Mathematical Algebra

The word "algebra" comes from Arabic: al-jebr because the subject has been studied and written about in something like the modern sense, by scholars who spoke Arabic in what is now the Middle East in the 9th century CE. Although the classical Greeks and several of their predecessors and contemporaries had studied the problems we now call "algebraic", these studies were aware of Speakers of European languages that are not from the classical sources but from Arabic writers. So that's why we use a term derived from Arabic.

Muhammad ben Musa al-Khwarizmi seems to have been the first person whose writing uses the term al-jebr. When he used the word refers to a technique for solving equations by performing operations like addition or multiplication on both sides of the equation - as taught in the first year high school algebra. Al-Khwarizmi, of course not use our modern notation with Latin letters for unknown and symbols like "+", "x" and "=". Instead, he expressed everything in common words, but in a way similar to our modern symbolism. 

The word al-jebr itself is a metaphor, as the usual meaning of the word referred to the setting or straightening out of the fractures. The same metaphor exists in Latin and related languages, as in the English word "reduce" and "reduced". Although now usually refer to doing something smaller, older meaning refers to making something simpler or more straight. The Latin root is the verb wards keeping, to drive - and thus to re-produce is to bring something back to a simpler from a more complex condition. In elementary algebra still talking about "reducing" fractions to lowest terms and simplifying equations. 

The essence of the study of algebra, so loose or "reduce" equations to the simplest possible form. The emphasis is on finding and describing explicit methods to perform this simplification. Such methods are known as algorithms - in honor of al-Khwarizmi. Different types of methods can be used. Guessing at solutions, for example, is a method. One can often by trying long enough, guess the exact solution of a simple equation. And if you have a guess that is close but not exactly changing that little guess you can get a better solution by an iterative process with each other. This is a perfectly acceptable way to "solve" equations to many practical purposes - so much that it is the method generally used by computers (where irrational numbers are only around anyway). Some approximation methods are very imaginative, like "Newton's method for finding roots of polynomial equations - but they are still mainly based on guessing a first rough answer. 

Another method for finding solutions of equations using the geometric design. One can construct geometric shapes, where the length of a particular segment is a solution to a given equation. This works well, for example, when square roots are needed, since the hypotenuse of a right triangle has a length which is the square root of the sum of the squares of the other two sides of the triangle. That is, if the lengths of the sides are a, b and c, then a2 + b2 = c2 and hence c = √ (a2 + b2). If a and b are integers, so is the sum of their squares. Algorithms to find the approximate square root of an integer known as C can be calculated approximately. But with a geometric construction could c there simply be to measure the length of the straight line segment. For future reference, note that an interesting problem is to find two numbers a and b such that for some given number d, d = a2 + b2. This is because if d is given, to find a and b makes it possible to find the square root of d from a geometric design. 

In addition to such laws and geometric methods, al-Khwarizmi was interested in methods for finding exact solutions of a series of steps that can be described in cookbook style - algorithms. This can be very successful for linear equations of the form ax + b = c (in modern notation), using only arithmetic operations of addition, subtraction, multiplication and division. Equally important, the operations symbolically - not just on specific figures, but the symbols stand for "a number". And so, one seeks to express the solution of a given equation in a symbolic form. 

An interesting question is how the idea to represent the equations in symbolic form arose. It is not easy to answer such questions, partly because the symbols have been used before their full and high usability has been recognized. For example, say the Greek geometer lines of their figures with individual characters, so it was natural to write what we now recognize as the Pythagorean theorem in the form a2 + b2 = c2. But the significance of this representation was somewhat blurred, since the distinction between a line and length of a line that was not fully appreciated. Actually applied themselves Greeks and other early mathematicians (eg in India) symbolic equations, al-Khwarizmi did not. (It is therefore likely that he knew of Greek mathematics, and much of his work was original if it is not always as advanced as the Greeks.) 

