63. (which is read Divided by) signifies that the former of the quantities between which it is placed is to be divided by the latter. Thus ab signifies that the quantity a is to be divided by b. The division of one quantity by another is frequently represented by placing the dividend over the divisor with a line between them, in which case the expression is called a a fraction. Thus signifies a divided by b (Art. 11); and b a is the numerator, and b the denominator, of the fraction; a+b+c also e+f+ g signifies that a, b, and c added together, are to be divided by e, f, and g added together. 64. A power in the denominator of a fraction is also expressed by placing it in the numerator, and prefixing the negative sign to its index; thus a1, a-2, a ̄3, a ̄", 65. The sign between two quantities signifies their difference. Thus a~a, is a-x, or x-a, according as a or x is the greater; and a ±x signifies the sum or difference of a and x. [66. The sign > between two quantities signifies that the former is greater than the latter, and the sign < that the former is less than the latter. The sign.. signifies therefore, and since or because.] 67. When several quantities are to be taken collectively, they are enclosed by brackets, as (), []. Thus (a − b + c) × (d − e) signifies that the quantity represented by a b c is to be multiplied by the quantity represented by d e. Let a stand for 6; b, 5; c, 4; d, 3; and e, 1; then abc is 65+4, or 5; and d -e is 3 1, or 2; therefore (ab+c) x (de) is 5 × 2, or 10. (abcd)× (ab - cd) or (ab - cd)" signifies that the quantity represented by abcd is to be multiplied by itself. Sometimes a line, called a vinculum, is drawn over quantities, when taken collectively. Thus ab + c d e means the same as (a − b + c) × (d − e). x 68. =3 (which is read Equals or is Equal to) signifies that the quantities between which it is placed are equal to each other; thus ax - by cd ad signifies that the quantity ax-by is equal to the quantity ed+ad. = 69. The square root of any proposed quantity is that quantity whose square, or second power, gives the proposed quantity. The cube root is that quantity whose cube gives the proposed quantity, &c. The nth root is that quantity whose nth power gives the proposed quantity. The signs or √, or V, V, V, &c. are used to express the square, cube, biquadrate, &c. nth, roots of the quantities before which they are placed. 3 Va2 = a, Va3 = a, √a^= a, &c. Va" = a. These roots are also represented by the fractions 1 1 1 2 2 3 4 Thus &c. placed a little above the quantities, to the right. 1 7 að, að, at, a, represent the square, cube, fourth, and 7th, root of a, respectively; a, a, a, represent the square root of the fifth power, the cube root of the seventh power, the fifth root of the cube, of a, respectively. (See Note 1. Appendix 1.) 70. If these roots cannot be exactly determined, the quantities are said to be irrational, or are called surds. 71. Points are made use of to denote proportion; thus a b c d signifies that a bears the same proportion to b that c bears to d. 72. The number prefixed to any quantity, and which shews how often it is to be taken, is called its coefficient. Thus, in the quantities 7ax, 6by, 3ds, the numerals 7, 6 and 3 are called the coefficients of ax, by, and d≈, respectively. When no number is prefixed, the quantity is to be taken once, or the coefficient 1 is understood. These numbers are sometimes represented by letters, which are also called coefficients. [Thus in the quantities pa3, qx2, ræ, we call p, q, and r the coefficients of a3, 2 and a respectively.] 73. Similar or like algebraical quantities are such as differ only in their coefficients; 4a, 6ab, 9a2, 3a2bc, &c. are respectively similar to 15a, 3ab, 12a, 15a2bc, &c. Unlike quantities are different combinations of letters; thus ab, a2b, abc, &c. are unlike. 74. A quantity is said to be a multiple of another, when it contains it a certain number of times exactly; thus 16a is a multiple of 4a, as it contains it exactly four times. 75. A quantity is called a measure of another, when the former is contained in the latter a certain number of times exactly; thus 4a is a measure of 16a. 76. When two numbers have no common measure but unity, they are said to be prime to each other. 77. A simple algebraical quantity is one which consists of a single term, as a2bc. 78. A binomial is a quantity consisting of two terms, as a+b, or 2a-3bx. A trinomial is a quantity consisting of three terms, as 2a + bd - 3c. [A multinomial is a quantity consisting of many terms, as a + bx + cx2 + dx3 + &c.] The following examples will serve to illustrate the method of representing quantities algebraically. AXIOMS. 79. If equal quantities be added to equal quantities, the sums will be equal. 80. If equal quantities be taken from equal quantities, the remainders will be equal. 81. If equal quantities be multiplied by the same, or equal quantities, the products will be equal. 82. If equal quantities be divided by the same, or equal quantities, the quotients will be equal. 83. If the same quantity be added to and subtracted from another, the value of the latter will not be altered. 84. If a quantity be both multiplied and divided by another, its value will not be altered. 85. ADDITION OF ALGEBRAICAL The addition of algebraical quantities is performed by connecting those that are unlike with their proper signs, and collecting those that are similar into one sum. Ex. 1. Add together the following unlike quantities; It is immaterial in what order the quantities are set down, if we take care to prefix to each its proper sign. When any terms are similar, they may be incorporated, and the general expression for the sum shortened. 1st. When similar quantities have the same sign, their sum is found by taking the sum of the coefficients with that sign, and annexing the common letters. The reason is evident; 5a to be added (Ex. 3), together with 4a to be added, makes 9a to be added; and 3b to be subtracted, together with 76 to be subtracted, is 106 to be subtracted. 2d. If similar quantities have different signs, their sum is found by taking the difference of the coefficients with the sign of the greater, and annexing the common letters as before. In the first part of the operation we have 7 times a to add, and 5 times a to take away; therefore upon the whole we have 2a to add. In the latter part, we have 3 times b to add, and 9 times b to take away; therefore we have upon the whole 6 times b to take away; and thus the sum of all the quantities is 2a-6b. If several similar quantities are to be added together, some with positive and some with negative signs, take the difference |