Undergraduate Seminar I
Undergraduate Seminar I MATH 2784
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Date Created: 09/17/15
IRRATIONALITY OF 7r AND 6 KEITH CONRAD Math 2784 University of Connecticut Date Sept 2 2009 1 2 3 4 CONTENTS Introduction Irrationality of 7139 Irrationality of 6 General Ideas Irrationality of rational powers of 6 References OOCTJgtJgtWMH IRRATIONALITY OF 7r AND 6 1 1 INTRODUCTION Numerical estimates for 7139 have been found in records of several ancient civilizations These estimates were all based on inscribing and circumscribing regular polygons around a circle to get upper and lower bounds on the area and thus upper and lower bounds on 7139 after dividing the area by the square of the radius Such estimates are accurate to a few decimal places Around 16007 Ludolph van Ceulen gave an estimate for 7139 to 35 decimal places He spent many years of his life on this calculation7 using a polygon with 262 sides With the advent of calculus in the 17 th century7 a new approach to the calculation of 7139 became available in nite series For instance7 if we integrate 7171 t47t6 t87t10 tlt1 1 1 f f 7 H fromt0totzwhenlxllt17we nd 3 5 7 9 11 11 arctanzziii H39 3 5 7 9 11 Actually7 this is also correct at the boundary point z 1 Since arctan1 7T47 11 specializes to the formula 1 2 7r 1 1 1 1 1 1 quotl 4 35 79 11 which is due to Leibniz lt expresses 7139 in terms of an alternating sum of the reciprocals of the odd numbers However7 the series in 12 converges much too slowly to be of any numerical use For example7 truncating the series after 1000 terms and multiplying by 4 gives the approximation 7r 314057 which is only good to two places after the decimal point There are other formulas for 7139 in terms of arctan values7 such as 7139 1 1 7 arctan 7 arctan 7 4 2 3 1 1 2 arctan 7 arctan 7 3 7 1 1 4 arctan 7 7 arctan 7 5 239 Since the series for arctanz is more rapidly convergent when x is less than 17 these other series are more useful than 12 to get good numerical approximations to 7139 The last such calculation before the use of computers was by Shanks in 1873 He claimed to have found 7139 to 707 places In the 1940s7 the rst computer estimate for 7139 revealed that Shanks made a mistake in the 528 th digit7 so all his further calculations were in error Our interest here is not to ponder ever more elaborate methods of estimating 7T but to prove something about the structure of this number it is irrational That is7 7139 is not the ratio of two integers The basic idea is to argue by contradiction We will show that if 7139 is rational7 we run into a logical error This is also the principle behind the proof that the simpler number 2 is irrational However7 there is an essential difference between the proof that 2 is irrational and the proof that 7139 is irrational One can prove 2 is irrational using some simple algebraic manipulations with a hypothetical rational expression for x2 to reach a contradiction But the irrationality of 7139 does not involve only algebra It requires calculus 2 KEITH CONRAD That is7 all known proofs of the irrationality of 7139 are based on techniques from analysis Calculus can be used to prove irrationality of other numbers7 such as e and rational powers of e excluding of course 60 1 The remaining sections are organized as follows In Section 27 we prove 7139 is irrational using some calculations with de nite integrals The irrationality of e is proved using in nite series in Section 3 A general discussion about irrationality proofs is in Section 4 and we apply those ideas to prove the irrationality of non zero rational powers of e in Section 5 2 TRRATIONALITY OF 71 The rst serious theoretical result about 7139 was established by Lambert in 1768 7139 is irrational His proof involved an analytic device which is never met in calculus courses in nite continued fractions A discussion of this work is in 17 pp 68778 Lambert7s proof for 7139 was actually a result about the tangent function When r is a non zero rational where the tangent function is de ned7 Lambert proved tanr is irrational Then7 since tan7T4 1 is rational7 7139 must be irrational or we get a contradiction The irrationality proof for 7139 we give here is due to Niven 3 and uses integrals instead of continued fractions Theorem 21 The number 7139 is irrational Proof For any nice function fz7 a double integration by parts shows f sinxdz 7fx cosm f sinx 7 f z sinmdm Therefore using sin0 07 cos0 17 sin7r 07 and cos7r 71 f sinxdz f0 f7r 7 f x sinsdm 0 0 ln particular7 if f is a polynomial of even degree7 say 