- MATHEMATICS AND COMPUTER EDUCATION DIOPHANTINE APPROXIMATION AND THE IRRATIONALITY OF CERTAIN NUMBERS Eric Joes ad Thomas J. Osler Mathematics Departmet Rowa Uiversity 201 Mullica Hill Road Glassboro, New Jersey 08028 osler@rowa.edu 1. INTRODUCTION I may early courses i aalysis we examie the cocept of irratioal umbers. Relatively easy proofs of irratioality are give for umbers such as 12,.j3, lfj, etc. The irratioality of the umber e is slightly more difficult to show, but ca be explaied to studets who have studied ifiite series. I the paper [2], this elemetary method is explaied i detail ad exteded to a iterestig list of familiar umbers. I this paper we itroduce aother method, kow as Diophatie approximatio that will demostrate the irratioality of certai umbers. Diophatie approximatio is surprisigly simple to explai, ad ca be itroduced to studets who have studied ifiite series. This would iclude good studets i calculus as well as real aalysis courses. It is also a appropriate topic for a course i umber theory. We radomly examied 20 popular real aalysis texts ad did ot fid oe that icluded this topic. '~. ''''''"... c.'"""' ~ '~1i.. :s; ;.~'.~ -.- ~. 1.. ";~'..... i.~, ~. i~;? m ". :~~ 2. DIOPHANTINE APPROXIMATION The subject of Diophatie Approximatio utilizes a ovel method of showig the ature of real umbers, i particular whether the umbers are ratioal, irratioal, trascedetal, or algebraic. Surprisigly, we will deteie whether a give umber, x, is irratioal by studyig a sequece of ratioal umbers P that approach x. (Throughout this paper, P will ~ ~ deote a ratioal umber where the umerator ad deomiator are itegers.) We require that the deomiators qmootoically approach ifiity. For example, cosider the sequece of ratioal umbers defied by P +l -=--, q 2 + 1 for =1,2,3,.... Clearly this sequece approaches the umber! as approaches ifiity. It is sequeces of this type that will be used i the 125
- MATHEMATICS AND COMPUTER EDUCATION followig discussio to prove the irratioality of certai umbers. I geeral, it is ot difficult to see that for ay umber x, ad a give deomiator q' we ca fid the umerator P such that P I x-- ~--. q 2q Assume that our umber x =f is a ratioal umber such that r ad s are atural umbers. Let us also suppose that P;I:. ~. This yields q s _r _ p _Irq-spi s q sq The umerator is a positive iteger, thus the smallest value it ca be is oe. We have the 1 r p 1 -< --- < sq - S q - 2q We see that s is idepedet of the approximatig fractio ~. We have derived the followig importat theorem. Theorem 1: Give x > 0 a ratioal umber, we choose a sequece of ratioal approximatios to x. Each term ofthis sequece P ;I:. x, both P q ad q are itegers ad the q grow mootoically to ifiity. The there exists a positive costat c such that P c -<X--. q q (The positive costat c referred to i this theorem is the umber t from the previous aalysis.) We will ow use Theorem 1 to prove the irratioality of certai umbers, as usual, by cotradictio. Suppose we have a sequece of ratioals P that approach x ad we ca show that q where r ~ co as ~ co. Ifx is ratioal, by Theorem 1, we would have the existece of a positive costat c such that 126
.,.'-... 7 a (j - MATHEMATICS AND COMPUTER EDUCATION C P 1 -<x-- <--. q q qr This meas that c < - 1. ThO IS IS o. ImpOSSI 'ble SIce the umber c is r idepedet of r ad the umbers r approach ifiity. Thus we coclude that x is irratioal. 3. THE IRRATIONALITY OF e We ow use Theorem 1 to prove that e is irratioal. Start by defiig e as the ifiite series e = "'CXJ_oJ...-. We choose for our approximatig L... -! sequece P = "'k _oj...-. It is importat to otice that our deomiator q L... - k! q =!o The we have. 1 Factog (+l)! ~ ICXJ 1 1 1 1 e-q:= k=+] k! = (+l)! + (+2)! + (+3)! +... we get P 1 (1 1 J e-q:= (+l)! 1+ (+2) + (+2)(+3) +... Ufortuately we caot sum the series i parethesis. However, we ca replace it by a geometric series which is larger. For this series we ca use P 1 ( 1 1 J e-q: < (+l)! 1+ (+2) + (+2)2 +... Recall that the geometric series 1+ r + r 2 + r 3 + 0 P 1 1 e - - <---I ------,- q (+l)! 1-1 (+2) Simplify the right side ofthe iequality to get P +2 e--<. q!(+l)2 0 0 =_1_. Therefore l-r Assume x =e is ratioal. From Theorem 1 we kow there exists a positive 127
