An irrational number, I, is therefore defined by the relation have the properties (i.) and (ii.), Art. 6; as √2. 9. Whatever value p, and therefore and m m+1 -, may have, there will always be numbers, integers or fractions, lying between any two numbers of the series and m n m+1 n But no such number can be selected which will not be passed by numbers of one or the other series, if p be sufficiently increased. Therefore there is no number in the system defined so as to include only integers and fractions, which is greater than every number of the series (1.) and less than every number of the series (2.); that is, which is approached by both series and not reached by either. Since, however, these series cannot meet, we conclude that there was a gap between them which could not be filled by any integer or fraction. Consequently by including irrational numbers in the number system, continuity has been introduced where before it was lacking. Negative Irrational Numbers. 10. If the fractions of the series which define an irrational number be negative, the number thus defined is called a Negative Irrational Number. Therefore a negative irrational number is defined by the relation 11. The positive and negative irrational numbers defined by the The Fundamental Operations with Irrational Numbers. 12. Addition. - Let I and I be two positive irrational numbers defined by the relations mi+1 (1) (2) 12 n2 If the corresponding rational numbers of the series which define I1 and I be added, we obtain the two series of rational numbers The numbers of these series have the properties (i.) and (ii.), Art. 6. For, m1 since increases as n1 increases, and increases as no in m1 N2 creases, therefore + me increases as n1 and no increase. For a similar n1 N2 be made less than any assigned number, say d; and than any assigned number, say d. Therefore, than d+d, = d. Therefore, the two series of numbers 1 N1 can be made less can be made less N2 determine a positive number which lies between them. This number is defined as the sum I1 + I2. That is, Exactly similar reasoning will apply if either or both of the irrational numbers be negative, or either be rational. - 13. Subtraction. The following definition of Subtraction of irrational numbers is a natural extension of the principle of subtraction for rational numbers. To subtract an irrational number from a rational or irrational number is equivalent to adding an equal and opposite irrational number. The Associative and Commutative Laws for Addition and Subtraction of Irrational Numbers. 14. These fundamental laws hold also for irrational numbers; that is I1 + I2 = I2+I1, 11+ 12+ Is, = 11 + (I2 + Is) = I1 + (Is + I2) = etc. The Associative Law can be proved in a similar manner. E.g., √2+√3 = √3+√2, √2 + √3 + ( − √5) = √2 + ( − √5) + √3. 15. Multiplication.- Let I1 and I be two positive irrational numbers defined by the relations creases, 11 therefore mi M2 mi + 1 n2 reason If the corresponding rational numbers of the series which define I1 and I be multiplied, we obtain the two series of rational numbers, The numbers of these series have the properties (i.) and (ii.), Art. 6. For, since increases as n1 increases, and increases as no in N2 increases as n1 and no increase. For a similar can be made less than any assigned number, however small. For, since decreases, it is always less than some positive finite m2 M2 + 1 rational number, say R; and since N2 positive finite rational number, say R1. mi + 1 n1 n2 N2 m1 can each be made less than any assigned number, say d. Therefore the given difference can be made less than dR + dR1, = d(R + R1). But d(R+ R1) can be made less than any assigned number, say 8, by taking d less than δ R+R1 determine a positive number which lies between them. defined as the product I1· I1⁄2. That is, This number is The following definition of multiplication of irrational numbers is consistent with the preceding result. The product of two irrational numbers is the product of their absolute values, with a sign determined by the laws of signs for the product of two rational numbers. The Associative, Commutative, and Distributive Laws for Multiplication of Irrational Numbers. 16. These fundamental laws hold also for irrational numbers. That is, I1I2 = I2I1; I1I2I3 = I1(I2I3) = etc.; (I1 ± 12) I3 = I1I3 ± I2I3. The proofs of these principles are similar to those given in Art. 14. 17. Reciprocal of an Irrational Number.—It can easily be proved that the reciprocal of the numbers of the series which define I have the properties (i.) and (ii.), Art. 6. number which lies between them. This number is defined as the reciprocal 1 of I. That is, m + 1 m 18. Division. - Division by an irrational number may be defined as follows: To divide any number by an irrational number, not 0, is equivalent to multiplying it by the reciprocal of the irrational number. From this definition it follows that the fundamental laws hold also for division of irrational numbers. 19. It follows from the preceding theory that the laws governing the fundamental operations with irrational numbers are the same as those governing these operations with rational numbers. E.g., √(√√3 − √5) = √2 − √3 +√5; (√2√3)3 = ( √2)3 ( √√3)3. CHAPTER XIX SURDS. 1. In Ch. XVI. we considered only roots whose radicands are powers with exponents equal to or multiples of the indices of the roots. In Ch. XVIII. we assumed the existence of roots of numbers which are not powers with exponents equal to or multiples of the indices of the required roots, and proved that such roots obey the fundamental laws of Algebra; as √2 × √3=√3× √2, etc. Such roots were called Irrational Numbers. 2. A Rational Number is a number which can be expressed as 2 x an integer or as a fraction; as 2,3 √(27 ao). y A Rational Expression is an expression which involves only rational numbers; as a + b, ab + √a2. 3. A Radical is an indicated root of a number or expression; as √7, √9, (a + b). A Radical Expression is an expression which contains radicals; as 27, √x + √y, √(a + √b). A Surd is an irrational root of a rational number; as √7, √α. Observe that √(1 + √7) is not a surd, since 1 +√7 is not a rational number. Notice the difference between arithmetical and algebraical irrationality. Thus, a is algebraically irrational; but if a = 4, then √a, = √√4, = 2, is arithmetically rational. Classification of Surds. 4. A Quadratic Surd, or a Surd of the Second Order, is one with index 2; as √3, Va. |