Will elementary algebra disappear with the use of new graphing calculators?.
What do we understand elementary algebra to be? Elementary algebra is the language with which we communicate the majority of mathematics. Thanks to algebra we can work with concepts at an abstract level and then apply them.
Elementary algebra begins as a generalization of arithmetic and then focuses on its own structure and greater logical coherence. From there comes the importance of the various uses of algebraic symbols. When we write A + B, we can be indicating the sum of two natural numbers, the sum of two algebraic expressions, or even the sum of two matrices. Thus there is, at first, representations and symbolism, and later the development of algorithms and procedures to work formally with algebraic expressions. But what we today understand to be algebra has been the fruit of the efforts of many generations that have been contributing their grains of sand in constructing this magnificent building.
It seems that the Egyptians already knew methods for solving first degree equations. In the Papyrus of Ahmes (1650 BC) one finds: "Calculate the value of an amount if that amount and a seventh part of that amount equals 19". But to solve such problems, they used arithmetic methods, such as, for example, the rule of false position. The Babylonians solved some simple systems of linear equations. Also, according to a discovery by Neugebauer in 1930, the Babylonians handled quadratic equations with great skill.
By the 6th Century BC the deductive method appeared in Greek mathematics, and by the 4th Century BC what can be called geometric algebra with such exercises as: "Given the sum and the product of the sides of a rectangle, find those sides". Some of that geometric algebra was treated by Euclid in his Elements.
But the most important of the Greek algebraists was Diophantus of Alexandria. Little is known of his life, even though on his tomb there is an inscription that translates to a linear equation that gives us some information about him. Diophantus can be considered to be the father of ancient algebra. His most important work was Arithmetic, but of its thirteen books only the first six have survived. It is not an algebra text, but a collection of problems about the application of algebra. The influence of Diophantus has been greater that is often believed. Even Pierre de Fermat (1601-1665) arrived at his famous last theorem when he tried to generalize a problem that he had seen in Diophantus' Arithmetic: "decompose a square given the sum of two other square".
The Indian mathematicians Brahmagupta (598-?) and Bhaskara (1114-1185) contributed general solutions of quadratic equations to the development of algebra, including those of two roots with one of them negative. One of the most distinguished members of the House of Wisdom in Baghdad was the mathematician and astronomer, Al-Khowarizmi (around 780-850). In his work, entitled Algebra, he studied in detail the six types of linear and quadratic equations that have positive roots. His way of solving them was essentially geometric, connected thus to the Greek algebra of Euclid.
But classic algebra, as we know it today, was developed more completely during the Renaissance, which is when the first break between ancient and classic algebra occurred. In 1494, the Italian Lucas Pacioli (1445-1514) published Summa Arithmetica in which he included the solution of first and second degree equations. In that work, a rudimentary symbolic algebra began to be used. Pacioli was very pessimistic about the possibility of solving cubic equations, and thought that cubic, not to mention quartic, equations were beyond the reach of algebra. Scipione Ferro (1465-1526) accepted the challenge of Pacioli and arrived at a formula for cubic equations of the form ax3 + cx + d = 0. He revealed his discovery to his disciple, Antonio de Fiore (1506-?). Niccolo Fontana (Tartaglia) (1500-1557) claimed to be able to solve cubic equations of the form x3+ mx2 = n. However, he apparently did not know how to solve those of the form ax3 + cx + d = 0. Challenged by Fiore, he managed to find solutions to these equations. Ludovico Ferrari (1522-1565) was able to find a procedure for solving fourth degree algebraic equations.
Girolamo Cardan (1501-1576) published Ars Magna in which he included the method for solving some kinds of cubic equations that Tartaglia had shown him, even though he promised not to make them public. It is today known as "solution by radicals". But if negative numbers were considered to be suspect in the Sixteenth Century, their square roots were deemed to be totally absurd. It was, however, the beginning of the appearance of what are now called complex numbers.
In 1572 Rafael Bombelli (1526-1573) published his treatise,
Algebra, in which he gave one more step in the solution of
cubic equations, expressing solutions in a form such as 
. We should thank Bombelli for
discovering that imaginary numbers play an important role in the
development of algebra.
The French mathematician François Vieta (1540-1603) proposed a new focus for the solution of cubic equations. He also began the study of the relationship between the roots and the coefficients of an equation. That work was completed by the Flemish mathematician Albert Girard (1590-1633) with the publication of Invention nouvelle en l'algèbre in 1629. Rene Descartes (1596-1650) in his La Géométrie: Livre Premier included a detailed system of instructions for solving quadratic equations, but with geometric methods like those used by the ancient Greeks.
Neither Cardan nor Ferrari were able to find solutions to fifth degree equations. It was not until 1824 that the young Norwegian mathematician Neils Abel (1802-1829) shook the entire scientific community by proving that it was impossible to solve fifth or higher degree equations by radicals. This did not mean that fifth or higher degree equations did not have solutions, it just meant that there did not exist an algebraic method that permitted obtaining the roots of an equation as a function of its coefficients. The discovery of Abel links to the pessimism of Lucas Pacioli and shows the failure of algebra when you go beyond the fourth degree.
Girard, facing the problem of taking square roots of negative numbers, was, in 1629, the first to dare to conjecture the following: "An equation of degree n has exactly n roots, as long as you count the impossible ones". Later, Euler (1707-1783), Lagrange (1736-1813) and D'Alembert (1717-1783), studied the same theorem, but without thinking that the roots could be complex numbers. It is to Gauss (1777-1855) that we owe the name of this fact, The Fundamental Theorem of Algebra. In his doctoral dissertation, finished in 1799, he critiqued the work of Euler, Lagrange and D'Alembert, and gave a proof of the theorem, based on geometric considerations, which was not completely convincing. In 1816 he published two new proofs and five years before his death he published the fourth proof attempting to find purely algebraic procedures.
At the beginning of the Eighteenth Century, the English mathematician Roger Cotes (1682-1716) and the French immigrant to England, Abraham de Moivre (1667-1754), reduced the solution of the equation zn -1 = 0, to the division of the circumference of a circle in n equal parts, thus showing a good understanding of complex numbers. But algebra continued to develop and in the Eighteenth and Nineteenth Centuries another rupture occurred, this time between classic algebra and modern algebra. This was due to the contributions, among others, of George Peacock (1791-1858). He initiated axiomatic thinking, that later would be developed by Augustus de Morgan (1806-1871), Sir William Rowan Hamilton (1805-1865), and George Boole (1815-1864). We should not forget the two most prolific algebraists of the time, the Englishmen Arthur Cayley (1821-1895) with his study of matrices and J.J. Sylvester (1814-1897) with the theory of invariants. Finally, Evariste Galois (1811-1832) with the creation group made that abstract concept the central idea of the theory of algebraic equations, signaling an end to the era of classic algebra.
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Gómez, P. & Waits, B. (Eds.) (1996). Roles of
calculators in the classroom.
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