2717 then repeating this calculation n times gives an AUWNmMMFFWL where Fltzgt we 7 W flt4gtltzgt 7 aware To prove 7139 is irrational7 we will argue by contradiction Assume 7T pq with non zero integers p and q Of course7 since 7139 gt 0 we can take p and q positive We are going to apply 21 to a carefully and mysteriously chosen polynomial fx and wind up constructing an integer which lies between 0 and 1 Of course no such integer exists7 so we have a contradiction and therefore our hypothesis that 7139 is rational is in error 7139 is irrational For any positive integer ii7 set ween wiww 7 TL 7 22 fnx i q n i n This polynomial depends on n and on 7Tl We are going to apply 21 to this polynomial and nd a contradiction when 71 becomes large But before working out the consequences of 21 for f fnx7 we note the polynomial fnx has two important properties 0 for 0 lt z lt 7T fnz is positive and when n is large very small in absolute value7 0 all the derivatives of fnx at z 0 and z 7139 are integers IRRATIONALITY OF 7r AND 5 3 To show the rst property is true the positivity of fnx for 0 lt z lt 7139 is immediate from its de ning formula To bound from above when 0 lt z lt 7139 note that 0 lt W72 lt 7139 so x7r 7 lt 7T2 Therefore m lMMf The upper bound tends to 0 as n 7 00 In particular the upper bound is less than 1 when 71 gets suf ciently large To show the second property is true we rst look at z 0 The coef cient of in fnx u m r n n is 5quot0j At the same time since fnx x p 7 qz n and p and q are integers the binomial theorem tells us the coef cient of can be written as 0771 for some integer 07 Therefore 39l m w7a Since fnx has its lowest degree non vanishing term in degree n 07 0 for j lt n so f gt0 0 forj lt n Forj 2 n jn is an integer so f gt0 is an integer by 24 To see the derivatives of fnx at z 7139 are also integers we use the identity fn7r 7 m Differentiate both sides j times and set x 0 to get 71j 53 7W0 for all j Therefore since the right side is an integer the left side is an integer too This concludes the proof of the two important properties of Now we look at 21 when f f Since all derivatives of fn at 0 and 7139 are integers the right side of 21 is an integer when f fn look at the de nition of Therefore f0 fnz sinzdm is an integer for every 71 Since fnz and sinx are positive on 071 this integral is a positive integer However when n is large sinx S S q7T2 n by 23 As 71 7 oo q7T2 n 7 0 Therefore f0 fnx sinxdz is a positive integer less than 1 when n is very large This is absurd so we have reached a contradiction Thus 7139 is irrational D This proof is quite puzzling How did Niven know to choose those polynomials fnx or to compute that integral and make the estimate 3 TRRATIONALITY OF 6 We turn now to a proof that e is irrational This was rst established by Euler in 1737 using in nite continued fractions We will prove the irrationality in a more direct manner using in nite series The idea of this proof is due to Fourier and it is short Theorem 31 The number 6 is irrational Proof Write For any n 6 111 1 11 2 3 n n1 n2 7111 11 1 1 T 2 3 n n n1 n2n1 39 4 KEITH CONRAD The second term in parentheses is positive and bounded above by the geometric series 1 rmrmi 1 n1 n12 n13 in Therefore 0lt57 i lt 21 31 711 771711 Write the sum 1 12 171 as a fraction with common denominator 71 say as pnnl Clear the denominator n to get 1 31 0ltnle7pn 7 71 So far everything we have done involves no unproved assumptions Now we introduce the rationality assumption lf 6 is rational7 then 7116 is an integer when n is large since any integer is a factor by n for large But that makes 7116 7p an integer located in the open interval 01717 which is absurd We have a contradiction7 so 6 is irrational D 4 GENERAL IDEAS Now its time to think more systematically The basic principle we need to understand is that numbers are irrational when they are approximated too well77 by rationals Of course7 any real number can be approximated arbitrarily closely by a suitable rational number use a truncated decimal expansion For instance7 we can approximate 2 141421356 by 14142 1414213 7 14142 7 10000 1000000 With truncated decimals7 we achieve close estimates at the expense of rather large denomi nators To see what this is all about7 compare the above approximations with 41 1414213 99 1393 42 7 141428571 7 141421319 70 985 where we have achieved just as close an approximation with much smaller denominators 697 the second one is accurate to 6 decimal places with a denominator of only 3 digits These rational approximations to x2 are7 in the sense of