6. The investigation of such rules for calculation is one of the two objects of Algebra. The other object, which is subservient to the former, is the discovery of the different operations which may be performed with the same given numbers, and shall produce the same ultimate numerical results as any given operations different from these shall produce, without regard to what those numbers may chance to be. Thus, if the square of the sum of a and b were sought in another form, it may be exhibited thus a × a + 2 × a × b + b × b. And the statement of this fact is thus written : (a + b) x (a + b) : = (a × a) + (2 × a × b) + (b × b.) Shorter modes of writing it will be exhibited presently; but here the simple symbols used in the arithmetic have been alone employed, for the purpose of showing the nature of algebraic notation in its earliest forms, and to illustrate the objects for which it was devised. 7. The discovery of formulæ for the solution of questions constitutes the algebraical problem; and the discovery of formulæ of transformation, or of those which give equivalent results independently of the particular value of the quantities which enter into their composition, constitute the algebraical theorem. 8. The motives which gave rise to the use of alphabetic letters as symbols of number in preference to any other system of symbols, arbitrarily selected for the same purpose, are principally the following. First, As they have no numerical signification in themselves, they are subject to no ambiguity, having in reference to numbers no other signification than they are defined to have in the outset of each problem, or either defined, or understood from general practice, to have in each theorem. Secondly, Being familiar to the eye, the tongue, the hand, and the mind, that is, having a well-known form and name, they are easily read, written, spoken, remembered, and discriminated from one another, which could not be the case were they mere arbitrary marks, formed according to the caprice of each individual who used them, and always different, as in such case they must almost of necessity be, at each different time that the same person required to use them. Thirdly, The order in which the letters are arranged in the alphabet, facilitates the classification of them into groups much more easy to survey and comprehend in the expressions which arise from the performance of any assigned operations, and thereby renders the investigator much less likely to omit any of them by an imperfect enumeration, than if they were composed of marks that were used for that purpose only, and selected for each individual occasion from the various combinations that could be formed of such simple linear elements as the hand could readily trace, and the eye readily distinguish from all other combinations. II. Definitions, Notation, and Fundamental Principles. THE principal symbols which are employed to designate the operations of algebra and arithmetic, and the relations which subsist between quantities, are the following. Their object is to abbreviate. I. 1. signifies addition, and is read plus. Thus 2 + 3 or a + b + c respectively signify that 3 is to be added to 2, and that b is to be added to a, and that then c is to be added to the sum of a and b. A quantity to which the symbol + is prefixed, is called a positive or affirmative quantity. 2. signifies subtraction, and is read minus. Thus, 31, or b―a, signify respectively that 1 is to be subtracted from 3, and a from b. The number to be subtracted is always placed after the symbol. A quantity to which the sign is prefixed is called a negative quantity 3. signifies the difference of the quantities between which it is placed; and is used either when it is not known or is not necessary to specify which is the greater of them. In this case a b, or b a, signify the same thing. 4. x is the symbol of multiplication, and is placed between the factors which are to be multiplied together. Sometimes a point. (placed at the lower part of the line, to distinguish it from the decimal point, which is placed at the upper part of the line,) is employed for the same purpose, and especially between the numerical factors, as 3. 5. 7, or 1. 2. 3. 4: and in the case of simple literal factors, the practice is now almost universal to drop all marks between the simple factors, and write them in consecutive juxta-position. Thus a × b xc xx, or a . b. c. x, or abcr designate the same thing, viz. the continued product of the numbers which a, b, c, and x are put to represent ↑. When one of the factors is a number, it is called a coefficient: thus in 2 × a × b or 2ab, the 2 is called the coefficient of ab, and in 53xyz, 53 is called the coefficient of xyz. When no coefficient is written, 1 is understood to be meant, the quantity being taken once. Also, in some cases where letters are put for numbers, the letters which represent given or known numbers are likewise called coefficients; as in 3 axz, 3a is called the coefficient of xz. In the case of a number being actually given, the coefficient is said to be a numeral coefficient; but when it is given in literal symbols, it is called a literal coefficient. Moreover it may be remarked that cases of algebraical investigation sometimes present themselves in which even the symbols of the unknown quantities are conveniently considered as coefficients: but these will be pointed out when they arise. Though, as is proved in the note, the order of the