Algebra

Speaking more precisely, that part of mathematics which considers the relations and properties of numbers by the aid of general symbols, usually letters (a, b, c,...x, y, z)(a, b, c,...x, y, z) and signs of operation (+, -, x, ...)(+, -, x, ...) and relation (<, >, =, ...)(<, >, =, ...). For example, the statement that the area of a rectangle is equal to the product of the number of units of length in the base multiplied by the number of units in the height, is algebraically expressed by the symbols A = bhA = bh, where AA stands for the number of square units of area; bb, for the number of linear units in the base; and hh, for the number of linear in the height. Similarly, the expression x**2x² + 5x = 14 + 5x = 14 states that there is a number (represented by x) such that its square, plus five times itself, is equal to 14. Algebraic symbolism and operations enter into nearly all branches of science, including the various subdivisions of mathematics. In certain cases, however, as in vector analysis, the letters are not restricted to the representation of numbers. In its broader sense algebra treats of equations, polynomials, continued fractions, series, number sequences, forms determinants, and new types of numbers. It considers the fundamental theorem that every integral equation f(x) with real coefficients has at least one real or complex root, indeterminate equations, general algebraic equation of the third and forth degrees, numerical higher equations, and it enters into such important branches as the calculus, trigonometry and the theory of functions.

Changes in Scope. If by the word algebra we mean that branch of mathematics by which we learn how to solve the equation x**2x² + 5x = 14 + 5x = 14, written in this way, the science begins in the 17th century. If we allow the equation to be written with order and less convenient symbols, it may be considered as beginning at least as early as the 3rd century. If we permit it to be stated in words and solved, for simple cases of positive roots, by the aid of geometric figures, the science was known to Euclid and others of the Alexandrian school as early as 300 B.C. If we permit of more or less scientific guessing in achieving a solution, algebra may be said to have been known nearly 2000 years B.C., and it had probably attracted the attention of the intellectual class much earlier. The scope of all early algebra was limited to a study of equations or to the solution of problems which at present would be sold by their aid. In the 16th century, after the advent of the printed book in Europe, the field was enlarged through the efforts of men like Christoff Rudolff, Robert Recorde, Rafael Bombelli and Christofer Clavius, becoming more of a generalized arithmetic, the fundamental operations with numbers being duplicated with rather crude algebraic symbols. The perfecting of symbolism in the 17th century greatly extended the domain of algebra and rendered possible the development of the higher branches of the subject.

Changes in Name. The name "algebra" is quite fortuitous. When Mohammed ibn Musa al-Khowarizmi (Mohammed, son of Moses, the Khowarezmite(Mohammed, son of Moses, the Khowarezmite), a native of Khowarezm, wrote in Baghdad, he gave to one of his works the name Al-jebr w'al-muqabalah. The title is sometimes translated as "restoration and equation", but the meaning was not clear even to the later Arab writers. Of late it has been thought that al-jebr is Arabic, while muqabalah is from Persian, and that each meant or referred to an equation. At any rate, al-Khowarizmi work was the first to bear the title "algebra", and the treatise was so important as to cause the name to be adopted, often with strange variations in spelling, by later writers. Various other names have been given to the science, such as arithmetica, Bija Ganita (Brahmagupta's Hindu treatise, the term meaning calculation with primitive elements), T'ien-yuen (Chinese, "celestial element")(Chinese, "celestial element"), Kigen seiho (one of the Japanese names, referring to "revealing unknowns")(one of the Japanese names, referring to "revealing unknowns"), Fakhri (al-Karkhi, who gave this name to his algebra in honour of his patron, Fakhr al-

Mulk), Regola de la cosa (Rule of the cosa, the unknown quantity)(Rule of the cosa, the unknown quantity), Ars magna ("great art", used by Cardan in 1545)("great art", used by Cardan in 1545), the German Die Coss and English Cossiked to the science of arte (both in the 16th century)(both in the 16th century). Most of these names referred to the science of the equation, and this is the meaning assigned to elementary algebra in certain European languages at present, the fundamental operations with literal expressions being then included under the term "arithmetic".

Summary of the changes in meaning. We may, therefore, summarize the leading steps in the growth of algebra as follows: (1) The period of the puzzle problem relating to numbers, with little or no symbolism; (2) The inclusion of the geometric problem of completing the square, thus leading to the finding of a line that would be represented by x x in a modern quadratic equation; (3) The development of a systematic although a crude symbolism, applied to the theory of numbers. as in the Arithmetica of Diophantus; (4) A more critical study of equations with some approach to scientific treatment, as in the period of Muslim ascendancy; (5) The rise of the theory of equations, beginning with the solution of the cubic and the biquadratic in the 16th century; (6) The development of the convenient symbolism, chiefly in the century 1550 - 1650, changing algebra from a crude theory of equations to an analytic subject concerned with algebraic numbers and polynomials; and (7) The modern development of higher algebra.

