The Progression Of Abstract Algebra

Algebra is not an easy subject for many people. It is full of letters, numbers, and rules mixed together to represent real life problems that are hard to swallow for anyone who doesn’t look at the world from the perspective of a mathematician. In his essay, “Wrong Answer: A Case Against Algebra II”, Nicholson Baker addresses this problem and explains why he thinks Algebra II should not be seen as a staple in the education of High school students.

The Pythagorean School, for example, contributed many ideas to the mathematic community, among them, studies of geometry and the theory of proof.5 Euclid also lived in the time of ancient Greece and became a prominent mathematician, as well as author of a book about geometry called, The Elements, considered the second best-selling book of all time.6 The works of Pythagoras and Euclid have become fundamental building blocks for any person with an eighth-grade understanding mathematics. With these advancements, as well as innovations from Archimedes and Apollonius of Perga, Islamic scholars translated their works and contributed even more, growing the worldwide understanding of mathematics.7 Greek progressions in geometry as well as the theory of proof contributed greatly to our ultimate understanding of contemporary mathematics, without which, our modern society would be

There was an unexpected explosion in the math and science world in the 17th century across Europe, known as the Age of Reasoning. Scientists such as Galileo, Brahe and Kepler continued to increase our knowledge on mathematics and science, especially the solar system which led to Kepler’s laws of planetary motion. Isaac Newton also discovered the laws of physics explaining Kepler’s Laws, and brought together the concepts now known as calculus. The invention of the logarithm by John Napier contributed to the advances of science and astronomy and was one of the most significant developments of this time. Rene Descartes development of analytical geometry and Cartesian coordinates allowed the orbits of the planets to be plotted. Other mathematicians such as Fermat and Pascal formulated theorems which extended our knowledge on number theory. Pascal is most famous for his Pascal triangle even though similar figures had been done by the Chinese and Persian mathematicians before him. Newton and Leibniz revolutionized mathematics by developing infinitesimal calculus. Much more credit should be given to many other mathematicians at this time, but as said before, this was a time of severe increase in mathematics and these are only a few of the most important discoveries. (15)

Hilbert's contributions to mathematics can be divided into five major areas: invariant theory, which he studied until his transfer to Gottingen; algebraic number field theory, which he studied from approximately 1984-1899; foundations of geometry and mathematics, studied from 1899-1903; integral equations, studied from about 1904 to 1909; physics, studied from about 1912-1914; and foundations of mathematics, studied after 1918 (Kimberling, n.d.). Hilbert's proofs of Gordan's problem, which dealt with invariant theory, catapulted Hilbert into the spotlight as a first-class mathematician (Hilbert,

In regards to algebra the advancement math can be perceived as a puzzle. Each piece plays a detrimental role in the discovery of the bigger picture. In algebra specifically there are 3 significant stages of progress that have evolved math into what we know today. The first phase of math to be discovered was rhetorical, which was later followed by syncopated and lastly symbolic. These three stages of progression also reflect on the advancement of S.T.E.M. based careers.

Algebra is the basis of all math and its advancement took place in three major steps. The first stage of algebra is rhetorical algebra dating back to before 275 BCE. Rhetorical algebra was a form of algebra that contained no symbols so to them +,-, and*, did not exist. It was usually explained orally or written on scribes. They were able to do things such as find squad and cubed roots, solve linear equations, and explain fractions. As well as recognizing irrational numbers and identifying that quadratics had two answers. The next phase in algebra’s evolution was Syncopated math. This was most popularly used between 275 BCE - 1600. Diophantus played a crucial role in this stage by being the first to develop algebraic notations and abbreviations.

She published philosophical works clarifying facts and generalizing the into laws to form a system of sociology and psychology. Years later, Germain was the first person to provide proof of Fermat’s Last Theorem, that has been a struggle for mathematicians for the past 200 years. “That if xn + yn = zn, then neither x, y, nor z can be a natural number if n is greater than 2, stated from Fermat's Last Theorem" (Barrow-Green). Germain used her proficiency on prime numbers to help to prove this theorem. Successful from her work, she published her deeds in a supplement to her acquaintance Adrien-Marie Legendre's work Théorie des nombres.” (The Remarkable Life of Sophie Germain)

Abstract: - The many concepts we currently have in the field of mathematics are thanks to great mathematicians from different cultures throughout time. An important era when great mathematical discoveries were made was during Medieval Times, or the Middle Ages. In this paper we discuss important discoveries and contributions that were made by three famous mathematicians of this time period including French Nicole Oresme, German Jordanus Nemorarius and Italian Leonardo Pisano, better known for his nickname of Fibonacci.

