ied 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 vã and vax, are called simple sirds; and those consisting of several terms, as v ab + cd and Va—*+be, compound surds. 16. When any quantity is to be taken more than once, the number is to be prefixed, which shews 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 ; 3** signisies, that the quantity ** is multiplied by 3, and the numeral coefficient is 3 ; also 51.** ta* denotes, that the quantity 7x +ais multiplied by 5, or taken 5 timeş. When no number is prefixed, an unit or i is always understood to be the coefficient : thus, I is the coefficient of 2 or of x ; for' a signifies the same as id, and x the same as 1x, since anỳ quantity, multiplied by unity, is still the samc. Moreover, if a and d be given quantities, and ** and 9 required ones ; then axi denotes, that x? is to be taken a times, or as many times as there are units, in a; and dy shews, that y is to be taken d times; so that the coefficient of ax* is a, and that of dy is d : suppose a 6 and d=4, then I* then will ax'6**, and dy=47.' Again, x, or de 2 notes the half of the quantity *, and the coefficient of * is į ; so likewise 4.x, or 3*, signifies of signifies of x, and the co- 4 efficient of 4x is 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 ** ta' and 7V.** ta'. But unlike quantities are those, which are expressed by different letters, or by the same letters under different powers: thus 2ab, a'b, habe, 5ab-, 4x', y, ya and za are all unlike quantities. 18. The double or ambiguous sign + signifies pluis or minus the quantity, which immediately follows it, and being placed between two quantities, it denotes their sum, or dif ference. Thus, atv -- b shews, that the quan 4 tity ✓ 4 b is to be added to, or subtracted from, a. 19. A general exponent is one, that is denoted by a letter ? instead of a figure ; thus, the quantity 2." has a general exponent, viz. m, which universally denotes the mth pow. er of the root x. Suppose m=2, then will X"=x if m 7 3, then will **=x}; if m=4, then will 4"=x4, &c. . In like manner, a=-6 expresses the mth power of ambe 2 ; 20. This root, viz. a--, is called a residual root, because its value is no more than the residue, remainder, or difference, of its terms a and bo It is likewise call ed Vat ab ; ed a binomial, as well as atb, because it is composed of two parts, connected together by the sign 25. A fraction, which expresses the root of a quantity, is also called an index, or exponent ; the numerator shews the power, and the denominator the root : thus at signifies the same as va ; and a+abiš the same as a ; vatab likewise at denotes the square of the cube root of the quantity a. Suppose a=64, then will aš=64*=4*= 16 ; for the cube root of 64 is 4, and the square of 4 is 16. Again, a + 5* expresses the fifth power of the biquadratic root of atb. Suppose a-9 and b=7, then will 6+61*=9771*=16*=2'=32 ; for the biquadratic root of 16 is 2, and the fifth power of 2 is 32. Also, a á signifies the nth root of a. If n=4, then will ht=a; if n = 5, then will at=aš, &c. Moreover, a toj denotes the mth power of the nth root 27 를 of atb. If m=3 and n=2, then will a+b\" =a-+61 , namely, the cube of the square root of the quantity a+b; . 1 n n in n and as an equals a, or ✓a, so a +-61= Va+8", namely, the nth root of the mth power of a+b. So that the mth power of the nth root, or 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 ** Suppose x=2, then will ** = 2* = 4; if x=3, then will ** = 33= 27. ADDITION, ADDITION. Andition, 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. Add the coefficients together, to their sum join the common letters, and prefix the common sign when neces sary. EXAMPLES * The reasons, on which these operations are founded, will readily appear from a little reflection on the nature of the quantities 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 za is 3 shillings, and sa' is 5 shillings, and their sum is 8 shillings. In like manner - ab and ab, 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 -, 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 + and 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 æ be supposed to represent a crown, and b a shilling ; then the sum of a and b can be neither za nor 2b, that is, peither 2 crowns por 2 shillings, but only I crown plus i 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 discovered, 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 varishes. |