factors in multiplication, so Quantities affected with the signs +and+ or - and, are said to have like signs; and those affected with and +, or + and -, are said to have unlike signs. It is manifest from the nature of addition and subtraction, that the disposition of the quantities as to order is immaterial; for a + b, or b + a, is the same thing, and a + b — c, a − c + b, or — c + b + a, express the same quantity, only under a different arrangement, as to relative position. It may be readily shown that it is immaterial in what order the factors are taken for the purpose of multiplication: that is, which is made the multiplicand and which the multiplier. For if a number of dots (or units) be placed horizontally equal in number to the units in the factor selected as the multiplicand, and this be repeated under this horizontal band till there are as many bands as there are units in the other factor: then the same number of dots considered as forming vertical columns will be constituted of as many times the number there is in the multiplier as there are units in the multiplicand, and representing therefore the result of a multiplicand with the order of the factors inverted. Thus if we take four times three, the dots will stand and if we turn the column which is vertical into a horizontal position, it becomes far as the value of the product is concerned, is immaterial; yet in the disposition of them as algebraic symbols, it is found convenient generally to arrange them in alphabetic order. Thus the quantity abcay is the same in point of value as bcxya, or any other arrangement that can be made of them, still the form abcxy is preferable, for many reasons, to any other that can be given to the same quantity. 5. is the symbol of division. Sometimes the dividend is put before and the divisor after the mark, and sometimes they are placed respectively above and below the line in the place of the two dots, after the manner of an arithmetical fraction. Thus ab or alike signify that the number a is to be divided by the number b. a i 6. = is the symbol for the words "is equal to," and is generally read equals." It is used to signify that the value of the aggregate of the terms which precede it is equal to the value of the aggregate of those which follow it. The whole expression is called an equation, and the quantities which stand to the left of the symbol are said to constitute the first side of the equation, and in like manner those which stand on the other constitute the second side of the equation. Thus ax + b =cxx + dx e is an equation, the first side of which e. is ax + b, and the second side cxx + dx 7. The symbols >, <, and sometimes, are used to express the inequality of the quantities between which they are placed. The opening of the symbol is always turned towards the greater quantity. Thus a > b signifies, that a is greater than b; and ƒ < g signifies, that ƒ is less than g. When it either is not known, or is not necessary to state which is the greater of the two quantities, which are nevertheless unequal, the symbol is used. 8. signifies that it is the ratio of the two quantities between which it stands which is the subject of consideration. Thus ay designates the ratio of x to y. It is read x to y, or the ratio of x to y. When two ratios are equal; that is, when some two numbers have the same ratio that two others have, it is expressed in one of these two ways : x:yu v orx: y = u: v The former is most usual, and adhered to in this Course. The phrases which they represent are x is to y as u is to v; or, a has the same ratio to y that u has to v or, the ratio of x to y, is the same as that of u to v. 9. It is often found necessary to class the quantities of which an expression is composed, into sets combined in some particular way. This classification is effected by enclosing them under a horizontal bar (called a vinculum), or between parentheses, or braces, or brackets. This is always done when the actual operations indicated are to be changed for others which shall produce equivalent results, and which are more easy to perform than those which were originally indicated. Nearly the whole of algebra consists in discovering these equivalent operations. Thus, referring to the example given in the Introductory Chapter, we might have put it in the more complicated but equivalent form, 6 x 10 + 7 x 10- 3 x 10-6 × 6 - 6 × 7+ 6 × 3: but by taking the form there given, the actual labour of computation is not above a fourth of that which would attend upon all these. Or, again, in general symbols, the expression (a + 26 3c) (4a-2b+3c) indicates that the sum of a and twice b being taken, and three times c being subtracted from the sum, and that to the difference of four times a and twice b, three times c is added, then the product of these two results is to be taken. This effect might be produced by other means much more complicated, but which are avoided by the contrivance indicated above. When the terms which are collected together are also employed to form the numerator and denominator of an expression in a fractional form, no ambiguity can arise, except the said numerator and denominator are composed of vinculated factors, in dropping the vinculum. Thus a + b (a + b) (a b) may be written simply 10. When several of the factors which compose a quantity are equal to one another, a considerable abbreviation of the trouble of writing, and of the space occupied, has been devised, by simply writing