Equation with roots reciprocals of those of f (x)=0 Discussion of reciprocal equations. Equation with roots squares of those of f (x)=0 Equation with roots exceeding by h those of ƒ (x) = 0 Equation with roots given functions of those of f(x)=0 Cubic equations. Cardan's Solution HIGHER ALGEBRA. CHAPTER I. RATIO. 1. DEFINITION. Ratio is the relation which one quantity bears to another of the same kind, the comparison being made by considering what multiple, part, or parts, one quantity is of the other. The ratio of A to B is usually written A: B. The quantities A and B are called the terms of the ratio. The first term is called the antecedent, the second term the consequent. 2. To find what multiple or part A is of B, we divide A by B; hence the ratio A: B may be measured by the fraction and we shall usually find it convenient to adopt this notation. A B In order to compare two quantities they must be expressed in terms of the same unit. Thus the ratio of £2 to 15s. is measured NOTE. A ratio expresses the number of times that one quantity contains another, and therefore every ratio is an abstract quantity. 3. Since by the laws of fractions, it follows that the ratio a b is equal to the ratio ma mb ; that is, the value of a ratio remains unaltered if the antecedent and the consequent are multiplied or divided by the same quantity. H. H. A. 1 4. Two or more ratios may be compared by reducing their equivalent fractions to a common denominator. a b and xy are two ratios. Now a ay b by and Thus suppose х bx y by the ratio a b is greater than, equal to, or less than the ratio x y according as ay is greater than, equal to, or less than bx. 5. The ratio of two fractions can be expressed as a ratio 6. If either, or both, of the terms of a ratio be a surd quantity, then no two integers can be found which will exactly measure their ratio. Thus the ratio √2:1 cannot be exactly expressed by any two integers. 7. DEFINITION. If the ratio of any two quantities can be expressed exactly by the ratio of two integers, the quantities are said to be commensurable; otherwise, they are said to be incommensurable. Although we cannot find two integers which will exactly measure the ratio of two incommensurable quantities, we can always find two integers whose ratio differs from that required by as small a quantity as we please. so that the difference between the ratios 559017 : 1000000 and √5: 4 is less than '000001. By carrying the decimals further, a closer approximation may be arrived at. 8. DEFINITION. Ratios are compounded by multiplying together the fractions which denote them; or by multiplying together the antecedents for a new antecedent, and the consequents for a new consequent. Example. Find the ratio compounded of the three ratios 2a: 3b, 6ab: 5c2, c: a : 9. DEFINITION. When the ratio a b is compounded with itself the resulting ratio is a2: b2, and is called the duplicate ratio of a : b. Similarly a3: 63 is called the triplicate ratio of a: b. Also a b is called the subduplicate ratio of a : b. : Examples. (1) The duplicate ratio of 2a : 3b is 4a2 : 962. (2) The subduplicate ratio of 49 : 25 is 7 : 5. (3) The triplicate ratio of 2x : 1 is 8x3: 1. 10. DEFINITION. A ratio is said to be a ratio of greater inequality, of less inequality, or of equality, according as the antecedent is greater than, less than, or equal to the consequent. 11. A ratio of greater inequality is diminished, and a ratio of less inequality is increased, by adding the same quantity to both its terms. and a-b is positive or negative according as a is greater or less than b. Similarly it can be proved that a ratio of greater inequality is increased, and a ratio of less inequality is diminished, by taking the same quantity from both its terms. 12. When two or more ratios are equal many useful propositions may be proved by introducing a single symbol to denote each of the equal ratios. The proof of the following important theorem will illustrate the method of procedure. where p, q, r, n are any quantities whatever. Let then whence pa" =pb"k", qc" = qd"k", re" =rf "k",...; pa" + qc" + re" + ..._pb"k" + qd"k" + rf"k" + ... pb"+qd"+rf"+... = pb" + qd" + rf" + ... By giving different values to p, q, r, n many particular cases of this general proposition may be deduced; or they may be proved independently by using the same method. For instance, a result of such frequent utility that the following verbal equivalent should be noticed: When a series of fractions are equal, each of them is equal to the sum of all the numerators divided by the sum of all the denominators. |