denominators7 much better than the ones we nd from decimal truncation To measure the quality77 of an approximation of a real number 04 by a rational number goq7 we should think not about the difference la 7pql being small in an absolute sense7 but about the difference being substantially smaller than 11 thus tying the error with the size of the denominator in the approximation In other words7 we want q lqaipl 10 a7 1 to be small in an absolute sense Measuring the approximation of Oz by pq using lqa 7 pl rather than la 7pql admittedly takes some time getting used to7 if you are new to the idea Consider what it says about our approximations to For example7 from 41 we have l10000 7 141421 135623 l1000000 714142131 562373 IRRATIONALITY OF 7r AND 6 5 and these are not small when measured against 110000 0001 or 11000000 000001 On the other hand from the approximations to 2 in 42 we have 170f7991005050 1985 71393l000358 which are small when measured against 170 014285 and 1985 001015 We see vividly that 9970 and 1393985 really should be judged as good77 rational approximations to 2 while the decimal truncations are bad77 rational approximations to The importance of this point of view is that it gives us a general strategy for proving numbers are irrational as follows Theorem 41 Let oz 6 R If there is a sequence of integers pmqn such that qnoz 7p 74 0 and lqnoz 7101 7 0 as n 7 00 then 04 is irrational In other words if 04 admits a very good77 sequence of rational approximations then 04 must be irrational Proof Since 0 lt lqnoz 7pnl lt 1 for large n by hypothesis we must have 1 74 0 for large n Therefore since only large n is what matters we may change terms at the start and assume 1 74 0 for all ii To prove 04 is irrational suppose it is rational oz ab where a and b are integers with b 74 0 Then a Ii 2 1 P M an b an bqn 39 Clearing the denominator q i Ina 7 pnb Since this is not zero the integer qna 7 pnb is non zero Therefore lqna 7 pnbl 2 1 so 1 This lower bound contradicts lqnoz 7 pnl tending to 0 D It turns out the condition in Theorem 41 is not just suf cient to prove irrationality but it is also necessary if 04 is irrational then there is such a sequence of integers pmqn whose ratios provide good rational approximations to 04 A proof can be found in 2 p 277 We will not have any need for the necessity except maybe for its psychological boost and therefore omit the proof Of course to use Theorem 41 to prove irrationality of a number 04 we need to nd the integers pn and q For the number e these integers can be found directly from truncations to the in nite series for e as we saw in 31 In other words rather than saying e is irrational because the proof of Theorem 31 shows in the end that rationality of e leads to an integer between 0 and 1 we can say e is irrational because the proof of Theorem 31 exhibits a sequence of good rational approximations to e In other words the proof of Theorem 31 can stop at 31 and then appeal to Theorem 41 While other powers of e are also irrational it is not feasible to prove their irrationality by adapting the proof of Theorem 31 For instance what happens if we try to prove e2 is irrational from taking truncations of the in nite series e2 200 2kkl Writing the truncated sum 220 2khl in reduced form as say anbn numerical data suggest bnez 7 an 6 KEITH CONRAD does not tend to 0 As numerical evidence the value of bnez 7 an at n 22 23 and 24 is roughly 002614488 and 3465 Since the corresponding values of bn have 12 16 and 17 decimal digits these differences are not small by comparison with 1bn so the approximations anbn to e2 are not that good Thus these rational approximations to e2 probably wont t the conditions of Theorem 41 to let us prove the irrationality of e2 5 TRRATIONALITY OF RATIONAL POWERS OF 6 We want to use Theorem 41 to prove the following generalization of the irrationality of e Theorem 51 For any integer a 74 0 e is irrational To nd good rational approxiations for a particular power of e good enough that is to establish irrationality we will not use a particular series expansion but rather use the interaction between the exponential function em and integration Some of the mysterious ideas from Niven7s proof of the irrationality of 7139 will show up in this context Before we prove Theorem 51 we note two immediate corollaries Corollary 52 When r is a non zero rational number e7 is irrational Proof Write r ab with non zero integers a and b If e7 is rational so is e17 e but this contradicts Theorem 51 Therefore e7 is irrational Corollary 53 For any positive rational