350. Irrational numbers obey the commutative, associative, and distributive laws of integers and rational numbers. This generalization is readily obtained on considering the approximate values of the given numbers. For example, the product of two irrational numbers ab is the limit of the product xy, x and y being rational numbers which have respectively the limits a and b. Hence in the product of two irrational factors we can interchange the factors, and thus is established the commutative law for irrationals. Similarly it may be shown readily that the laws governing the fundamental operations with irrational numbers are the same as those governing these operations with rational numbers. E. g., 1. v3±√7 = ±√7+v3. 2. √2+(−√5)+ √ 11 = √2 + √11+ (−√5). 3. √7÷(√5 ÷ √13) = √ 7 ÷ √5 × √ 13. 4. (√215) = (√5) (v2). V + 351. Equality.-Of two numbers a and b, defined by series I and II, 8349, the first is greater than, equal to, or less than the second according as This definition is to be justified exactly as the definitions of the fundamental operations on irrational numbers were justified in 2348. Some important principles of limits follow immediately from the results established in the preceding paragraphs. For by definition, 348, CHAPTER VI SURDS 352. In 311 the student has learned that there are two notations in use for expressing the root of an expression, one notation using the radical signs and the other fractional exponents. Though it is not necessary to have two ways of writing the same thing, yet, because each notation has special advantages in certain cases, the two notations are retained. It has been shown in Chapter V, Book III, that the same laws I-V of 26, 7, which govern the fundamental operations on integers, zero, the negative number, and the fraction, govern the operations with roots, in both forms of notations mentioned, 350. 353. A radical is an indicated root of a number or quantity; as √5, 3√8, √x+y, 3√x2 (a+b)3. 3 5 A radical expression is an expression composed of radicals; as 3√71, √a+√b, (Va+√b) +3⁄4√c. 3 A surd is a root of a rational quantity which can not be found exactly; as 16, Va, 3√2. 3 One should notice that 1/1 +33/5 is not a surd, since 1+33√5 is an irrational number. The distinction between arithmetical and algebraical irrationality is important. Thus, V is algebraically irrational; but in case x = 16, then √x=√16 = 4 is arithmetically rational. 5 3 Thus, √, '√4 are surd numbers. Expressions like √9, 3⁄4√/8, etc., are written in the form of surds. Expressions like Vx, √xy, etc., are often called surds, although, of course, they are such only in case x and y are commensurable quantities whose roots can not be found exactly. In the preceding definitions a distinction is made between the terms incommensurable, irrational expression, and surd. According to the definition just given 3 3√3+1/5, 3√ √7, √TM, √ e = Napierian base, are not surds; however, they are irrational and incommensurable. This limited meaning of the term surd is not only convenient, but is being used more and more by authors. ORDERS OF SURDS 354. A quadratic surd or a surd of the second order, is one with index 2; as 15, vx. A cubic surd, or a surd of the third order, is one with index 3; as 1/4, 1/11, √ x + y. 3 3 3 A biquadratic surd, or a surd of the fourth order, is one with index 4; as *7, *Vx (x + y). Similarly, surds are classified according to their indices, as Quintic, . . . n-tic., as the case may be. A simple monomial șurd number is a single surd number, or a rational multiple of a single surd number; as v2, 13√5. A simple binomial surd number is the sum of a rational number and a simple surd number or of two simple surd numbers; as 3+√5, √3 + 3√7. 2 3. 355. The rules for operations with surds follow from the principles and theorems of Chapter III, 303, etc. We restate for convenience each principle as occasion for its use in this chapter arises. As in evolution, Chapter II, the positive values only of the radicals are considered and likewise the principal roots only. NOTE. In operations involving surds, arithmetical numbers contained in the surds should be resolved into their prime factors. REDUCTION OF SURDS TO THEIR SIMPLEST FORM 356. A surd is in its simplest form if the radicand is integral, and does not contain a factor whose exponent is equal to or a multiple of the index of the root; as 1 3, 31 xy2, "V x”, 3 A surd can be reduced to its simplest form by applying one or more of the following principles: 357. RULE 1.-A rational quantity can be expressed in the form of a surd, by raising it to a power whose exponent is equal to the index of the surd desired. 358. RULE II.-The coefficient of a surd may be introduced under the radical sign, by first reducing it to the form of the surd (Rule I), then multiplying according to principle I, 356. (3) (4) (5) 31/43.5 = 3√320. a V x = V a2 • V ́ x = = Va2x. xV2a-x2 Vx2 V2a- x2 = √2 ax2 — x1. x(x,y) = (x})}(x-y)} = = [(x})3 (x − y)3]! = [x2 (x − y)3]1. [x} (x—y)]1 [1357; 1356, 1] 359. RULE III.—Conversely, any quantity may be made the coefficient of a surd, if the radicand is divided by the quantity raised to a power whose exponent is equal to the index of the surd. Thus, (1) 160 = √(2)o15 = √/2o √/15 = 2 √/15. (2) √x1y2= √(x*y2)x = √x1y2Vx=x2y√x. [2356, I, V] [8356, I, V] (3) 1/16a2-16a2x2=V ́16a2(1—x2)=√/16a2v 1—x2-4a√/1—x2. 2n 2n (4) "√xn+1y2n +3="V\/(x"y2")(xy3)="V_x"y3n "Vxy3=xy2 "√xy3. 360. RULE IV.—If the radicand is a fraction, the surd may be reduced to its simplest form by multiplying both terms of the fraction by such a quantity as to make the denominator a perfect power of the same degree as the surd, and then proceed as in 1359, using also principle II, 356. 361. RULE V.-Since IV, 8356, is true in all cases, we know that the index of a surd can be lowered if the expression under the radical sign is a perfect power corresponding to some factor of the original radical index. [2356, IV] 3 or "V 362. A surd is in its simplest form: (1) when the radicand is not itself, or does not contain, a factor which is a perfect power of the required root; (2) when the radicand is integral; (3) when the index of the surd is the lowest possible. 363. It is usually supposed in any piece of work, that all the surds will finally be left in their simplest form. |