Algebra - Artificial Numbers

ALGEBRA - ARTIFICIAL NUMBERS 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 a is as natural as 2, and V 5 as naturally comes into use as 5, although we cannot look at an object a of a time nor can we pick up a book V 5 times. The term is convenient, however, even though it may be as inappropriate as "imaginary" in connection with V/ -3, or "fraction" (in its prim itive sense) in connection with 2/4. The preceding discussion leads to a much larger and more important question as to the meaning of number itself. (See NUMBER; NUMBERS, THEORY OF; COMPLEX NUMBERS.) Algebraic Scale.-If we construct a scale of integers, I, 2, 3, 4, ..., and count backwards (to the left) by repeatedly subtracting I, we have ... 4, 3, 2, I, o. The next count on the scale carries us beyond the point marked o on the scale, to 1, 2, 3, 4, ... in the opposite direction, just as we count I, 2, 3, 4, ••• below zero on a thermometer. For obvious reasons we find it con venient 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 this sign the 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 I, .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 and - , and such surd numbers as - V2 and - 5, so these are as real in certain situations as the integers themselves.

If 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 o has no meaning, the case in which b=o 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 V2 and ,/ 7 do not come within the definition of rational numbers, and are called irrational numbers. In element ary algebra we also meet with certain numbers represented by the symbol a\/ -1, and these are called imaginary numbers. Numbers like a+bV -1, are called complex numbers. (See NUM BER; COMPLEX NUMBERS; FRACTIONS; NUMBERS, THEORY OF.) 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 find ing of powers) and evolution (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 consid ered as a monomial, and so in other similar cases involving signs of aggregation (parentheses, brackets, etc.). If an algebraic expres sion 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 the expression. If the value of either factor, say a, is known and is to remain the same through out the discussion of the 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 repre sented by the first letters of the alphabet (a, b, c, ...), and variables by the last letters (... x, y, z), but this is not a uni versal rule especially in physical formulas. In an equation, say 2a-x=4, x usually represents a number to be found-"the un known quantity"-the first letters of the alphabet representing numbers supposed to be known. In. the monomial we may speak of any factor as the coefficient of the rest of the expression, but it is customary to speak of 2 as a numerical coefficient, and of 2a as the coefficient of the coefficient being the first factor or factors. For example, in the expression is the coefficient of and 2(04) is the coefficient of and it is also allowable to speak of as the coefficient of and so on. In the expression 3 is the exponent of x, and similarly in the case of xm. 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 (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.).

Function.

An expression like 2x+5 is called a function of x and is said to "depend upon" x for its value. Similarly, is a function of x and of 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(-2)= -2-3=-5, f(o)=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 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 frac tions were adapted from arithmetic after the introduction of the improved symbolism of the i7th 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 9th century the time consumed in reducing artificial fractions to lowest terms, and in operations involving polynomials was excessive.

Ratio.—Inpractical work a ratio is considered simply as a fraction, although fundamentally the ratio of the 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 concerned 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 pro portion as a fractional equation, and deals with all ratios as simple fractions, ignoring the distinction between algebraic and arithmetical fractions, and the fact that the ratio may be irrational.

Uses of Irrational Numbers.

Inthe solution of quadratic equations, excepting those artificially constructed to give only in tegral 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 x= V2. 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 in 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 ap proximation. If it is stated that x==-4-- V243, nothing is gained by writing this result as al_-9V3. An expression like V 2 and ;,/ 17, but (for no very good reason) not like V3+V 2, is known as a surd, from a mediaeval Latin translation of an Arabic rendering of the Greek ao-yos (al'ogos, irrational). Until recently the finding of the square root of a binomial'surd, as of 7-4\13 or of 12+2V35, 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 7r, elementary algebra is little concerned. (See NUMBER.) Factors.—Thefactorizing 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 the mechanical arts its legitimate place is not large. The needs of the pupil are usually met by the cases of monomial factors, and of the binomial factors of 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, how ever, it is often necessary to enlarge this conception, and to speak of x— I- as having the factors x+ and x--!; of x+V a a a and x-2-0 as factors of x2-1-x(Va-2Vb)-2Vab; and of Va, V b, and Vc, as factors of Vabc.

The Equation.

Expressionsof equality are of several types. For our present purposes it is necessary to consider only three: (t) 2+3=5 or, after the operation has been performed, 5=5; (2) ad-a=2a, (3) x+2=5. The first is a numerical relation between known numbers, and does not represent an algebraic equation; the second represents a relationship that is true for all values of a, and hence is called an identity, another example being (a+b)2=a2-1-2ab-l-b2; the third is an algebraic equation. true in this case for one and only one value of x, this value being known as the root of the equation. An identity is often indicated by the symbol instead of =. In the present discussion we consider only elementary algebraic equations with rational and integral terms. The general treatment with a consideration of the existence of a root is given in the article on EQUATIONS.

