ALGEBRAIC SIGNS. 75 sion of 16 by 4; that is, 16 divided by 4, or the 4th 260. SIGN OF INVOLUTION.-This is a small numeral or letter placed a little above and at the right of the quantity to which it belongs; and denotes a certain power of that quantity. 261. A power of a quantity means the quantity itself, or its product by itself a certain number of times. 262. The sign of involution attached to a quantity is called its index or exponent. In multiplying 4 once by 4 (4 × 4), 4 is taken twice as a factor. The product may be written 42, and is then called the second power or square of 4. The small 2 is the index or exponent of the second power of any quantity. When 4 is taken thrice as a factor (4 × 4 × 4), the product may be written 43, and is then called the third power or cube of 4; and a small 3 is the index of that power. 263. The quantity itself is considered its first power, its index as such being 1. Thus 41 is the same as 4. This index is seldom written, being usually omitted as understood. Sometimes it has to be reckoned, and added to or subtracted from other indices. 41 = 4; 42 = 16; 43 = 64; 44 = 256; 45 = 1024. The last is the fifth power of 4. 264. Thus, in involution, the quantity is to be multiplied by itself once less than the number of times denoted by its index, which expresses the number of times the quantity is taken as a factor. To get 43, we multiply 4 twice by itself, or write it thrice as a factor (4 × 4 × 4). 265. So, instead of writing aa, we write a2, meaning a multiplied once by itself; for aaa, we write a3, meaning a multiplied twice by itself. The first power of a is a1, or, simply, a. This is called involution, or raising a quantity to one of its powers. 266. SIGN OF EVOLUTION, or radical sign.-The sign √, together with the quantity before which it is placed, denotes some root of that quantity. A root is a quantity, one of whose powers is the quantity to which the radical sign is prefixed. 267. The sign √, or 2, denotes the square root, that root whose square (or second power) is the quantity before which it is placed. Examples.-The expression √9, called the square root of 9, means 3, or that quantity whose square is 9. The square of 3, or 32, is 9. So, a is that quantity whose Nax √a= = a. The square root of a2, or a2, is that quantity whose square is a2; that is, a ; square is a. for aa = a2. The sign, denotes the third root, or cube root; that root whose cube is the quantity before which it is placed. Examples. The 125 is 5, for 53, that is, 5 × 5 × 5, = 125. 63 is b, for b, raised to the third power, is b3. 268. Roots are sometimes expressed by fractional indices. Then, the upper number denotes the power, the lower figure the root. Thus, a is the same as a, meaning the square root of the first power of a; at is the cube root of a; a3 is the same as Va2, meaning the cube root of a2. 269. CO-EFFICIENTS.-The number prefixed as a factor to any quantity is called its co-efficient. Examples. In the expressions 3a, 7bc, d, the numbers 3, 7, are co-efficients. 270. When no co-efficient is expressed, the co-efficient 1 is understood, and has sometimes to be reckoned: la is the same as a. 271. The co-efficient is a multiplier, and shows how often the quantity following it is to be taken. Thus 8xy means 8 times the quantity xy. 272. Letters may sometimes be considered as coefficients. In the expressions ab, cd2, 5xy, a, c, 5x may be considered as co-efficients, meaning a times b, c times d2, 5x times y. 273. Algebraic expressions may often be simplified by adding or subtracting co-efficients of the same quantity. Thus 5a + 7a may be condensed into one term, (5+7) a, or 12a: so 136-6b can be reduced to one term, (136) b, or 7b. In like manner, ax + bx may be brought to one term, (a + b) x ; cy - dy may be expressed as (cd) y. Ratio-Proportion. 274. A Ratio is the relation of one quantity to another of the same kind, in respect to their magnitude. The relation of 2 to 3, in regard to their magnitude, is this-2 is, or contains, 2 times the third part of 3; that is, 2 is 2 times 1, and 3 is 3 times 1. Expressed fractionally, 2 is of 3. The ratio of 2 to 3 is therefore said to be , or 2:3; for ratio is expressed both ways. In like manner, the ratio of 3 to 2 is 3:2; 3; or 1; for is the same as 14. We see at once that 3 is one and a half of 2. 275. As is the same as 2 divided by 3, and the same as 3 divided by 2, the upper number being dividend, the lower, divisor; the ratio of one number to another is found by dividing the former by the latter. That operation shows how often the first contains the second. Examples.-Find the ratio of 12 to 4. 12 ÷ 4, or = 123. It is manifest that 12 contains 4 three times. Find the ratio of 11:4. 11÷4, or 11,23. 11 contains 2 times 4 and 2 of 4,—the ratio required is therefore 11: 4, 11, or 24. 4 Find the ration of 9 to 12., brought to lowest terms, is . The ratio required is therefore 3: 4, or . And it is plain that 9 contains 3 times the 4th part of 12. 276. The first term of a ratio is called the antecedent (or, going before term); the second term, the consequent (or, following term). 277. A proportion is formed by the terms of two equal ratios, properly arranged. The ratio of 2 to 7 is equal to that of 6 to 21; for are equal to 21 These numbers, then, form a proportion, the numerators being the antecedents of the two ratios, while the denominators are the consequents. proportion is written 27:6:21, or = This and is usually expressed, 2 is to 7 as 6 is to 21; that is, 2 has the same proportion to 6 as 7 has to 21; which is manifest, each antecedent being of its consequent. If four lines, A, B, C, D, are of such magnitude that the ratio of A to B is equal to that of C to D, these lines form a proportion, and AB:: C: D. If the length of A were such that it contained one and a quarter of B, then C would contain one and a quarter of D. Quantities which form a proportion are often said to be proportional. 278. A fourth proportional to three quantities, is a quantity which with them forms a proportion; that is, two equal ratios. Thus 10 is a fourth proportional to 2, 4, and 5, for In the Rule of Three, we find a fourth proportional to three given terms. 279. A third proportional to two quantities, is a quantity such that one of the given quantities is to the other as the latter is to the third proportional. Thus 25 is a third proportional to 4 and 10, for 4 10 10 : 25. 280. A mean proportional between two quantities, is a quantity such that one quantity is to the mean proportional as the latter is to the other quantity. 10 is a mean proportional between 4 and 25. 281. When four quantities form a proportion, the product of the 1st and 4th is equal to the product of the 2d and 3d; or, as usually expressed, the product of the extremes is equal to the product of the means. The 1st and 4th are the extremes; the 2d and 3d are the Also, the quotient of the first by either of the means, is equal to the quotient of the other mean by the fourth. If 4 10 6: 15, then; and = 18. 10 THE EQUATION. 282. The Equation is an algebraic expression, denoting the equality of two quantities. It is very useful for exhibiting, in a compendious form, rules or formulæ |