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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.

The numbers used in counting objects have long been known as natural numbers. Numbers that are not naturally used in counting objects are commonly called artificial numbers, a term that is open to certain objections. From various points of view the number 3/4 is as natural as 2, and sq.root of 5 as naturally comes into use as 5, although we cannot look at an object 3/4 of a time nor can we pick up a book sq.root of 5 times. The term is convenient, however, even though it may be as inappropriate as "imaginary" in connection with sq. root of -3, or "fraction" (in its primitive sense) in connection with 2/2, 5/2, 1/2/3/4. The preceding discussion leads to a much larger and more important question as to the meaning of number itself.

Further, on the algebraic scale we can represent such fractions as 1/2 and -3/4, and such surd numbers as -sq. root of 2 and 3 sq. roots of 5, so these are as real in certain situations as integers themselves.

In the expression a/b, in which a and b are natural numbers, does not denote a natural number, it is called a fraction, but since division by zero has no meaning, the case in which b = 0 is excluded. Integers and fractions are classified as rational numbers. More generally speaking, a rational algebraic fraction is the quotient of any integral function by any other integral function. Certain numbers such as -sq. root of 2 and 3 sq. roots of 5 do not come within a definition of rational numbers, and are called irrational numbers. In elementary algebra we also meet with certain numbers represented by the symbol a sq.root of -1, and these are called imaginary numbers. Numbers like a+b sq. root of -1 are called complex numbers.

Uses of Irrational Numbers.

In the solution of quadratic equations, excepting those artificially constructed to give only integral values of the unknown quantity, the roots are generally irrational, not being expressible as the quotient of a/b, where a and b are integers. This is seen in the simple quadratic x²=2, where . Since early arithmetic was concerned largely with rational numbers, the irrational ones were generally assigned to algebra, the branch of mathematics in which they were needed. The purpose of placing them there was soon obscured, however, the result being a much more extensive treatment of the subject than was warranted by any practical considerations. If, in a physical problem, there is need for solving the equation x²=243, it is important to find the value of x to a definite degree of approximation. If it is stated that , nothing is gained by writing this result as . An expression like and , but (for no very good reason), not like , is known as a surd, from a medieval Latin translation of an Arabic rendering of the Greek al'ogos (irrational). Until recently the finding of the square root of a binomial surd, as of or of , was a familiar operation in elementary algebra, and there is still good cause for complaint that the work in surds is excessive. With the properties of such transcendental numbers as e and , elementary algebra is little concerned.

Factors.

The factorizing of polynomials has a place in the theory of equations and in the advanced study of polynomials, but its value in elementary algebra is slight. In the algebra that the pupil will use in the sciences or in mechanical arts its legitimate place is not large. The need of the pupil are usually met by the cases of monomial factors, and of the binomial factors of the expressions of the type x²+(a+b)x+ab and x²-y².

The original idea, carried over from arithmetic, was that factors should be integral and rational. In practical use, however, it is often necessary to enlarge this conception, and to speak of x²-I/a² as having the factors x+I/a and x-I/a, of and as factors of ; and of as factors of .

The numbers used in counting objects have long been known as natural numbers. Numbers that are not naturally used in counting objects are commonly called artificial numbers, a term that is open to certain objections. From various points of view the number 3/43/4 is as natural as 22, and sq.root of 5sq.root of 5 as naturally comes into use as 5 5, although we cannot look at an object 3/43/4 of a time nor can we pick up a book sq.root of 5 timessq.root of 5 times. The term is convenient, however, even though it may be as inappropriate as "imaginary" in connection with sq. root of -3-3, or "fraction" (in its primitive sense) in connection with 2/2, 5/2, 1/2/3/42/2, 5/2, 1/2/3/4. The preceding discussion leads to a much larger and more important question as to the meaning of number itself.

Algebraic Scale.

Algebraic Scale.

If we construct a scale of integers, 1, 2, 3, 4, ...,1, 2, 3, 4, ..., and count backwards (to the left) by repeatedly subtracting 1, we have ... 4, 3, 2, 1, 0.... 4, 3, 2, 1, 0. The next count on the scale carries us beyond the point marked 00 on the scale, to 1, 2, 3, 4, ... 1, 2, 3, 4, ... in the opposite direction, just as we count 1, 2, 3, 4, ...1, 2, 3, 4, ... below zero nn a thermometer. For obvious reasons we find it convenient to speak of these symbols on the other side of zero as representing numbers, even though we cannot look at an object "three-less-than-zero" or "three-on-the-other-side-of-zero" times. We were led to these particular artificial numbers by subtracting as we went down the scale, and so we come to designate them a minus sign, thus giving to that sign a qualitative meaning (say "negativeness") instead of the operative one (subtraction). Such numbers are imaginary in some senses and natural in others. If we wish to emphasize the positive nature of 1, 2, ..., 1, 2, ..., we may write them as +1, +2, ..., +1, +2, ..., although even without any sign they are considered positive.

