ed 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 ax, are called simple of one term only, as ✓ a and surds; and those consisting of several terms, as ab + cd 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, za signifies twice a, or a taken twice, and the numeral coefficient is 2; 3x2 signifies, that the quantity ** is multiplied by 3, and the numeral coefficient is 3; also 5 x2 + a2 denotes, that the quan tity√x+a2 is multiplied by 5, or taken 5 times. When no number is prefixed, an unit or I is always understood to be the coefficient: thus, I is the coefficient of a or of x; for a signifies the same as 1a, and x the same as 1x, since any quantity, multiplied by unity, is still the same. 2 Moreover, if a and d be given quantities, and x and y required ones; then ax 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 a6 and d=4, then then will ax6x, **, and dy=4y. Again, w, or Ix -, de 2' notes the half of the quantity x, and the coefficient of 3x is; so likewise x, or, efficient of x is. I , signifies of x, and the co 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 2 ✓ x2 + a* and 7 x2 + a*. But unlike quantities are those, which are expressed by different letters, or by the same letters under different powers: thus 2ab, a2b, 2abc, 5ab*, 4x, y, y and z* are all unlike quantities. 2 2 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, at 2 a I tity -b is to be added to, or subtracted from, a. 4 1 19. A general exponent is one, that is denoted by a letter instead of a figure: thus, the quantity " has a general exponent, viz. m, which universally denotes the mth power of the root x. Suppose m2, then will x"=x2; if m 3, then will x=x2; if m=4, then will &"=x, &c, 771 m xx In like manner, ab expresses the mth power of ab, 20. This root, viz. 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 call ed 1 ed a binomial, as well as a + b, because it is composed of two parts, connected together by the sign -. 3 21. 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 a signifies the same as a; and atabs the same as ✓atab; likewise af denotes the square of the cube root of the quantity a. Suppose a=64, then will at=641=4*= 16; for the cube root of 64 is 4, and the square of 4 is 16. 5 Again, a + b expresses the fifth power of the biquadratic root of a+b. Suppose a=9 and b=7, then will 5 5 5 a+b=9+7=164=25=32; for the biquadratic root of 16 is 2, and the fifth power of 2 is 32. Also, a signifies the nth root of a. If n=4, then will aa; if n=5, then will a=at, &c. L m Moreover, a+b" denotes the mth power of the nth root m of a+b. If m=3 and 1= 2, then will a+b1" =a+bl namely, the cube of the square root of the quantity a+b; I and as a" aequals a, or " √a, so a+b = a+b1", 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 x*. Suppose x=2, then will x=224; if x=3, then will x* = 33=27. T ADDITION. 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 like signs. RULE.* Add the coefficients together, to their sum join the common letters, and prefix the common sign when necessary. 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 za 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 -7ab, 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 1 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 b can be neither za nor 2b, that is, neither 2 crowns nor 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 máy 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 & balance, and the paradox vanishes. |