this manner. Square roots were long believed to be two-valued, cubic roots three-valued, and so forth, and Abel was the first to link these two seemingly disparate ways of calculating the number of values in an arithmetic sequence (Schwartz, 2006). Abel's fourth aspect of impossibility proof included thorough and extremely explicit "supercomputing" analyses of operations of five quantities of two or five values. The formula-centered form of mathematical was coined to describe the predominant mode in the 18th century, which relied heavily on a solid framework of systematic and implicit manipulation techniques(Abel, 1881). Except when confronted with a few anomalies, relevant findings were not easily overturned due to the often bulky apparatus. This was demonstrated by the fact that perhaps the framework contained statements based on "generality" and findings that were only considered to be true "in particular." NIELS HENRIK ABEL’s BINOMIAL THEOREM 12 Scandinavian statisticians, such as Abel and Degen, said something about "general" or "common" arguments, concepts, or laws in the early nineteenth century. These often alluded to Cauchy's methods called "the generality of algebra" in 1821 and outlawed in his latest digitization of evaluation (Abel, 1881). These arguments will be particularly useful in the context of formal and general interactions between analytical expressions. These points will be applicable to formal and everyday connections between analytical words, regardless of numerical actions, when estimated values for the quantities were added. Abel's experiments on elliptic and higher transcendence phenomena contain examples of such a style of general reasoning. One small explanation will have to suffice as an introduction to this peculiar (by today's standards) way of thinking (Abel, 1881). Abel submitted his critically acclaimed, so-called Paris mémoir to the Academy of art des sciences in 1826, wherein he introduced a general, algebraic approach to the problem of incorporating algebraically related differentials. Conclusion Given its broad approach and a massive proportion of general, concept-centered conclusions, this paper performed well enough in the formula-centered format, with paragraphs of extended manipulation techniques preceding one another. In this sense, one particular procedure that appears regularly in the paper is particularly significant. Abel's assessment of his opposing team case, as well as his need for illustrations to prove his point, are remarkable in this setting. In the published edition, Abel was quite careful, saying only that one of the assumptions in the claimed "theorem" is very valid. Then the other, namely the inference to the left, "appears not to be so," and that "the theorem stated in the above quotations is therefore deficient in this situation." NIELS HENRIK ABEL’s BINOMIAL THEOREM 13 The system provides insight into the wording used in the excellent examples. Abel's use of words showed more than just admiration for Cauchy when he referred to Theorem C's "exception." When adapted to the formula-centered method, Abel's discourse enabled mathematicians to be using their presuppositions to escape ludicrous situations. When implemented to the formula-centered process, Abel's discourse enabled mathematicians to be using their presuppositions to escape ludicrous situations. Likewise, when Abel mentioned "restrictions" without elaborating, he was alluding to the very same general validity of outcomes. NIELS HENRIK ABEL’s BINOMIAL THEOREM 14 References Kuhn, T. S. (2012). The structure of scientific revolutions. University of Chicago press. Sorensen, H. K. (2005). Exceptions and counterexamples: Understanding Abel's comment on Cauchy's Theorem. Historia Mathematica, 32(4), 453-480. https://doi.org/10.1016/j.hm.2004.11.010 Sorensen, H. K. (2010). The mathematics of Niels Henrik Abel: continuation and new approaches in mathematics during the 1820s. Aarhus: RePoSS: Research Publications on Science Studies, 11. Abel, N. (1881). Sur une propriété remarquable d’une classe très étendue de fonctions transcedentales. Oeuvres completés, 2. Schwartz, R. K. (2006). Abel and the Insolubility of the Quintic. Cauchy, A. L., & Bottazzini, U. (1821). Cours d'analyse de l'École royale polytechnique: premiére partie: analyse algébrique. Editrice CLUEB. NIELS HENRIK ABEL’s BINOMIAL THEOREM 15