Niels Henrik is perhaps the greatest mathematician to have come from Norway. Though his father was a philosophy teacher, he and Abel’s mother managed to acquire a large amount of debt due to scrupulous spending. Thus, much of Abel’s youth was spent well acquainted in poverty. His poverty continued to adulthood when his father died, and his studies had to be supported by his professors and through the Norwegian government. Before his university, he was granted a fellowship to travel and study and Europe, and the year and a half which he spent in Germany, Italy, and France was one of the happiest periods in his life. Yet, upon return to Norway, Abel could only attain temporary positions. The last years of his life were beset by many difficulties, but it was during this period of poverty and struggle that he developed his most famous proof of the insolvability of the quintic function. Though he died tragically young at twenty- seven, before any recognition, by tuberculosis, he left a bountiful legacy for mathematics. It was not until his death that the world became aware of his genius, for his proof of the insolvability of the

This abstract mathematical concept was conceived in the 17th century in order to drastically decrease the amount of time required for the multiplication of large numbers. The first to publish ideas on this particular subject is a Scottish mathematician named John Napier in his text Description of the Marvelous Canon of Logarithms in the year 1614. Years later, a Swiss mathematician named Joost Bürg published his work Arithmetic and Geometric Progression Tables which directly connected geometric and arithmetic

The history of algebra dates back to 1800 BC, in countries such as India, China, Greece, Persia and England. With the understanding that algebra and algorithms have a huge impact on our lives today, we will take a look at the history of algebra and algorithms, notable people that helped in their creation and the things that influenced what we know today about algebra and algorithms. This paper will go into depth in the influences of Islam on the creation and growth of algebra and algorithms as separate branches of mathematics. We will look at the Golden age of Islam and how it influenced many scientists, astronomers, geographers and physicians as well as mathematicians. Also we will take a look at one notable mathematician of that time, Al-Khwarizmi who created algebra and algorithms, who is known as the father of algebra.

Omar was also a poet, philosopher, and astronomer. Omar’s works were translated in 1851, which was research on Euclid’s axioms. In the medieval period, he expanded on Khwarizmi’s and the Greeks mathematic works. He only worked with cubic equations only and focused on geometric and algebraic solutions of equations. In 1145AD, Al-Khwarizmi’s book was translated by Robert Chester, which made it possible for algebra to be introduced to Europe. After algebra was introduced in Europe, European mathematicians developed and expanded on algebra concepts. Even though algebra began in the Arabic countries, once European mathematicians obtained the information of algebra, they became the leaders of mathematical discoveries in the world (“Mathematics”).

Fermat's proof is as follows. Let n be prime, and suppose it is equal to x2 -y2 that is, to (x+y)(x-y). Now, by hypothesis, the only basic, integral factors of n and n and unity, hence x+y=n and x-y=1. Solving these equations we get x=1 /2 (n+1) and y=1 /2(n-1). -He gave a proof of the statement made by Diophantus that the sum of the squares of two numbers cannot be the form of 4n-1. He added a corollary which I take to mean that it is impossible that the product of a square and a prime form 4n-1[even if multiplied by a number that is prime to the latter], can be either a square or the sum of two squares. For example, 44 is a multiple of 11(which is of the form 4 x 3 - 1) by 4, therefore it cannot be expressed as the sum of two squares. He also stated that a number of the form a2 +b2, where a is prime b, cannot be divided by a prime of the form 4n-1. -Every prime of the form 4n+1 is accurate as the sum of two squares. This problem was first solved by Euler, who showed that a number of the form 2(4n+1) can be always showen as the sum of two squares, of course it was Mr. Pierre de Fermat. -If a, b, c, are integers, a2 + b2= c2, then ab cannot be a square. Lagrange solved this. - The determination of a number x such that x2n+1 may be squared, where n is a given integer which is not squared. Lagrange gave a solution of this also. -There is only one integral solution of the equation x2 +4=y3. The required

What followed were centuries of attempts and frustrating failures, as mathematicians the world-over endeavored to challenge and solve Fermat’s marginal notation. In 1718, the prominent European mathematician Leonhard Euler proved the

It was Euclid’s Elements that inspired mathematicians to take it one step further and solve the things that Euclid had left out, such as squaring the circle with only a compass and a straightedge. The difficulty of these problems led mathematicians to change the way that they