Further, on the algebraic scale we can represent such fractions as 1/2 1/2 and -3/4-3/4, and such surd numbers as -sq. root of 2 -sq. root of 2 and 3 sq. roots of 53 sq. roots of 5, so these are as real in certain situations as integers themselves.

In the expression a/ba/b, in which aa and bb are natural numbers, does not denote a natural number, it is called a fraction, but since division by zero has no meaning, the case in which b = 0b = 0 is excluded. Integers and fractions are classified as rational numbers. More generally speaking, a rational algebraic fraction is the quotient of any integral function by any other integral function. Certain numbers such as -sq. root of 2-sq. root of 2 and 3 sq. roots of 53 sq. roots of 5 do not come within a definition of rational numbers, and are called irrational numbers. In elementary algebra we also meet with certain numbers represented by the symbol a sq.root of -1sq.root of -1, and these are called imaginary numbers. Numbers like a+b sq. root of -1a+b sq. root of -1 are called complex numbers.

Algebraic Expressions.

Algebraic Expressions.

An expression consisting of a single letter, or made up either of letters or of letters and numerals, combined so as to represent some or all of the operations of addition, subtraction, multiplication, division, involution (the finding of powers) and evolutions (the finding of roots), is an algebraic expression. If it does not involve addition or subtraction, it is a monomial; but in the expression a-(b+c), -(b+c)a-(b+c), -(b+c) is considered as a monomial, and so in other similar cases involving signs of aggregation (parenthesis, brackets, etc.). If an algebraic expression is not a monomial, it is a polynomial, the binomial (two-term) and trinomial (three-term)being special types. In algebra, the letters of an expression represent numbers of some kind. In the monomial ab, a and b are factors of expression. If the value of either factor, say aa, is known and is to remain the same throughout the discussion of an expression, it is called a constant; but if it may have any value we please to give it and change from one value to another, it is called a variable. Constants are often represented by the first letters of the alphabet (a, b, c, ...)(a, b, c, ...) and variables by the last letters (..., x, y, z) (..., x, y, z), but this is not a universal rule especially in physical formulas. In an equation, say 2a-x=4, x2a-x=4, x usually represents a number to be found - "the unknown quantity" - the first letters of the alphabet representing numbers supposed to be known. In the monomial 2ax**22ax², we may speak of any factor as the coefficient of the rest of expression, but it is customary to speak of 2 2 as a numerical coefficient, and of 2a2a as the coefficient of x**2x², the coefficient being the first factor or factors. For example, in the expression 1/2(a+b)1/2(a+b)x**2y**3x²y³, 1/2, 1/2 is the coefficient of (a+b)(a+b)x**2y**3x²y³, and 1/2(a+b)1/2(a+b) is the coefficient of x**2y**3x²y³, and it is allowable to speak of 1/2(a+b)1/2(a+b)x**2x² as the coefficient of y**3y³ and so on. In the expression x**3x³, 3is3 is the exponent of xx, and similarly in the case of x**m.x^m. In the expression mxmx, the coefficient mm (if it be an integer) represents the number of times that xx is taken as an addend; while in the expression x**mx^m, m, m (if it be an integer) represents the number of times that xx is taken as a factor. In each case the meaning is later extended to permit of mm being any kind of number (fractional, surd, imaginary, etc.)

Function.

An expression like 2x+5 is called a function of x and is said to "depend upon" x for its value. Similarly, 2x+y³ is a function of x and y. For brevity we may, in any discussion, write f(x) for the function of x, and f(x,y) for the function of x and y, and so on for other variables. We may then, in discussing x-3, for example, say that f(i)=i-3=-2, f(7)=7-3=4, f(-2)=-2-3=-5, f(0)=0-3=-3, and so on, according to the value of x which we substitute in x-3.

The Elementary Operations.

From the standpoint of actual use, whether in the natural sciences or in pure mathematics, there is little need for the ordinary operations involving polynomials, therefore a brief treatment of the four fundamental operations, limited chiefly to binomial operators, with a slight reference to the theory of roots, is all that is essential to the further study of the science of algebra. The operations upon algebraic fractions were adapted from arithmetic after the introduction of the improved symbolism of the 17th century. They later became more complicated owing to the doctrine of formal discipline, the result leading to an expenditure of time quite out of proportion to the use made of them. In the 19th century the time consumed in reducing artificial fractions to lower terms, and in operations involving polynomials was excessive.

Ratio.

In practical work a ratio is considered simply as a fraction, although fundamentally the ratio of a circumference to the diameter of a circle is a transcendental number and not a fraction, as we define the term. Practical work in a laboratory or workshop is not considered with irrationals as such; it seeks for precision within certain designated limits, recognizing that all measurement is approximate. On this account and because of the immaturity of the pupils, elementary algebra looks upon a proportion as a fractional equation, and deals with all rations as simple functions, ignoring the distinction between algebraic and arithmetical fractions, and the fact that ratio may be irrational.

Algebraic Scale.

