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This Topic Will Be Treated Again in Our Next Issue. and in the Issue After e: The Master of All BRIAN J. MCCARTIN Read Euler, read Euler. He is the master of us all. --P. S. de Laplace aplace's dictum may rightfully be transcribed as: "Study Thus, e was first conceived as the limit e, study e. It is the master of all." L e = lim (1 + l/n)*' = 2.718281828459045" (1) Just as Gauss earned the moniker Princeps mathemati- n~ao corum amongst his contemporaries [1], e may be dubbed although its baptism awaited Euler in the eighteenth cen- Princeps constantium symbolum. Certainly, Euler would tury [2]. One might naively expect that all that could be concur, or why would he have endowed it with his own gleaned from equation (1) would have been mined long initial [2]? ago. Yet, it was only very recently that the asymptotic The historical roots of e have been exhaustively traced development and are readily available [3, 4, 5]. Likewise, certain funda- mental properties of e such as its limit definition, its series =(1+1/n) ~ = ~ e~. representation, its close association with the rectangular hy- perbola, and its relation to compound interest, radioactive decay, and the trigonometric and hyperbolic functions are ev= e l---7--. ' (2) k=o (v+ k)! l=o too well known to warrant treatment here [6, 7]. Rather, the focus of the present article is on matters re- with & denoting the Stirling numbers of the first kind [9], lated to e which are not so widely appreciated or at least was discovered [10]. have never been housed under one roof. When I do in- Although equation (1) is traditionally taken as the defin- dulge in reviewing well-known facts concerning e, it is to ition of e, it is much better approximated by the limit forge a link to results of a more exotic variety. [ (n + 2)n+2 (n + l)n+l ] Occurrences of e throughout pure and applied mathe- e = n--+~lim (n + 1),,+i r/n (3) matics are considered; exhaustiveness is not the goal. Nay, the breadth and depth of our treatment of e has been cho- discovered by Brothers and Knox in 1998 [11]. Figure 1 dis- sen to convey the versatility of this remarkable number and plays the sequences involved in equations (1) and (3). The to whet the appetite of the reader for further investigation. superior convergence of equation (3) is apparent. In light of the fact that both equations (1) and (3) pro- The Cast vide rational approximations to e, it is interesting to note Unlike its elder sibling ~r, e cannot be traced back through that 87~ = 2.71826 provides the best rational approxirna- 323 the mists of time to some prehistoric era [8]. Rather, e burst tion to e, with a numerator and denominator of fewer than into existence in the early seventeenth century in the con- four decimal digits [12]. (Note the palindromes!) Consider- text of commercial transactions involving compound inter- ing that the fundamental constants of nature (speed of light est [5]. Unnamed usurers observed that the profit from in- in vacuo, mass of the electron, Planck's constant, and so terest increased with increasing frequency of compounding, on) are known reliably to only six decimal digits, this is re- but with diminishing returns. markable accuracy indeed. Mystically, if we simply delete 10 THE MATHEMATICAL INTELLIGENCER 2006 Springer Science+Business Media, Inc. 1 i i i i i i 6 o 0 0 0 0 2.7 l e 2.6 2.5 2.4 = Eq. 1 o= Eq. 3 2.3 2.2 2.1 2 i i I i I 1 I I I 2 4 6 8 10 12 14 16 18 20 n Figure I. Exponential sequences. the last digit of both numerator and denominator then we ~, 2k+2 t~ s732, the best rauonal" approximation to e using fewer e = (5) k=0 (2k)! han three digits [13]. Is this to be regarded as a singular property of e or of base 10 numeration? Figure 2 displays the partial sums of equations (4) and (5) In 1669, Newton published the famous series represen- and clearly reveals the enhanced rate of convergence. A va- tation for e [14], riety of series-based approximations to e are offered in [15]. Euler discovered a number of representations of e by con- e---= 2 ~t1 = tinued fractions. There is the simple continued fraction [16] k 0 1 1 1 1 1 1+1+ ~.(1+ ~-.(1+ ~-.(1+ ~-.(1+ .'')))), (4) e = 2 + 1 , (6) 1+ 2+ 11 established by application of the binomial expansion to 1+ 1 1+ 4+ l+.