- MATHEMATICS AND COMPUTER EDUCATION- costat c, idepedet of P such that q C P +2 -< e-- ~----! q!(+l)2. ld +2 ThIS yle s C < 2 which is impossible sice + 2 ~ 0 as ~ 00 (+l) ( +1)2 Thus our assumptio is wrog ad e is irratioal. 4. ANALYSIS AND GENERALIZATION OF THE PROOF Now we will geeralize the previous proof of the irratioality ofe. I order to do this we will preset a geeral class of umbers defied by series which will allow us to repeat the previous steps. We defie a geeral umber x as the ifiite series, x =L oo a =O a subject to the followig restrictios: a. a =1 or-i b. a =11 ck where the ck are itegers k=o c. ck < ck+1 d. limk~oo ck =00 Step A. Just as before, we allow the approximatig sequece to be P = L:=o a. Notice that our deomiator q = a' q ak Step B. Subtract the approximatig sequece from x ad expad the remaiig series x _ P =L:=+1 a k = a + 1 + a + 2 + a + 3 +... q ak a+1 a+2 a+3 Step C. Factor the first term from the series. 128
- MA THEMATICS AND COMPUTER EDUCATION Step D. Fid a geometric series that is greater tha the previous series. Because the sequece of itegers c are mootoically icreasig, ad 8 == ±1, we have 1 x_ P <_1_(1+ + 1 2 + 1 3 +...J. Step E. Sum the geometric series. q a+l c+2 (C +2) (C+2) x _ p < 1 [c+ 2 ]. q CIC2"' C C +l C+2-1 Step F. Assume that x is ratioal. The by Theorem 1 there exists a positive C such that C = C < x- P < 1 [C +2]. a clc 2", c q cl c2",cc+1 c +2-1 Step G. Fid a cotradictio to prove x is irratioal. This meas C <--1 [ --'-'-'~ c +2 ] c+1 C+2-1 which is impossible sice [ C +2 ] ~ 1 ad _1_ ~ 0 as ~ 00. c+2-1 c+l 5. FAMILIAR NUMBERS THAT ARE IRRATIONAL Here we list several values of familiar fuctios that the above method will show to be irratioal. Examples: All the followig umbers with M == 1, 2, 3,... are irratioal. e llm = ~ I =o!m ~ ' 00 (_1) sm(li M) == I 2 I' =o(2+1)!m + (1) cos(l! M) = I 00-2' =O (2)!M CL) 1 smh(ll M) =I 2 I' ad =O (2 + l)!m + CL) 1 cosh(ll M) =L 2. =O (2)!M 129
t Explaatio: We will ow show the umber e = I with =o!m M = 1,2,3,... meets the ecessary restrictios from sectio 4. Clearly 8 k =I ad restrictio a is met. We ca see aj = M, az = 2M 2, a3 =3!M 3, etc. Additioally, ci = M, c2 = 2M, c3 == 3M,..., ck =km. Thus, restrictios b, c, ad d are met. Thus ellm with M =1,2,3,... is prove to be irratioal. We leave the remaiig umbers to the reader to examme. 6. FINAL REMARKS A umber is said to be algebraic of degree d if it is the root of a polyomial equatio with iteger coefficiets ofdegree d, ad ot the root of ay polyomial equatio of lesser degree. If a umber is ot algebraic it is called trascedetal. While irratioal umbers were recogized as early as 600 BCE, the existece of trascedetal umbers was ot demostrated util the ieteeth cetury. A extesio of our Theorem I kow as Liouville's theorem ca be used to demostrate that certai umbers are trascedetal. Here the iequality i our theorem is replaced by c P I -< x-- :S;-. qd q 2q The trascedetal ature of certai umbers x is the proved by cotradictio i much the same way as we proved irratioality i this paper. Liouville's theorem is oly slightly more difficult to prove tha our Theorem I ad a clear explaatio ca be foud i [I, pp. 104-107]. - MATHEMATICS AND COMPUTER EDUCATIONllM REFERENCES 1. R. Courat, ad H. Robbis, What is Mathematics?, A Elemetary Approach to Ideas ad Methods (revised by Ia Stewart), Oxford Uiversity Press, Oxford, ISBN: 0-19510519-2 (1996). 2. N. Stugard ad T. 1. Osler, "A collectio of umbers whose proof of irratioality is like that of the umber e", Mathematics ad Computer Eduatio, Vol. 40, pp. 103-107, ISSN: 0730-8639 (2006). 130