How did Al Khwarizmi contribute to modern algebra?
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Here's what the Encyclopedia Brit has to say about him.
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Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer at the Parlement of Toulouse, southwestern France, and a mathematician who is given credit for the development of modern calculus. With his insightful theorems Fermat created the modern theory of numbers. The depth of his work can be gauged by the fact that many of his results were not proved for over a century after his death, and one of them, the Last Theorem, took more than three centuries to prove.\n\nBy the time he was 30, Pierre was a civil servant whose job was to act as a link between petitioners from Toulouse to the King of France and an enforcer of royal decrees from the King to the local people. Evidence suggests he was considerate and merciful in his duties.\n\nSince he was also required to act as an appeal judge in important local cases, he did everything he could to be impartial. To avoid socializing with those who might one day appear before him in court, he became involved in mathematics and spent as much free time as he could in its study. He was so skilled in the subject that he could be called a professional amateur.\n\nHe was mostly isolated from other mathematicians, though he wrote regularly to two English mathematicians, Digby and Wallis. He also corresponded with French mathematician, Father Mersenne who was trying to increase discussion and the exchange of ideas among French mathematicians. One was Blaise Pascal who, with Fermat, established a new branch of math - probability theory.\n\nBesides probability theory, Fermat also helped lay the foundations for calculus, an area of math that calculates the rate of change of one quantity in relation to another, for example velocity and acceleration. In particular, he is the precursor of differential calculus with his method of finding the greatest and the smallest ordinates of curved lines, analogous to that of the then unknown differential calculus.\n\nFermat himself was secretive and, since he rarely wrote complete proofs or explanations of how he got his answers, was mischievously frustrating for others to understand. He loved to announce in letters that he had just solved a problem in math but then refused to disclose its solution, leaving it for others to figure out.\n\nFermat's passion in math was in yet another branch - number theory, the relationship among numbers. While he was studying an ancient number puzzle book, he came up with a puzzle of his own that has been called Fermat's Enigma. Mathematicians worked for over three centuries to find its answer, but no one succeeded until 1994. Andrew Wiles, an English mathematician, created a proof and published it 330 years after Fermat's death in 1665.\n\nAlthough he carefully studied and drew inspiration from Diophantus, Fermat inaugurated a different tradition. Diophantus was content to find a single solution to his equations, even if it was a fraction. Fermat was only interested in integer solutions to his diophantine equations and he looked for all solutions of the equation. He also proved that certain equations had no solution, an activity which baffled his contemporaries.\n\nHe studied Pell's equation and Fermat, perfect, and amicable numbers. It was while researching perfect numbers that he created Fermat's theorem.\n\nHe created the principle of infinite descent and Fermat's factorization method.\n\nHe created the two-square theorem, and the polygonal number theorem, which states that each number is a sum of 3 triangular numbers, 4 square numbers, 5 pentagonal numbers, ...\n\nHe was the first to evaluate the integral of general power functions. Using an ingenious trick, he was able to reduce this evaluation to summing geometric series. The formula that resulted was a key hint to Newton and Leibniz when they independently developed the fundamental theorems of calculus.\n\nAlthough Fermat claimed to be able to prove all his arithmetical results, few of his proofs (if he had them) have survived. And c
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Algebra (from Arabic: الجبر, al-ğabr) is a branch of mathematics concerning the study of structure, relation and quantity. Elementary algebra is often taught in secondary education and gives an introduction to the basic ideas of algebra: studying what happens when numbers are added or multiplied, and how to make polynomials and find their roots.\n\nAlgebra is much broader than elementary algebra and can be generalized. Rather than working directly with numbers, one can work with symbols, variables, or elements of some set. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.\n\nTogether with geometry and analysis, algebra is one of the three main branches of mathematics.\n\nAlgebra (from Arabic: الجبر, al-ğabr) is a branch of mathematics concerning the study of structure, relation and quantity. Elementary algebra is often taught in secondary education and gives an introduction to the basic ideas of algebra: studying what happens when numbers are added or multiplied, and how to make polynomials and find their roots.\n\nAlgebra is much broader than elementary algebra and can be generalized. Rather than working directly with numbers, one can work with symbols, variables, or elements of some set. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.\n\nTogether with geometry and analysis, algebra is one of the three main branches of mathematics.
