Since 2 lies between 1.9881 and 2.0164 (that is, between 1.412 and 1.422), the 2 lies between 1.41 and 1.42, i.e., 1. 41 <√2< 1.42. This method of procedure can be continued indefinitely. may be summarized as follows: These results It follows from tables (a) and (b) that there can be found two numbers, one greater and the other less than √2, which differ from each other by as little as we please, and which therefore differ from √2, which lies between them, by as little as we please. Observe that the numbers of the one series, which are always less than √2, continually increase toward √2, while the numbers of the other series, which are always greater than √2, continually decrease toward √2. Either of these two values is an approximation to √2. In the proof of the general case, which now follows, it is necessary to represent the two values between which the required root lies at any stage of the work in terms of common fractions instead of decimal fractions. Thus, could have been written Let D 1.414 <√2 < 1.415 be a fraction, in which N and D (either or both) are not qth powers of positive integers. Evidently the powers D increase without limit. Therefore, whatever positive value may have, there will always be two consecutive powers of the above series between N D < is found to lie, wherein k1 is 0 or any positive integer. Then since lies between and 9 N lies between (I.) The method can evidently be carried on indefinitely; that is, we can 1 is found to lie at any stage of 1 As p increases without limit, 10p 10p the work evidently differ by decreases without limit (Ch. III., § 4, Art. 19). Since, therefore, these two numbers can be made to differ from each other by less than any assigned number, however small, the which lies between them, will differ from either of them by less than any assigned number, however small. 谔 IN Either of these numbers is an approximate value of D 5. It is important to keep clearly in mind that, although approximate values have been obtained for √2, and in general for have as exact values as have integers and fractions. IN these numbers Thus, √2 × √2=2, by definition of a root. Now no approximate value of √2 multiplied by itself gives exactly 2. Therefore the number which multiplied by itself gives 2 must have an exact value. This exact value, to be sure, cannot be expressed in terms of integers and fractions. ki ki k2 k1+1 ki, k2 + 1 + wherein p= 1, 2, 3, ......., ∞o. These two series have the following properties: (i.) The numbers 10 102 + wherein p = 1, 2, 3, of the first series increase as p increases, but remain always less than the numbers k1 k2 + + + 10 102 10p' (1) k1 k2 K+1 + + + 10 102 (2) 10p of the second series; and the numbers of the second series decrease as p increases, but remain always greater than the numbers of the first series. That is, the numbers of the one series more and more nearly approach the numbers of the other series, but never meet them. (ii.) The difference between a number of the one series and the corresponding number of the other series can be made less than any assigned number, however small, by taking p sufficiently great. 7. Two series of numbers which possess the properties (i.) and (ii.), Art. 6, are said to have a common limit, which lies between them. Two such series therefore define the number which is their common limit. This number is approached by both series and not reached by either. N The two numbers between which the lies can be reduced to D Let us designate 10 by n. Then since they may be represented by respectively. In the theory which follows, we shall let m n and 8. An Irrational Number is a number which cannot be expressed either as an integer or as a fraction, but which can be inclosed between two fractions ultimately differing from each other, and therefore from the inclosed number, by less than any assigned number however small. 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 m m+1 and n -, may have, there will always be numbers, integers or fractions, lying between any two numbers of the series and m n m + 1 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 relations are called equal and opposite. The absolute value of an irrational number is its value without regard to quality. The Fundamental Operations with Irrational Numbers. 12. Addition. - Let I and I2 be two positive irrational numbers 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, since my increases as n1 increases, and m2 increases as no in n1 m1 n2 creases, therefore + m2 increases as n1 and no increase. For a similar n1 n2 can be made less than any assigned number, however small. For 1 be made less than any assigned number, say d; and can be made less 1 1 = + 21 n2 1 1 than any assigned number, say d. Therefore, + can be made less 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, I1 + I2 + I3, = I1 + (I2 + 13) = I1 + (I3 + I2) = etc. |