Linear Equations.—Considerations of analytic geometry (q.v.) have led to the use of the term linear equation to mean an equation of the first degree having any number of variables. The oldest part of elementary algebra, so far as known, relates to the solution of linear equations in one unknown, the type form reduc ing, through the final stage (in modem symbols) of px=q, to x=k. This type of equation is found in the Ahmes (Rhind) Papyrus (c. 17oo-1600 B.c.) and is the one found most frequently for a period of more than 3,5oo years, varying merely in the language and symbols employed. Such number puzzles have been used by substantially all writers on algebra, and in modern times they have found a wide range of practical applications in the sciences, in industry and in commerce.

Linear equations of the type ax+by+c=o and dx-Pb'yd-c'=o generally have common values for x and y; that is, they are simultaneous. For example, the two equations x+3y— =o and 2x+9y+i=o have in common the roots x=4, y= 1. Each equation is, by itself, indeterminate, having an infinite number of roots. For example, the first of these equations, x+3y---1=o, is satisfied by x=o, y=i-, and similarly by the pairs of values (I, o), (2, —is), (3, --I), (4, —I), and so on; but there is only one pair which satisfies both of the equations. In modern textbooks this is made clearer by means of graphs, each equation being represented by a straight line, and these lines having, in general, one point in common. This introduction of the elements of analytic geometry into elementary algebra is helpful in under standing the meaning of roots. For example, the two equations 3x-7y= 6 and 2X- 4iy = 4 are satisfied by any number of pairs of values, their graphs being coincident; while 3x-7y = 6 and 2x-4iy =5 cannot be satisfied by any pair of values, their graphs being parallel. (See ANALYTIC GEOMETRY.) Simultaneous linear equations are solved by several methods, the equations as given in the textbooks being artificially con structed so as to illustrate each. These methods are sufficiently discussed in such works. If, however, only a single method is to be given for solving a pair of simultaneous equations, this method being made mechanical by much practice, that of substitution is the most satisfactory in cases that actually arise in science or industry. This method consists in finding in either equation the value of one unknown in terms of the other, and then in sub stituting this value in the other equation. For advanced classes the method of determinants (q.v.) is interesting and valuable.

Simultaneous linear equations with more than two unknown quantities have an interest as puzzles and some value in develop ing skill in manipulating algebraic expressions, and they also have value in certain technical fields and in certain commercial prob lems. During the centuries there have been developed various devices for solving specially constructed types, but such devices and types lack generality and hence they have little value except as recreations. In actual practice with real problems the method of determinants is the best. The question of the possibility of solution of given sets should be considered, as in the case of simultaneous equations with two unknowns.

Quadratic Equations.

Anequation of the type ax2-1-bx+c =o in which b and c may have any finite numerical values, and a may have any finite value except zero, is a quadratic equation (from the Latin quadratus, a square). Such equations are fre quently needed in solving scientific problems. There are several methods of solving such an equation, including the following: (I) Reducing to the form ax2-1-bx-1-c=o; resolving into factors (px+q) (p'x+q')=o; equating each factor to zero; and solving the two resulting linear equations; (2) Completing the square ; (3) Using a general formula, usually developed by the second method. Of these, (t) is too difficult in a practical case like 1.47x2-0.36x+14.o2=o, however easy it may be in an artificial one like x2-5x+6=o; (2) is a traditional method, essentially due to the Alexandrian school (c. 3oo B.c.) ; it requires the memo rizing of a process—the completion of the square; and it is arith metically difficult in practical cases; (3) requires the memorizing of a formula, but in problems that are practical it is usually the easiest of the three.

Simultaneous quadratic equations can be solved by elementary algebra (the biquadratic not being there given) only in cases artificially constructed to admit of easy solution. The analytic explanations given in textbooks may profitably be supplemented by the use of the graph. This will serve to show the difference between a case like 2x-3y=5, and a case like i r. In the former substitution reduces the problem to the solution of a quadratic ; in the latter case, to the solution of an equation of the fourth degree. The nature of such equations may profitably be shown by the aid of graphs.

Ratio and Proportion.

Asalready stated, this subject is best treated in connection with fractional equations. Since in elemen tary algebra the letters represent numbers, and since it is neces sary to assume at this stage that operations with irrational num bers are subject to the ordinary laws that obtain with integers, the subject offers no new features. The range of application to outdoor measurement and to simple physical problems is exten sive. With this method of treatment such special names as ante cedent, consequent, means, and extremes lose their importance in practical problems. Moreover, the old notation, a : b :: c : d, may profitably give place to the more familiar and more easily under stood symbols representing a fractional equation.

Variation.—.Thesubject of variation, which formerly had a symbolism of its own, is more conveniently and clearly treated as a topic under ratio and proportion or, what is substantially the same, by the ordinary method of equations; that is, variation may be expressed by the aid of the equation x=ky (direct variation) or by xy=k (inverse variation).