number r 74 1 lnr is irrational Proof The number lnr is non zero lf lnr is rational then Corollary 52 tells us eh is irrational But eh r is rational We have a contradiction so lnr is irrational D The proof of Theorem 51 will use the following lemma which tells us how to integrate e mfz when f is any polynomial Lemma 54 Hermite Let f be a polynomial of degree in 2 0 For any number a m m a we dz 2 WW 7 Z We 0 j0 j0 Proof We compute f e mfz da by integration by parts taking u f and do e mdz Then do f d and i ie m so we dz 7mm errz dz Repeating this process on the new inde nite integral we eventually obtain e mfx da ieiw fjz Now evaluate the right side at z a and z 0 and subtract D Remark 55 It is interesting to make a special case ofthis lemma explicit When f x n a positive integer the lemma says a 1 n nd iii 71 739 1W Oe z z n 1an n j a 10 IRRATIONALITY OF 7r AND 5 7 Letting a 7 oo 71 is xed7 the second term on the right tends to 07 so fooo fix dm 711 This integral formula for n is due to Euler Now we prove Theorem 51 Proof We rewrite Hermite7s lemma by multiplying through by e 51 e Wm dz a f70 7 Z f7a 0 j0 j0 Equation 51 is valid for any number a and any polynomial Let a be a non zero integer at which 6 is assumed to be rational We want to use for f a polynomial actually7 a sequence of polynomials with two properties 0 the left side of 51 is non zero and when n is large very small in absolute value7 0 all the derivatives of the polynomial at z 0 and z a are integers Then the right side of 51 will have the properties of the differences qnoz 7 pl in Theorem 417 with 04 e and the two sums on the right side of 51 being pn and q Our choice of f is 52 fwc where n 2 1 is to be determined Note the similarity with 22 in the proof of the irra tionality of 7Tl In other words7 we consider the equation z x 7 a n a 2n Zn 53 ea Wm dz ea 2 f5 0 7 Z fygtltagt 0 j0 j0 We can see 53 is non zero by looking at the left side The number a is non zero and the integrand e wfnx e mznw 7 a nl on the interval 06 has constant sign7 so the integral is non zero Now we estimate the size of 53 by estimating the integral on the left side Since a 1 n 7 1 n 67zfn d a2n1 67ayy V dy7 0 0 71 we can bound the left side of 53 from above a a 2n1 1 ea e mfnw dm 3 e ay dy 0 l 0 As a function of n this upper bound is a constant times lalZYLnl As 71 7 007 this bound tends to 0 I I To see that7 for any n 2 17 the derivatives f 70 and f 7a are integers for every j 2 07 rst note that the equation fna 7 m fnx tells us after repeated differentiation that 71j 73a j0 Therefore it suf ces to show all the derivatives of fnz at z 0 are integers The proofthat all MR0 are integers is just like that in the proof of Theorem 217 so the details are left to the reader to check The general principle is this for any polynomial g which has integer coef cients and is divisible by x all derivatives of gznl at z 0 are integers The rst property of the fn7s tells us that lqne 7 pnl is positive and tends to 0 as n 7 00 The second property of the fn7s tells us that the sums pn 230 f 7a and qn 8 KEITH CONRAD 230 a on the right side of 53 are integers Therefore the hypotheses of Theorem 41 are met7 so 6 is irrational What really happened in this proof We actually wrote down some very good rational approximations to 6 They came from values of the polynomial mac 2 Me Indeed7 Theorem 51 tells us FnaFn0 is a good77 rational approximation to 6 when n is large The dependence of on a is hidden in the formula for The following table illustrates this for a 27 where the entry at n 1 is pretty bad since F10 0 n mung 7 Fn2l 1 4 2 15562 3 43775 4 09631 5 01739 6 00266 7 00035 8 00004 If we take a 17 the rational approximations we get for e e by this method are different from the partial sums 220 1kl Although the proofs of Theorems 21 and 51 are similar in the sense that both used es timates on integrals7 the proof of Theorem 21 did not show 7139 is irrational by exhibiting a sequence of good rational approximations to 7139 The proof of Theorem 21 was an inte ger between 0 and 177 proof by contradiction No good rational approximations to 7139 were produced in that proof It is simply harder to get our grips on 7139 than it is on powers of 6 REFERENCES 1 El Hairer and GI Wanner7 Analysis by its History77 SpringerVerlag7 New York7 1996i 2 Ki Ireland and Mi Rosen7 A Classical Introduction to Modern Number Theory77 2nd edi7 Springer Verlag7 New York7 1990 3 I Niven7 A simple proof that 7r is irrational7 Bull Amer Math Soc 53 19477 509
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