In modern notation, polynomial equations can be classified in relation to the highest power of a variable occurring in them. We call a linear equation, if the highest power is one, because its graph is a straight line. If there is only one variable, such an equation, which is the most common form ax + b = 0 If the highest power is two, the equation is called the "square" and have the form AX2 + bx + c = 0 (Why the Latin prefix quad, usually associated with the number 4, going on here? Simply because the word "square" is the Quadra in latin.) Despite missing a symbolic representation of equations, al-Khwarizmi actually did quadratic formula says that there are two solutions to the last equation can be written as x = (-b ± √ (b2-4ac)) / 2a. He also realized that the equation has solutions for all in the form of "real" numbers only if the quantity we now call the discriminant, b2-4ac, is not negative. 

The highest power of an unknown, occurring in a given polynomial equation is known as the degree of the equation. Although al-Khwarizmi did not seem to have studied equations of degree 3 is called cubic equations, a more famous successor, who lived about 250 years later, were: Omar Khayyam (ca. 1050-1123), a Persian. Like its predecessor, has Khayyam not work with symbolic expressions of equations. But he was able to produce solutions using geometric constructions (with conic sections), provided a positive solution exists. He also thought, mistakenly, that such solutions could not be found by algebraic (algorithmic) methods of the sort al-Khwarizmi used. 

The next major step forward in solving equations began when researchers in Western Europe began to study and appreciate the work of people like al-Khwarizmi and Khayyam. Most prominent among these researchers was Leonardo of Pisa, better known as Fibonacci (ca. 1180-1250). He showed that algorithmic (as opposed to geometric) methods could be used to find solutions to some cubic equations. Fibonacci had a much more obscure contemporary Jordanus Nemorarius, there is limited knowledge from several books attributed to him on arithmetic, mechanics, geometry and astronomy. He made a more systematic use of letters to stand for "variable" (not necessarily known) quantities, equations, but the importance of this technique was still not much appreciated. 

With the advent of the Renaissance, advances in mathematics began to accelerate. One of the first big names was a German, Regiomontanus (1436-76). Although he produced less original work than others, he was much read in the classic works of both the Greeks and the Muslim world. In particular, he had studied the arithmetic of Diophantus of Alexandria (who was active around 250 CE) in the original Greek, and even considered publishing a Latin translation, though he never got it. Diophantus was in some respects more advanced than any other mathematician prior to the Renaissance, and among the problems he thought was what is now called Diophantine equations. The relevance of these problems will be explained in due time. 

Something more original than Regiomontanus was a Frenchman, Nicolas Chuquet who

died around 1500. He used expressions involving nested radicals relatively close to the modern style, such as √ (14 - √ 180)), to represent the solutions of 4 degree equations. 

The real breakthrough came in the work of several Italians in the 16th century. In 1545 Girolamo Cardano (1501-76) was published explicit algebraic solutions (that is, using arithmetic operations and extraction of roots) of both cubic and quartic (4th degree) equations. Cardano, however, did not find the solutions themselves. The result for the cubic's were known before 1541 by Niccolo Tartaglia (ca. 1500-57), but apparently discovered even earlier by Scipione Del Ferro (ca. 1465-1526). Cardano admitted that he had not found the solution, but apparently he broke a promise to Tartaglia to keep the results secret. (Just as now, was before the publication of new scientific findings are the subject of great prestige.) Regarding quartic, Cardano, the solution was discovered by Ludovico Ferrari (1522-65), but in his [Cardano's] request. 

Such rapid progress naturally raised the question of solutions to equations of 5 degree (quintics) and higher education, either by algebraic way (using arithmetic operations and radicals), or at least by means of geometric constructions (using only ruler and compass). Surprisingly, it was established, nearly 300 years later, the solutions of either kind were not possible in general, ie for all cases. This was done independently by two young men, Niels Henrik Abel (1802-29) in 1824 and Everest Galois (1811-32) in 1832. Galois 'result is especially important because it is based on entirely new methods of abstract algebra - the theory of groups - and in fact Galois' ideas thoroughly permeate the theory of algebraic numbers, be discussed. 

Despite this astonishing negative result) is only a few years earlier Carl Friedrich Gauss (1777-1855 had identified in his thesis in 1798 that polynomial equations of degree n must have exactly n solutions to some very specific meaning. This result was so important, it became known as the basic algebra.