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Euclid. a Hellenistic scholar living in Egypt in the 3rd Centiry BC.\n\nHis magnum opus was Elements, which wikipaedia summarise as follows:\n\nAlthough Elements is a geometric work, it also includes results that today would be classified as number theory. The contents of the work are as follows:\n\nBooks 1 through 4 deal with plane geometry:\n\nBook 1 contains the basic properties of geometry: the Pythagorean theorem, equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area).\n\nBook 2 is commonly called the "book of geometrical algebra," because the material it contains may easily be interpreted as algebra.\n\nBook 3 deals with circles and their properties: inscribed angles, tangents, the power of a point.\n\nBook 4 is concerned with inscribing and circumscribing triangles and regular polygons.\n\nBooks 5 through 10 introduce ratios and proportions:\n\nBook 5 is a treatise on proportions of magnitudes.\n\nBook 6 applies proportions to geometry: Thales' theorem, similar figures.\n\nBook 7 deals strictly with number theory: divisibility, prime numbers, greatest common divisor, least common multiple.\n\nBook 8 deals with proportions in number theory and geometric sequences.\n\nBook 9 applies the results of the preceding two books: the infinitude of prime numbers, the sum of a geometric series, perfect numbers.\n\nBook 10 attempts to classify incommensurable (in modern language, irrational) magnitudes by using the method of exhaustion, a precursor to integration.\n\nBooks 11 through 13 deal with spatial geometry:\n\nBook 11 generalizes the results of Books 1–6 to space: perpendicularity, parallelism, volumes of parallelepipeds.\n\nBook 12 calculates areas and volumes by using the method of exhaustion: cones, pyramids, cylinders, and the sphere.\n\nBook 13 generalizes Book 4 to space: golden section, the five regular (or Platonic) solids inscribed in a sphere
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i love algebra...so yes
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Algebra, by Gelfand and Shen\n\nis textbook I taught myself algebra from in 3rd grade.  It's clear, concise, and elegant.
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"Descartes's chief contributions to mathematics were his analytical geometry and his theory of vortices, and it is on his researches in connection with the former of these subjects that his mathematical reputation rests. "\nhttp://www.maths.tcd.ie/pub/HistMath/People/Descartes/RouseBall/RB_Descartes.html\n\n\n"The most notable contribution that Descartes made to mathematics was the systematization of analytic geometry (see Geometry: Analytic Geometry). He was the first mathematician to attempt to classify curves according to the types of equations that produce them. He also made contributions to the theory of equations. Descartes was the first to use the last letters of the alphabet to designate unknown quantities and the first letters to designate known ones. He also invented the method of indices (as in x2) to express the powers of numbers. In addition, he formulated the rule, which is known as Descartes's rule of signs, for finding the number of positive and negative roots for any algebraic equation."\nhttp://encarta.msn.com/encyclopedia_761555262/Rene_Descartes.html\n\n\nDescartes made a number of important contributions to mathematics and physics, among the most enduring of which was his foundation (with Galileo) of what is now known as analytic geometry. That is, broadly speaking, the use of geometrical analysis to solve complex algebraic problems, and vice versa. It is difficult to overestimate the importance for the history of mathematical physics of this bringing together of the sciences of geometry and algebra. \nhttp://www.iep.utm.edu/d/descarte.htm\n\n\n" 'This one thing [analytic geometry] is of the highest order of excellence, marked by the sensuous simplicity of the half dozen or so greatest contributions of all time to mathematics. Descartes remade geometry and made modern geometry possible. '(E. T. Bell) \n\nInventor of analytic and coordinate geometry. Cartesian coordinates. "\nhttp://scidiv.bcc.ctc.edu/Math/Descartes.html
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Yes, i learned basic algebra in junior high.\nI see no reason why a high school senior would be incapable of learning it.
