... The sign of continuation, is used for the words and so on, or and so on to. ... Thus 1, 2, 3, 4, means 1, 2, 3, 4, and so on, and x1, x2, xs, means x1, x2, x3, and so on to n. Хр ALGEBRAIC EXPRESSIONS. 23. A collection of algebraic symbols, that is of letters, figures, and signs, is called an algebraical expression. The parts of an algebraical expression which are connected by the signs + or are called the terms. Thus 2a-3 bx +5 cy2 is an algebraical expression containing the three terms 2 a, - 3 bx, and + 5 cy2. 24. When two terms contain the same letters, every one of which is raised to the same power in both, they are called like terms. Thus 3 ab2x3 and 7 ab2x3 are like terms; but although 3 a2bx3 and 7 a2b2x3 contain the same letters, they are not like terms, for all the letters are not raised to the same power. 25. A monomial expression is one which contains only one term, and a multinomial expression is one which contains more than one term. Thus 5 ab2cx is a monomial expression, and a + b is a multinomial expression. An expression which consists of two terms is often called a binomial expression, and an expression which consists of three terms is called a trinomial expression. Monomial and multinomial expressions are sometimes called respectively simple and compound expressions. SYMBOLS OF AGGREGATION. 26. The parentheses (), the square brackets [ ], and the braces {}, are used to indicate that all operations denoted by signs within the enclosure are to be consummated before any operation without it is performed. Thus (a + b) c means that b is to be added to a and that the result is to be multiplied by c; again, (a + b)3 means that b is to be added to a, and that the cube of the result is to be formed; also (a + 2b)(c-3 d) means that 2b is to be added to a and that 3 d is to be taken from c, and that the first of these results is then to be multiplied by the second. A line called a vinculum is often drawn over the expression which is to be treated as a whole: thus a b c is equivalent to a-(b-c), and Va+b is equivalent to √a+b √(a+b). When no vinculum or bracket is used, a radical sign refers only to the number or letter which immediately follows it: thus √2 a means that the square root of 2 is to be multiplied by a, whereas √2 a means the square root of 2 a ; also √a + x means that x is to be added to the square root of a, but √a +x means the square root of the sum of a and x. The line between the numerator and denominator of a+b a fraction acts as a vinculum, for is the same as 12 11⁄2 (a+b). NOTE. It should be carefully noticed that every term of an algebraical expression must be added or subtracted as a whole, as if it were enclosed in brackets. Thus, in the expression a+bc - d÷e+f, b must be multiplied by c before addition, and d must be divided by e before subtraction, just as if the expression were written a + (bc) — (d ÷ e) + f. when a = 4, b = 3, c = 1, d=0. The results are: (ii.) (a+b)3 (2 b - 3c)2 = (4+3)3 (2 × 3 − 3 × 1)2 = 73 × 32 = = 343 × 9 = 3087. (iii.) a + b + ca = 48 + 31 + 1a = 64 + 3 + 1 = 68. (iv.) 3⁄4/{7a3 + (b + c)3 + d3} = 3⁄43/{7 × 43 + 43 + 03} = {7 × 64 +64 + 0} = (448 +64) = 3/512 = 8. EXAMPLES V. 1. Write down the values of 24, 33, 43, 44, √64, 3/64, √/16, 3/125, †625, and √32. If a = 2, b = 3, c = 4, and d = 5, find the numerical values of 3, c = 4, and d = 5, find the numerical values of a2b2 8. + 27 12. ab3ct a4bc2d 20 11. a2b2c3ab2c2d. If a 5, b = 3, c = 1, d= 0, find the numerical values of 13. (2a +5b) (3 b − 6 c). 14. (a+2b) (c + 2 d). 15. (3a-4b)3 – 2 (3 b − 6 c)2 + 2(ad + bc)2. 16. 4a3 +4b3+ 4 c3 − 3(b + c) (c + a) (a + b). 17. 5(a + c)3 (b − c)2 — ž (a − 2 d)3 (b + 3 c)2. 18. Show that x2- 5x + 6 is equal to zero, if x = 2 or if x = 3. 25. 22. √2 bc + 3 a. 23. Vác +. 24. /(2 a2 + b2 - 8 c2). 26. √(a+b)(3 ab+2bc). 27. Find the numerical value of (a + b)2 (x + y)2 − 4 (ax + by) (bx + ay), 28. Find the value of √s(s − a)(s – b)(s – c), when a = 9, 30. Find, when a = 8, b = 5, c = 3, the numerical value of √{2 b2c2 + 2 c2a2 + 2 a2b2 — aa — b1 — c4}. 31. Verify that a2 b2 and (a + b)(a - b) are equal to one another (i.) when a = 6, b = 3; (ii.) when a = = 9, b = 4; and (iii.) when a = 12, b = 7. and 32. Verify that the expressions a3 — b3, (a − b) (a2 + ab + b2), (a − b)3 + 3 ab (a − b), (a + b)3 — 3 ab(a + b) — 2 b3 are all equal to one another (i.) when a = 3, b = 2; (ii.) when a = 6, b = 3; and (iii.) when a = 5, b = 2. POSITIVE AND NEGATIVE QUANTITIES. 27. All concrete quantities must be measured by the number of times each contains some unit of its own kind. Now a sum of money may be either a receipt or a payment, it may be either a gain or a loss; motion along a straight line may be in either of two opposite directions; a period of time may be either before or after some particular epoch; and so in very many other cases. Thus many concrete magnitudes are capable of existing in two diametrically opposite states. 28. Whatever kind of quantity we are considering, +4 will stand for what increases that quantity by 4 units, and 4 will stand for whatever decreases the quantity by 4 units. If we are calculating the amount of a man's property (estimated in dollars), +4 will stand for what increases his property by $4, that is +4 will stand for $4 that he possesses, or that is owing to him; so also - 4 will stand for whatever decreases his property by $4, that is - 4 will stand for $4 that he owes. If, on the other hand, we are calculating the amount of a man's debts, +4 will stand for whatever increases his debts, that is +4 will now stand for a debt of $4; so also 4 will now stand for whatever decreases his debts, that is — 4 will stand for $4 that he has, or that is owing to him. If we are considering the amount of a man's gains, +4 will stand for what increases his total gain, that is +4 will stand for a gain of 4; so also 4 will stand for what decreases his total gain, that is - 4 will stand for a loss of 4. If, however, we are calculating the amount of a man's losses, +4 will stand for a loss of 4, and 4 will stand for a gain of 4. Again, if the magnitude to be increased or diminished. is the distance from any particular point, measured in |