the common factor in its place, and a small figure above it and to the right hand, expressing how many times that factor occurs in the product. Thus, instead of a ×a×a×a or aaaa, it is usual to write a1, the a signifying the common factor, and the 4 the number of times of its occurrence. So likewise, instead of the expression 3aaabbxxxx, is written 3a3b2x2; and (a + x) (a + x) (a + x) is written (a + x)3: and so on for any number or form of the component equal factors of an expression. The number affixed is called the index or exponent, and the quantity is said to be raised to the power denoted by that exponent. Thus a signifies the fourth power of a, and 4 is called the index or exponent of the power of a: and a" is called the nth power of a, and n is the exponent of that power. In conformity with this notation, when no index is annexed to a quantity, 1 is understood: thus, by a, the first power of a is meant, that is, a1. It will obviously follow, that when two different powers of the same quantity are to be multiplied together, the symbolical result may be written as a power whose exponent is the sum of the exponents of the several factors. Thus a1× a2 × a1× a3 = a1+2+1+3 = a+10; for it is the same thing as aaaa × aa×a×aaa, that is, aaaaaaaaaa or a1o. On the same principle a2b3c multiplied by a3bc5 may be more briefly written a2+363+1+5, or still more briefly a3b1c. Similarly, in general a" × a" × a"... = a*+n+, where the dots after the quantities express the continuation of the same class of quantities to which they are respectively annexed to any assignable extent. It moreover follows, that the quotient of one power of a quantity by another is symbolically expressible by a power of that quantity whose exponent is equal to that denoted by the remainder left after subtracting the exponent of the divisor from that of the dividend. For since multiplication increases the number of a8 factors, division will decrease them. Thus, since a3 × a3 = a3, so also = a3; The two modes of notation just employed to designate the division of one power of a quantity by another power of the same quantity may be used indifferently. By remarking, however, the corresponding forms in some particular cases, we shall be led to some simple but important conclusions. And first, When m=n, we have = 1, and a”-” = a', which are of course iden tical in signification. Hence we learn that ao always signifies 1, whatever a may be. Indeed a signifying 1 a", if m = o we have 1 ao = 1, or 1 × ao = 1, or 1 time a taken of no power * whatever. The result is therefore perfectly consistent with first principles and the adopted notations. When m<n, then if we put m = n — r, we have 14 an an =a"-n-y = =a'. But this also signifies, when n = m + r†, the following expression, 1 Hence, a = = We learn from this, that when in αν any assigned series of operations upon the powers of a quantity, we arrive at an index of the form r, then the expression signifies the reciprocal of the rth 11. As we have occasion to calculate the powers of given numbers, considered as roots, so we have often to find the roots of given numbers considered as powers. The operations are considered as the inverse of each, and are denoted by inverse notations. 6 Thus to cube the quantity aa or a2 we have aa × aa × aa or a2. a2. a2 or a¤; so likewise to find the cube root of as we have to separate it into three equal factors. This operation, in all such cases, is indicated by dividing the exponent of the power by the exponent of the root: thus the cube root of a is a3, where 6 is the exponent of the given quantity, and 3 the exponent of the root. In thé same manner the square root of a b1 c2 d−4, is a b c d3, or a3 b2 c d ̄‡. If there be a numerical coefficient, the root of that is either actually extracted or indicated like the rest; as in the square root of 25a1 we may either put 251a2 or 5a2. Most commonly the numerical root is actually extracted when the given number admits of an accurate root, but indicated when the value cannot be assigned in a finite form. It will be obvious from the signification given above to this notation, that the numerator of a fractional exponent expresses the power to which a root is raised, and the denominator the root which is to be taken of that result. It also follows from the nature and relation of roots and powers, that it is immaterial whether we first extract the indicated root, and then raise it to the indicated power, or conversely, we first raise the indicated power, and then The term dimension is often employed instead of the word power. It is derived from the analogy which the dimensions of line, square, and cube in geometry, when they are expressed numerically, have to the first, second, and third powers. Beyond the third power, geometry furnishes nothing analogous to the powers of quantities; and hence the terms fourth, fifth, &c. dimensions though generally used, are hardly accurate. This is the same as the former, mnr, having r added to both sides; and hence m + r = n―r+r=n. When the index of the power is not divisible by the index of the root, the fractional form of the index which results is retained; as in the cube root of a2 67, we write it a3 3; or sometimes the given quantity is conceived to be put under the form a2 66 b1, or simply, a2 b6 b, and the root written a3 62 3. The ultimate purpose of the particular inquiry in question is the only guide to the Algebraist in his choice amongst these notations. |