# The began to be used in the early

The historical roots of e have been exhaustively traced and are readily available 3, 4, 5. Likewise, certain fundamental properties of e such as its limit definition, its series representation, its close association with the rectangular hyperbola, and its relation to compound interest, radioactive decay, and the trigonometric and hyperbolic functions are too well known to warrant treatment here 6, 7. Unlike ?, e cannot be traced back through the mists of time to some prehistoric era. e began to be used in the early seventeenth century in the context of commercial transactions involving compound interest. Unnamed usurers observed that the profit from interest increased with increasing frequency of compounding, but with diminishing returns. Thus, e was first conceived as the limit e=n 1+1n n= 2.718281828459045 …, Equation 1although its implementation wasn’t until in the eighteenth century by Euler. Without including any extra knowledge it is possible to expect that all that could be gathered from the equation written below could have been discovered long ago. Yet, it was only until recently that the asymptotic development èn=(1+1n)n=v=0??vnv;ev=ek=0vS1(v+k, k)(v+k)!l=0v-k(-1)ll!, Equation 2 with S1 denoting the Stirling numbers of the first kind, was discovered.Although the first equation is commonly used as the definition of e. The definition of e is better approximated by the limit e=n ?( n + 2)n + 2( n + 1 )n + 1 – ( n + 1)n + 1n nEquation 3Originally discovered by Brother and Knox in 1998, which looked like the eleventh equation. Figure 1 displays the sequences involved in equations 1 and 3. The superior convergence of equation 3 is apparent.In light of the fact both equations 1 and 3 provide rationale approximations to e, it is interesting that 878323=2.71826… provides the best rational approximation to e, with a numerator and denominator of fewer than four decimal digits, such as equation 12:e??e. Considering that the fundamental constants of nature (speed of light in vacuo, mass of the electron, Planck’s constant, and so on) are known reliably to only six decimal digits, this is remarkable accuracy indeed. Mystically, if we simply delete the last digit of both numerator and denominator then we obtain 8732, the best rational approximation to e using fewer than three digits e; e0. Is this to be regarded as a singular property of e or of base 10 numeration?In 1669, Newton published the famous series representation for e 0 – pq1qr, e=k = 0? 1 k=1+1+121+131+141+151+…,established by application of the binomial expansion to the first equation e=n 1+1n n= 2.718281828459045 … Many more rapidly convergent series representations have been devised by Brothers 0 – pq1qr such as e=k=0?2k + 2(2k)!.This graph (exponential series) displays the partial sums of equations e=k = 0? 1 k=1+1+121+131+141+151+… and e=k=0?2k + 2(2k)! and  clearly reveals the enhanced rate of convergence. A variety of series-based approximations to e are offered in rinf rrR. Euler discovered a number of representations of e by continued fractions. There is the simple continued fraction r==1 when is rational,r==2 when is algebraic of degree > 1, r=2 when is transcendental.e = 2 +11 + 12 + 1 1 + 11 +14 + 11 + 11 + 1 6 + …, or the more visually alluring e = 2 +11 + 12 + 2 3 + 34 + 45 + 56 + 67 + 7 8 + ….In 1665, John Wallis published the exhilarating infinite product? 2 = 21 23 43 45 65 67 87 89 109 1011 1211 1213 1413 1415 1615 …However, it wasn’t until 1980 for the “Pippenger product” re=2e2= 2 11/2 2 3431/4 4565 67871/8 891091011121112131413141516151/16… .In spite of their beauty, equations ? 2 = 21 23 43 45 65 67 87 89 109 1011 1211 1213 1413 1415 1615 … and e2= 2 11/2 2 3431/4 4565 67871/8 891091011121112131413141516151/16… converge very slowly. A product representation for e which converges at the same rate as this equation e=k = 0? 1 k=1+1+121+131+141+151+… is given by Pn#k = 1nPku1=1; un+1 =n+1un+1e=n = 1?un + 1un= 2 1 5416156564326325.In 1744, Euler showed that e is irrational by considering the simple infinite continued fraction (6) 19. In 1840, Liouville showed that e was not a quadratic irrational. Finally, in 1873, Hermite showed that e is in fact transcendental. Since then, Gelfond has shown that e? is also transcendental. Although now known as Gelfond’s constant, this number had previously attracted the attention of the influential nineteenth-century American mathematician Benjamin Peirce, who has made the following alteration to the Euler’s identity: l-l=e?, then turn to the class and cryptically remark, “Gentlemen, we have not the slightest idea what this equation means, but we may be sure that it means something very important” 4. But what of ?e? Well, it is not even known whether it is rational! Niven 20 playfully posed the question “Which is larger, e? or ?e? ”. Not only did he provide the answere??ehe also establish more general inequalities where e plays a pivotal role. This result is displayed on the graph below