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It *is* awfully hard to write algebra expressions on Yahoo Answers, isn't it? Oh well, we do what we can.\n\nAddressing your first question, let's write it out like a long division problem.  You'll have to imagine the horizontal bar above the larger polynomial, because I can't format it properly here:\n\nx^2 + 3x - 2  )  x^4 + x^3 - 5x^2 + 13x - 6\n\nOur first step is to look at the first term in each polynomial, and divide the smaller into the larger.  x^4 / x^2 = x^2, so the first term we write in our quotient (above the bar) is x^2.  You can write it directly above the x^4, if you wish, or above the 5x^2 term, your preference.\n\nNow, we're going to multiply that "partial quotient" by the entire divisor (x^2 + 3x - 2), which gives us x^4 + 3x^3 - 2x^2.  Write that "partial product" below the first three terms of the original dividend (x^4 + x^3 - 5x^2).  Our next step is to subtract the partial product: (x^4 + x^3 - 5x^2) - (x^4 + 3x^3 - 2x^2) = -2x^3 - 3x^2.  We then drop down the next term from the dividend, the 13x term, and repeat:\n\n-2x^3 / x^2 = -2x (add that term to our quotient)\n\n-2x * (x^2 + 3x - 2) = -2x^3 - 6x^2 + 4x\n\n(-2x^3 - 3x^2 + 13x) - (-2x^3 - 6x^2 + 4x) = 3x^2 + 9x\n\nFinally, we drop down the last term from the dividend, -6, and repeat once more.  I'm sorry that I can't write this out the way your math book can... I really hope Yahoo Answers creates some sort of way of inputting formulas, or at least monospacing text entry!\n\n3x^2 / x^2 = 3 (so we add a +3 term to our quotient)\n\n3 * (x^2 + 3x - 2) = 3x^2 + 9x - 6\n\n(3x^2 + 9x - 6) - (3x^2 + 9x - 6) = 0... no remainder!\n\nSo our final quotient is x^2 - 2x + 3, with no remainder.\n===\nNow, when doing the 2nd problem, you'll want to write the larger polynomial, the dividend, as x^4 - 3x^3 + 0x^2 + 6x - 5.  We incorporate that extra 0x^2 term so that we never find ourselves trying to subtract unlike terms.\n\nSorry there's no better way to write this out on Yahoo Answers.  Hope that helped anyway!
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www.algebra-online.com\nwww.sosmath.com/algebra/algebra.html\nmathworld.wolfram.com/topics/Algebra.html\nmathforum.org/algpow\nhttp://education.yahoo.com/homework_help/math_help/chapter_list?id=minialg1gt\nwww.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra\nmath.about.com/od/collegealgebra\n\nGOOD LUCK!! :) Happy studying!
