and the sixth root 2. In like manner, the fourth root of 81 is 3, the fifth root of 1024 is 4, &c. The process of finding any root of a quantity is called evolution; and the quantity is said to have that root extracted. The second root is also called the square root, and the third the cube root (Art. 14), 16. XII. The mark ✔ prefixed to a quantity indicates that a root of that quantity is to be extracted: this mark is called the radical sign. The number by which the root is distinguished is written over the radical sign to the left, and is called the index of the root: thus, 25 means the second root of 25; 27, the third root of 27. In using the radical sign to denote the second root, the index 2 is usually omitted: thus, for the second root of 25, we may The roots of any number, a, 25, or 25. either write The 2d root of a isa, or √ a; 17. XIII. Whatever number of factors aquantity contains besides its coefficient, of so many dimensions it is said to be: thus, bc, 7ax, 3x2, are quantities of two dimensions, because each of them contains two factors in addition to its coefficient. In like manner, xyz, 5ax2, are quantities of three dimensions; xy3, 12a2x2, are quantities of four dimensions, and so on. On the same principle, a, 15x, are quantities of one dimension. Instead of "one dimension," ," "two dimensions," &c. the terms first order or degree, second order or degree, &c. are often used: thus, 7ax is said to be of the second order or second degree; 5ax2, of the third order or third degree, &c. Sometimes a letter is considered as belonging to the coefficient, and then it is not reckoned in estimating the dimensions or degree of the quantity: thus, in the example 5ax2, if we consider 5a as the coefficient, the quantity will be of two dimensions only. 18. XIV. An expression which does not consist of parts connected by either of the signs + or, is called a simple or mononomial quantity: thus, 7 and x are simple quantities. In like manner, 6x3x4 is a simple quantity: also, 12ax, be and Ꮖ 19. XV. An expression which consists of two or more simple quantities connected by the signs +, -, is called a compound quantity and the simple quantities of which it is composed are called its terms: thus, 6 + 3 + 4 is a compound quantity, and its terms are 6, 3, and 4: also, 20. XVI. A compound quantity consisting of two terms is called a binomial, A binomial whose second term is negative is sometimes called a residual quantity. A quantity consisting of more terms than two is called a multinomial or polynomial quantity. The names trinomial and quadrinomial are sometimes given to quantities consisting of three and four terms 21. XVII. A parenthesis enclosing an expression is called a vinculum, and denotes that the sign or index on the outside of it affects the result of all the operations indicated by the several signs and indices within it. Thus, 18-(5+7) means that 7 is first to be added to 5, and then the sum taken from 18, which leaves 6; without the vinculum, 18-5+7 would mean that 5 is to be taken from 18, and 7 added to the remainder, which would give 20: also, 16-(15-9)= 10, but 16-15-9-8. In like manner, (12+5) ×7, or 7 (12+5)= 119, but 12+5×7 = 47. And (10+3)2 169, but 10+32=19; and √(25+11)=6, but 25+11=16. Also, (xy)2 would mean that x is to be multiplied by y, and the second power of the product taken; whereas xy2 would mean that the second power of y was to be taken, and x multiplied by it; or, supposing x=5 and y=3, (5×3)2 =225, but 5 X 32 = 45. The learner must carefully observe, that in all such cases the sign on the outside of the vinculum affects no one of the terms within the vinculum, each of which has, besides, a sign of its own; that is, in the expressions 18-(5+7), and 16—(15—9), the terms 5 and 15 are understood to have the sign + (Art. 3). A bar drawn over the quantities is also used as a vinculum: the bar is sometimes placed vertically: thus, Brackets [], and the brace {}, are used instead of the parenthesis when we have occasion to place one vinculum outside of another; as, [(a+b)x+(a—b) x]2. 22. XVIII. Like quantities are those in which the parts expressed by letters are the same: thus, 54ay, 6ay, and -19ay, are like quantities. But 7ay, and 7a2y, are unlike; because the factor a occurs twice in the latter, and only once in the former, for (Art. 14) 7a2y =7aay. In the same manner, 17a2x, and 5xa2, are like quantities (Art. 11);; but 4a2x, and 6a3x, are unlike. 23. Schol. When two or more like quantities are connected by the signs +,-, they may be incorporated into one term, by performing on their coefficients the operations indicated by their signs, and to the result subjoining the common letter: thus, +5x+3x+7x=15x: for the first term contains five times x to be added, the next contains three times x to be added, and the next contains seven times x to be added, making in all fifteen times x to be added, that is to say, +15x. In like manner, -5x-3x7x=-15x; for, in this case, the res are all to be subtracted. Also, 6ab+7ab—5ab—3ab+4ab=+9ab: for 6ab with 7ab added will make, as above, +13ab; and 5ab subtracted will leave +8ab; 3ab further subtracted will leave +5ab; and 4ab added to this will give +9ab. In like manner, 5ax2+2ax2-8ax2-4ax2+3ax2- 2ax2; and 10xy-12xy+xy=-xy. Since (Art. 7) it is indifferent in what order the terms are taken, we may first unite all the positive quantities into one term, and all the negative into another, and then subtract the latter from the former: thus, 5ax2+2ax2-8ax2-4ax2+3ax2-10ax2-12ax2-20x2 24. XIX. The mark> or <, placed between two quantities, denotes that the one to which the opening of the lines is turned is greater than the other. Hence a> b means that a is greater than b; and x < y, that x is less than y. 25. XX. When it is not known which of two quantities is the greater, the mark placed between them indicates that the less is to be subtracted from the greater: thus, xy indicates that if, of the two quantities, x be the greater, y is to be subtracted from x; but if y be the greater, a is to be subtracted from y; that is, the expression xy is equivalent to x- -y when x > y, and to y―x when x <y. 13 PRAXIS ON THE DEFINITIONS. 26. The conversion of literal into numerical expressions will be found a useful exercise on the meaning of the symbols and operations of Algebra. The letters are supposed to denote particular numbers, which are substituted for the letters, and the same operations performed upon them that are performed upon the letters. Thus, let it be required to find what the expression abc a+b2+ac-abx+ -2axy 5dx will be in numbers, when a=4,b=5, c=3, d= 2, x=1, and y=0. Substituting the numbers for the letters, the expression becomes which is the value of the expression on the above supposition respecting the letters. EXERCISES. 1. What is a+b-c, when a=5, b=4, and c=6? Ans. 3. 2. What is the same expression, when a = 7, b = 2, c=8? Ans. 1. 3. What is the same expression, when a = 10, b = 6, c=12? Ans. 4. C |