Speaking more precisely, that part of mathematics which considers the relations and properties of numbers by the aid of general symbols, usually letters (a, b, c,...x, y, z) and signs of operation (+, -, x, ...) and relation (<, >, =, ...). For example, the statement that the area of a rectangle is equal to the product of the number of units of length in the base multiplied by the number of units in the height, is algebraically expressed by the symbols A = bh, where A stands for the number of square units of area; b, for the number of linear units in the base; and h, for the number of linear in the height. Similarly, the expression x**2 + 5x = 14 states that there is a number (represented by x) such that its square, plus five times itself, is equal to 14. Algebraic symbolism and operations enter into nearly all branches of science, including the various subdivisions of mathematics. In certain cases, however, as in vector analysis, the letters are not restricted to the representation of numbers. In its broader sense algebra treats of equations, polynomials, continued fractions, series, number sequences, forms determinants, and new types of numbers. It considers the fundamental theorem that every integral equation f(x) with real coefficients has at least one real or complex root, indeterminate equations, general algebraic equation of the third and forth degrees, numerical higher equations, and it enters into such important branches as the calculus, trigonometry and the theory of functions.

Changes in Scope. If by the word algebra we mean that branch of mathematics by which we learn how to solve the equation x**2 + 5x = 14, written in this way, the science begins in the 17th century. If we allow the equation to be written with order and less convenient symbols, it may be considered as beginning at least as early as the 3rd century. If we permit it to be stated in words and solved, for simple cases of positive roots, by the aid of geometric figures, the science was known to Euclid and others of the Alexandrian school as early as 300 B.C. If we permit of more or less scientific guessing in achieving a solution, algebra may be said to have been known nearly 2000 years B.C., and it had probably attracted the attention of the intellectual class much earlier. The scope of all early algebra was limited to a study of equations or to the solution of problems which at present would be sold by their aid. In the 16th century, after the advent of the printed book in Europe, the field was enlarged through the efforts of men like Christoff Rudolff, Robert Recorde, Rafael Bombelli and Christofer Clavius, becoming more of a generalized arithmetic, the fundamental operations with numbers being duplicated with rather crude algebraic symbols. The perfecting of symbolism in the 17th century greatly extended the domain of algebra and rendered possible the development of the higher branches of the subject.

Changes in Name. The name "algebra" is quite fortuitous. When Mohammed ibn Musa al-Khowarizmi (Mohammed, son of Moses, the Khowarezmite), a native of Khowarezm, wrote in Baghdad, he gave to one of his works the name Al-jebr w'al-muqabalah. The title is sometimes translated as "restoration and equation", but the meaning was not clear even to the later Arab writers. Of late it has been thought that al-jebr is Arabic, while muqabalah is from Persian, and that each meant or referred to an equation. At any rate, al-Khowarizmi work was the first to bear the title "algebra", and the treatise was so important as to cause the name to be adopted, often with strange variations in spelling, by later writers. Various other names have been given to the science, such as arithmetica, Bija Ganita (Brahmagupta's Hindu treatise, the term meaning calculation with primitive elements), T'ien-yuen (Chinese, "celestial element"), Kigen seiho (one of the Japanese names, referring to "revealing unknowns"), Fakhri (al-Karkhi, who gave this name to his algebra in honour of his patron, Fakhr al-

Mulk), Regola de la cosa (Rule of the cosa, the unknown quantity), Ars magna ("great art", used by Cardan in 1545), the German Die Coss and English Cossiked to the science of arte (both in the 16th century). Most of these names referred to the science of the equation, and this is the meaning assigned to elementary algebra in certain European languages at present, the fundamental operations with literal expressions being then included under the term "arithmetic".

Summary of the changes in meaning. We may, therefore, summarize the leading steps in the growth of algebra as follows: (1) The period of the puzzle problem relating to numbers, with little or no symbolism; (2) The inclusion of the geometric problem of completing the square, thus leading to the finding of a line that would be represented by x in a modern quadratic equation; (3) The development of a systematic although a crude symbolism, applied to the theory of numbers. as in the Arithmetica of Diophantus; (4) A more critical study of equations with some approach to scientific treatment, as in the period of Muslim ascendancy; (5) The rise of the theory of equations, beginning with the solution of the cubic and the biquadratic in the 16th century; (6) The development of the convenient symbolism, chiefly in the century 1550 - 1650, changing algebra from a crude theory of equations to an analytic subject concerned with algebraic numbers and polynomials; and (7) The modern development of higher algebra.