Sometimes a theorem is so important that it becomes known as a fundamental theorem in mathematics. This is the case for the Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra states that every polynomial equation of degree n, greater than or equal to one, has exactly n complex zeros. In fact, there are many equivalent formulations: for example that every real polynomial can be expressed as the product of real linear and real quadratic factors. The Fundamental Theorem of Algebra can also tell us when we have factored a polynomial completely but does not tell us how to factor a polynomial completely. Carl Friedrich Gauss was the first person to completely prove the theorem. In this paper, we will explore the theorem in more depth.
Early studies of equations by al'Khwarizmi, an Islamic mathematician, only allowed positive real roots and so the Fundamental Theorem of Algebra was not relevant. It wasn’t until the 1500’s that an Italian mathematician by the name of Cardan was able to realize that one could work with “complex numbers” rather than just the real numbers. This discovery was made in the course of studying a formula, which gave the roots of a cubic equation. Bombelli, in his book Algebra, published in 1572, produced a set of rules for manipulating these "complex numbers”. In 1637, Descartes, the father of analytic geometry, observed that one could imagine for every equation of degree n, you would get n roots but these roots would not be a real number.
In 1629, a Flemish mathematician, Albert Girard, published a book called L’invention nouvelle en l’ Algebre. In his book, he claimed that there were always n solutions for equations of degree n. However he did not assert that solutions are of the form a + bi, w...
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...eneral theorem on the existence of a minimum of a continuous function.
Two years after Argand's proof, Gauss published a second proof of the Fundamental Theorem of Algebra. Gauss used Euler's approach but instead of operating with roots, Gauss operated with indeterminates. This proof was considered complete and correct. A third proof was written by Gauss and like the first, was topological in nature. In 1849 Gauss produced his 4th proof of the Fundamental Theorem of Algebra. This was different in that he proved that a polynomial equation of degree n with complex coefficients has n complex roots.
As you can see, the path to proving the Fundamental Theorem of Algebra started way back. Many mathematicians tried and progress was gradually made. Eventually, Gauss was the first to produce a complete theorem although it is thought that Gauss finished what Euler started.
The first proof, The Way of Motion, is about how things change in the world and how things are put into motion. Since you cannot infinitely regress backwards, there must be a first unmoved mover. This is understood to be God.
Geometry, a cornerstone in modern civilization, also had its beginnings in Ancient Greece. Euclid, a mathematician, formed many geometric proofs and theories [Document 5]. He also came to one of the most significant discoveries of math, Pi. This number showed the ratio between the diameter and circumference of a circle.
There was no discussion in regard to the use of computers and technology in this article, however by assumption there must have been high mathematical equations which must have needed the use of advanced technology like we have today.
In the book Practica Geometriae, geometry problems seemed to be his main focus. The book was arranged into 8 chapters with theorems based on Euclid's Elements and On Divisions. One can say that the authors of the books and him worked togetherbecause of the great influence he received from them. Once people found out about Fibonacci being a ge...
Through his studies, Galileo concluded that Copernicus was right. He was not the first to accept the theory. A German mathematician, Johannes Kepler, had already accepted the theory and was working on proving it with math. Galileo and Kepler wrote to each other about their findings regarding Copernicus’s theory. With his fancy new telescope, Galileo was able to further solidify the Copernican theory. However, even with mounting evidence many people could not jump on board an...
Michael Guillen, the author of Five Equations that Changed the World, choose five famous mathematician to describe. Each of these mathematicians came up with a significant formula that deals with Physics. One could argue that others could be added to the list but there is no question that these are certainly all contenders for the top five. The book is divided into five sections, one for each of the mathematicians. Each section then has five parts, the prologue, the Veni, the Vidi, the Vici, and the epilogue. The Veni talks about the scientists as a person and their personal life. The Vidi talks about the history of the subject that the scientist talks about. The Vici talks about how the mathematician came up with their most famous formula.
Analogous to Descartes' pursuit of finding a groundwork for his structure of thought, the scientific method calls for rigorous proofs of every part of the scientific framework, especially of its foundation. One result of this pursuit was that of giving calculus a solid structure and basis in the 19th century. Ev...
10 J.V. Field, Galileo Galilei. School of Mathmatics and St. Andrews, Scotland, August 1995; available from http://www.history.mcs.standrews.ac.uk/history/mathmatics/galileo.html;Internet.
It is said that without the help of previous discoveries in the world of math and science, we humans would not have the advancements that we see today. These discoveries did not simply fall from the sky, they were uncovered by geniuses across the globe. There are many famous mathematicians that have contributed to today’s world in almost every field of work. We have made significant groundbreaking discoveries with the help of great people such as Albert Einstein, Isaac Newton, and Aristotle. One prominent mathematician that contributed plenty to today’s world is Galileo Galilei.
He made advances in trigonometry, geometry and calculus. He is also credited for analyzing the infinite series. He made many innovations such as introducing the term ‘continued fraction’ and using the symbol for infinity for the first time. He is also said to be the initiator of the number line.
...icated and it was formed “on a continued fraction for the tanx function.” (Constant, 2014). Later on, in 1794 pi squared was also proved to be irrational by mathematician Legendre. It was not until 1882, that German mathematician Ferdinand von Lindemann proved pi to be transcendental. According to Wolfram MathWorld, a transcendental number is “a number that is not the root of any integer polynomial, meaning that it is not an algebraic number of any degree.” Steve Mayer writes in his article “The Transcendence of Pi” that the proof that indicates pi to be transcendental is not commonly known even though it is not difficult.
Under the understanding that math is created by humans as a method of comprehending the world, suspension of disbelief is essential. There is, nonetheless, grounds for debate on this topic as there are other individuals who claim mathematics is not created but discovered. This side of the argument understands that math is integrated into everything already and that humans have discovered or found ways to understand its existence. A main contender for such arguments is the Golden Ratio that is found in nature as well as among human creation. This infinite number has been discovered over and over again, firstly by mathematicians such as Phidias, Leonardo Fibonacci, and Luca Pacioli and represents the number of a divided line into a larger and smaller part, which when divided into one another equals the “whole length divided by the longer part”.
Carl Friedrich Gauss is revered as a very important man in the world of mathematicians. The discoveries he completed while he was alive contributed to many areas of mathematics like geometry, statistics, number theory, statistics, and more. Gauss was an extremely brilliant mathematician and that is precisely why he is remembered all through today. Although Gauss left many contributions in each of the aforementioned fields, two of his discoveries in the fields of mathematics and astronomy seem to have had the most tremendous effect on modern day mathematics.
The concept of impossible constructions in mathematics draws in a unique interest by Mathematicians wanting to find answers that none have found before them. For the Greeks, some impossible constructions weren’t actually proven at the time to be impossible, but merely so far unachieved. For them, there was excitement in the idea that they might be the first one to do so, excitement that lay in discovery. There are a few impossible constructions in Greek mathematics that will be examined in this chapter. They all share the same criteria for constructability: that they are to be made using solely a compass and straightedge, and were referred to as the three “classical problems of antiquity”. The requirements of using only a compass and straightedge were believed to have originated from Plato himself. 1
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculator science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry. Euclid, who lived around 300 BC in Alexandria, first stated his five postulates in his book The Elements that forms the base for all of his later Abu Abd-Allah ibn Musa al’Khwarizmi, was born abo...