a-c; suppose a=12, &=6, and c=9, then will 14. These literal expressions, namely, and с called algebraic fractions; whereof the upper parts are called the numerators, and the lower the denominators: thus, a 15. Quantities, to which the radical sign is applied, are called radical quantities, or surds; whereof those consisting of one term only, as ✔a and V a x, are called simple surds; and those consisting of several terms, as Vab+cd and a22+c, compound surds, 16. When any quantity is to be taken more than once, the number is to be prefixed, which shows how many times it is to be taken, and the number so prefixed is called the numeral coefficient: thus, 2a signifies twice a, or a taken twice, and the numeral coefficient is 2; 3x signifies, that the quantity is multiplied by 3, and the numeral coefficient is 3; also 5 Vx2+a denotes, that the quantity is multiplied by 5, or taken 5 times. When no number is prefixed, an unit or 1 is always understood to be the coefficient; thus, 1 is the coefficient of a or of x; for a signifies the same as la, and x the same as lx, since any quantity, multiplied by unity, is still the same. Moreover, if a and d be given quantities, and x and y required ones; then ar* denotes, that is to be taken a times, or as many times as there are units in a; and >% shows, that is to be taken d times; so that the coefficient of axis n y and that of dy is d. Suppose a 6 and d=4, then will ax =6x2, and dy=4y. 1x Again, x, or, denotes half of the 2' quantity x, and the coefficient of x is ; 3x x, or 4 so likewise signifies of x, and the coefficient of x is 4 17. like quantities are those, that are represented by the same letters under the same powers, or which differ only in their coefficients: thus, 3a, 5a, and a are like quantities, and the same is to be understood of the radicals Vx+a* and 7 -t-u2. But unlike quantities are those, which are expressed by different letters, or by the same letters under different powers thus 2ab, ab, 2abc, 5ab2, 4x2, y, y*, and za are all unlike quantities. : 18. The double or ambiguous sign + signifies plus or minus the quantity, which immediately follows it, and being placed between two quantities, it denotes their sum, or dif ference. Thus, atv + b is to be added to, or subtracted from a. m 19. A general exponent is one, that is denoted by a letter instead of a figure: thus, the quantity x has a general exponent, namely, m, which universally denotes the mth power of the root x. Suppose m=2, then will x=x; if m= 3, then will x like manner, m 3 m X if w 4, then will xx, &c. In י expresses the mth power of a—b. 20. This root, namely, a—b, is called a residual root, because its value is no more than the residue, remainder, or difference of its terms a and b. It is likewise called a binomial, as well as a+b, because it is composed of two parts, connected together by the sign 21. A fraction, which expresses the root of a quantity, is also called an index, or exponent; the numerator shows the power, and the denominator the root: thus a signifies the same as ✔a; and a+ab3⁄43, the same as3√a+ab; likewise 2 3 denotes the square of the cube root of the quantity a. Suppose a=64, then will a3=643—4—16; for the cube root of 64 is 4, and the square of 4 is 16. Again a+ ic root of a+b. expresses the fifth power of the biquadratSuppose a 9 and 6-7, then will a+b| *=9+7 | * =16] *=2=32; for the biquadratic root of 16 is 2, and the fifth power of 2 is 32. Moreover a+ denotes the mth power of the nth root m 3 of a+b. · If m=3 and n=2, then will a+b=a+o]3, namely, the cube of the square root of the quantity a+b; and as m m a* equals"√ a1, or √a, so a+i|” = a+b, namely, the nth root of the mth power of a+b. So that the mth power of the nth root, and the nth root of the mth power of a quantity are the very same in effect, though differently expressed. 22. An exponential quantity is a power, whose exponent is a variable quantity, as x*. Suppose x=2, then will x* =2=4; if x=3, then will x*=33=27. ADDITION. Addition, in Algebra, is connecting the quantities together by their proper signs, and uniting in simple terms such as are similar, In addition there are three cases. CASE I. When like quantities have Hike signs, RULE.* Add the coefficients together, to their sum join the com-mon letters, and perfix the common sign when necessary. * The reasons, on which these operations are founded, will readily appear from a little reflection on the nature of the quan⚫tities to be added, or collected together. For with regard to the first example, where the quantities are 3a and 5a, whatever a represents in one term, it will represent the same thing in the other; so that 3 times any thing, and 5 times the same thing, collected together, must needs make 8 times that thing. As if a denote a shilling, then 3a is 3 shillings, and 5a is 5 shillings> and their sum is 8 shillings. In like manner —2ab and —7abt or-2 times any thing and —7 times the same thing, make —9 times that thing.. As to the second case, in which the quantities are like, but the signs unlike; the reason of its operation will easily appear by reflecting, that addition means only the uniting of quantities together by means of the arithmetical operations, denoted by their signs and, or of addition and subtraction; which being of + contrary or opposite natures, one coefficient must be subtracted from the other, to obtain the incorporated or united mass. As to the third case, where the quantities are unlike, it is plain, that such quantities cannot be united into one, or otherwise added than by means of their signs. Thus, for example, if a be supposed to represent a crown, and b a shilling; then the sum of a and * can be neither 2a nor 2b, that is, neither 2 crowns nor 2 shillings, but only 1 crown plus 1 shilling, that is, a+b. In this rule the word addition is not very properly used, being much too scanty to express the operation here performed. The business of this operation is to incorporate into one mass, or algebraic expression, different algebraic quantities, as far as an actual incorporation or union is possible; and to retain the algebraic marks for doing it in cases, where an union is not possible. When we have several quantities, some affirmative and others negative, and the relation of these quantities can be diş * Subtract the less coefficient from the greater, to the remainder prefix the sign of the greater, and annex their common letters or quantities. covered, in whole or in part; such incorporation of two or more quantities into one is plainly effected by the foregoing rules. It may seem a paradox, that what is called addition in algebra should sometimes mean addition, and sometimes subtraction. But the paradox wholly arises from the scantiness of the name, given to the algebraic process, or from employing an old term in a new and more enlarged sense. Instead of addition, call it incorporation, union, or striking a balance, and the paradox vanishes. |