Algebraic Scale. If we construct a scale of integers, 1, 2, 3, 4, ..., and count backwards (to the left) by repeatedly subtracting 1, we have ... 4, 3, 2, 1, 0. The next count on the scale carries us beyond the point marked 0 on the scale, to 1, 2, 3, 4, ... in the opposite direction, just as we count 1, 2, 3, 4, ... below zero n a thermometer. For obvious reasons we find it convenient to speak of these symbols on the other side of zero as representing numbers, even though we cannot look at an object "three-less-than-zero" or "three-on-the-other-side-of-zero" times. We were led to these particular artificial numbers by subtracting as we went down the scale, and so we come to designate them a minus sign, thus giving to that sign a qualitative meaning (say "negativeness") instead of the operative one (subtraction). Such numbers are imaginary in some senses and natural in others. If we wish to emphasize the positive nature of 1, 2, ..., we may write them as +1, +2, ..., although even without any sign they are considered positive.

Algebraic Expressions.

An expression consisting of a single letter, or made up either of letters or of letters and numerals, combined so as to represent some or all of the operations of addition, subtraction, multiplication, division, involution (the finding of powers) and evolutions (the finding of roots), is an algebraic expression. If it does not involve addition or subtraction, it is a monomial; but in the expression a-(b+c), -(b+c) is considered as a monomial, and so in other similar cases involving signs of aggregation (parenthesis, brackets, etc.). If an algebraic expression is not a monomial, it is a polynomial, the binomial (two-term) and trinomial (three-term)being special types. In algebra, the letters of an expression represent numbers of some kind. In the monomial ab, a and b are factors of expression. If the value of either factor, say a, is known and is to remain the same throughout the discussion of an expression, it is called a constant; but if it may have any value we please to give it and change from one value to another, it is called a variable. Constants are often represented by the first letters of the alphabet (a, b, c, ...) and variables by the last letters (..., x, y, z), but this is not a universal rule especially in physical formulas. In an equation, say 2a-x=4, x usually represents a number to be found - "the unknown quantity" - the first letters of the alphabet representing numbers supposed to be known. In the monomial 2ax**2, we may speak of any factor as the coefficient of the rest of expression, but it is customary to speak of 2 as a numerical coefficient, and of 2a as the coefficient of x**2, the coefficient being the first factor or factors. For example, in the expression 1/2(a+b)x**2y**3, 1/2 is the coefficient of (a+b)x**2y**3, and 1/2(a+b) is the coefficient of x**2y**3, and it is allowable to speak of 1/2(a+b)x**2 as the coefficient of y**3 and so on. In the expression x**3, 3is the exponent of x, and similarly in the case of x**m. In the expression mx, the coefficient m (if it be an integer) represents the number of times that x is taken as an addend; while in the expression x**m, m (if it be an integer) represents the number of times that x is taken as a factor. In each case the meaning is later extended to permit of m being any kind of number (fractional, surd, imaginary, etc.)

The numbers used in counting objects have long been known as natural numbers. Numbers that are not naturally used in counting objects are commonly called artificial numbers, a term that is open to certain objections. From various points of view the number 3/4 is as natural as 2, and sq.root of 5 as naturally comes into use as 5, although we cannot look at an object 3/4 of a time nor can we pick up a book sq.root of 5 times. The term is convenient, however, even though it may be as inappropriate as "imaginary" in connection with sq. root of -3, or "fraction" (in its primitive sense) in connection with 2/2, 5/2, 1/2/3/4. The preceding discussion leads to a much larger and more important question as to the meaning of number itself.

Algebraic Scale. If we construct a scale of integers, 1, 2, 3, 4, ..., and count backwards (to the left) by repeatedly subtracting 1, we have ... 4, 3, 2, 1, 0. The next count on the scale carries us beyond the point marked 0 on the scale, to 1, 2, 3, 4, ... in the opposite direction, just as we count 1, 2, 3, 4, ... below zero n a thermometer. For obvious reasons we find it convenient to speak of these symbols on the other side of zero as representing numbers, even though we cannot look at an object "three-less-than-zero" or "three-on-the-other-side-of-zero" times. We were led to these particular artificial numbers by subtracting as we went down the scale, and so we come to designate them a minus sign, thus giving to that sign a qualitative meaning (say "negativeness") instead of the operative one (subtraction). Such numbers are imaginary in some senses and natural in others. If we wish to emphasize the positive nature of 1, 2, ..., we may write them as +1, +2, ..., although even without any sign they are considered positive.

Further, on the algebraic scale we can represent such fractions as 1/2 and -3/4, and such surd numbers as -sq. root of 2 and 3 sq. roots of 5, so these are as real in certain situations as integers themselves.

In the expression a/b, in which a and b are natural numbers, does not denote a natural number, it is called a fraction, but since division by zero has no meaning, the case in which b = 0 is excluded. Integers and fractions are classified as rational numbers. More generally speaking, a rational algebraic fraction is the quotient of any integral function by any other integral function. Certain numbers such as -sq. root of 2 and 3 sq. roots of 5 do not come within a definition of rational numbers, and are called irrational numbers. In elementary algebra we also meet with certain numbers represented by the symbol a sq.root of -1, and these are called imaginary numbers. Numbers like a+b sq. root of -1 are called complex numbers.

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.