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Garrett Birkhoff\nFrom Wikipedia, the free encyclopedia\nJump to: navigation, search\n \nGarrett Birkhoff.Garrett Birkhoff (January 19, 1911, Princeton, New Jersey, USA - November 22, 1996, Water Mill, New York, USA) was an American mathematician.\n\nThe son of the mathematician George David Birkhoff, Garrett began the Harvard University BA course in 1928 after less than seven years of prior formal education. Upon completing his Harvard BA in 1932, he went to Cambridge University in England to study mathematical physics but switched to studying abstract algebra under Philip Hall. While visiting the University of Munich, he met Carathéodory who pointed him towards two important texts, Van der Waerden on abstract algebra and Speiser on group theory.\n\nBirkhoff held no Ph.D., a qualification British higher education did not emphasize at that time, and did not even bother obtaining an MA. Nevertheless, after being a member Harvard's Society of Fellows,1933-36, he spent the rest of his career teaching at Harvard. From these facts can be inferred the number and quality of Birkhoff's papers published by his 25th year.\n\nDuring the 1930s, Birkhoff, along with his Harvard colleagues Marshall Stone and Saunders MacLane, substantially advanced American teaching and research in abstract algebra. In 1941 he and MacLane published A Survey of Modern Algebra, the first undergraduate textbook in English on the subject. Birkhoff and MacLane (1967) is a more advanced text on abstract algebra. A number of papers he wrote in the 1930s, culminating in his monograph, Lattice Theory (1940; the third edition remains in print), turned lattice theory into a major branch of abstract algebra. His 1935 paper, "On the Structure of Abstract Algebras" founded a new branch of mathematics, universal algebra. Birkhoff's approach to this subject owed little to Alfred North Whitehead's 1898 monograph bearing the same name.\n\nDuring and after WWII, Birkhoff's interests gravitated towards what he called "engineering" mathematics. During the war, he worked on radar aiming and ballistics. This weapons-related work culminated in his texts on fluid dynamics, Hydrodynamics (1950) and Jets, Wakes and Cavities (1957). Birkhoff, a friend of John von Neumann, took a close interest in the rise of the electronic computer. His research and consulting work (notably for General Motors) began to employ computational methods, such as numerical linear algebra and the representation of smooth curves via cubic splines.\n\nBirkhoff published more than 200 papers and supervised more than 50 Ph.Ds. He was a member of the National Academy of Sciences and the American Academy of Arts and Sciences.
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Like many mathematical ideas, symbolic algebraic notation arose almost simultaneously in many parts of the world as the need was found for it. It seems it was generally accepted to the greatest degree in the 14th century. See http://www.cs.xu.edu/math/math147/02f/algebra/algebra.html for more information.
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The formula is credited to Heron of Alexandria in the 1st century, and a proof can be found in his book Metrica. It is now believed that Archimedes already knew the formula, and it is of course possible that it has been known long before.\n\nLet the sides of a triangle have lengths a,b and c. Introduce the semiperimeter p = (a + b + c)/2 and the area S. Then Heron's formula asserts that\n S2 = p(p - a)(p - b)(p - c)\n\nW.Dunham analyzes the original Heron's proof in his Journey through Genius.\n\nFor the right triangle with hypotenuse c, we have S = ab/2. We'll modify the right hand side of the formula by noting that\n p - a = (- a + b + c)/2, p - b = (a - b + c)/2, p - c = (a + b - c)/2\n\nIt takes a little algebra to show that\n \n(a + b + c)(- a + b + c)(a - b + c)(a + b - c)\n= 2a2b2 + 2a2c2 + 2b2c2 - (a4 + b4 + c4)\n\nFor the right triangle, this expression is equal to 16S2 = 4a2b2. So we have\n 4a2b2= 2a2b2 + 2a2c2 + 2b2c2 - (a4 + b4 + c4)\n\nTaking all terms to the left side and grouping them yields\n (a4 + 2a2b2 + b4) - 2a2c2 - 2b2c2 + c4 = 0\n\nWith a little more effort\n (a2 + b2)2 - 2c2(a2 + b2) + c4 = 0\n\nAnd finally\n [(a2 + b2) - c2]2 = 0\nRemark\n\nFor a quadrilateral with sides a, b, c and d inscribed in a circle there exists a generalization of Heron's formula discovered by Brahmagupta. In this case, the semiperimeter is defined as p = (a + b + c + d)/2. Then the following formula holds\n S2 = (p - a)(p - b)(p - c)(p - d)\n\nSince any triangle is inscribable in a circle, we may let one side, say d, shrink to 0. This leads to Heron's formula.\n\nFor more information you can go to www.wikipedia.com  \nhttp://mathworld.wolfram.com/HeronsFormula.html\nhttp://www.cut-the-knot.org/pythagoras/herons.shtml\n\nI hope this helps! =-)
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Differential equations.\nLinear algebra.\nNumerical analysis.
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http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut35_polyfun.htm\n\nhttp://www.sosmath.com/algebra/algebra.html\n\nI found these to be helpful.
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Try this:\n\nhttp://www.algebra.com/
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looks like straightforward algebra, and isn't really relevant whether the formulas are electrical