__.k~_~1 equation (1). Many more rapidly convergent series repre- 1+ r~... sentations have been devised by Brothers [14] such as or the more visually alluring [5] BRIAN J. McCARTIN studied Applied Mathe- matics at the University of Rhode Island and e = 2 + 1 (7) went on to a Ph.D. at the Courant Institute, 1 + 2+-- 2 8 3+ New York University. He was Senior Research 4+ 4 5+6+ 0 6 7 Mathematician for United Technologies Re- 7+8+... search Center and Chair of Computer Science at RPI/Hartford before joining Kettenng Univer- In 1655, John Wallis published the exhilarating infinite sity in 1993. He received the Distinguished product Teaching Award from the Michigan Section of rr 2244668810101212141416 the MAA in 2004. Along the way, he also ................. (8) 2 133557799111113131515 eamed a degree in music from the University of Hartford (1994). To fill in the details see However, the world had to wait until 1980 for the "Pip- http://www.ketted ng,edu/- bmccarti. penger product" [17] Applied Mathematics e Kettering University 2 Flint, MI,' 48504-4808 USA \l J\ 3! \5 5 7 7/\ 9 9 11 11 13 13 15 ~/ '''. (9) 9 2006Springer Science+Business Media, Inc, Volume 28, Number 2, 2006 1 1 2.8 i i i i i i O O Q o Q t o 2.6 e 2.4 2.2 = Eq. (4) o = Eq. (5) 1.8 1.6 1.4 1.2 1 I f I I I I 0 1 2 3 4 5 6 n Figure 2. Exponential series. In spite of their beauty, equations (8) and (9) converge Barel-e Transcendental very slowly. A product representation for e which converges As noted above, e is transcendental, but just barely so [21, at the same rate as equation (4) is given by [18] 22]. In 1844, Liouville proved that the degree to which an algebraic irrational number can be approximated by ratio- U 1 ----- 1; Un+ 1 = (T/ 4- 1)(Un -I- 1) nals is limited [23], so that, from the point of view of ratio- [I un+ 1 2 5 16 65 326 nal approximation, the simplest numbers are the worst [24]. e ........... (10) n=l U*I 1 4 15 64 325 Liouville used this property--that if an irrational number is rapidly approximated by rationals then it must be transcen- An Interesting Inequalit-e dental-to construct the first known transcendental number, In 1744, Euler showed that e is irrational by considering the the so-called Liouville number L = 10 -1! + 10 -2! + 10 -3! 4- simple infinite continued fraction (6) [19]. In 1840, .. 10-m! + -. In 1955, Klaus Roth provided the defini- Liouville showed that e was not a quadratic irrational. Finally, tive refinement of Liouville's theorem by finding the ulti- in 1873, Hermite showed that e is in fact transcendental. mate limit to which algebraic irrationals may be approxi- Since then, Gelfond has shown that e ~ is also transcen- mated by rationals [19]. For this work, he was awarded the dental. Although now known as Gelfond's constant, this Fields Medal in 1958. number had previously attracted the attention of the influ- To phrase these results quantitatively, for any real num- ential nineteenth-century American mathematician Benjamin ber 0 < s~ < 1 and any fraction p/q in lowest terms, let R Peirce, who was wont to write on the blackboard the fol- denote the set of all positive real numbers r for which the lowing alteration of Euler's identity [2, 5]: inequality '= ~e ~, (11) 1 0 < I~ -- P l < qr (14) then turn to the class and cryptically remark, "Gentlemen, we have not the slightest idea what this equation means, but we possesses at most finitely many solutions. Then, define the may be sure that it means something very important" [4]. Liouville-Roth constant (irrationality measure) as But what of ~r~? Well, it is not even known whether it is rational! Niven [20] playfully posed the question "Which is r(~ ~- inf r, (15) larger, e ~r or rre?''. Not only did he provide the answer r~R e ~ > ITe, (12) that is, the critical rate threshold above which {: is not ap- proximable by rationals. he also established the more general inequalities Then, the above may be summarized as follows [22]: /3>a->e~ c~/3>/3~; e->/3>a>0~/3~>~/3, (13) {i: when {: is rati~ where e plays a pivotal role. This result is displayed graph- r(~ = when {: is algebraic of degree > 1, (16) ically in Figure 3. 2 when s~ is transcendental. 12 THE MATHEMATICALINTELLIGENCER xY-yX=O 10 I i I I I I I 9=i "/ >0 7 I 6 >, 5 4 <0 <0 3 (e,e) 2 >0 0 i I I I I I I I 0 1 2 3 4 5 6 7 8 10 x Figure 3. e" > ~'~. 2.8 I I I i I ,e..- e 2.6 2.41 2.2 2 1.8 1.6 1.4 = i i f i 0 1000 2000 3000 4000 5000 6000 n Figure 4.
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