An infinite constant R. Ramanujam, The Institute of Mathematical Sciences, Chennai Here is a problem for you. My account in the bank has Rupee 1.00 now and earns 100 percent interest per year. If the interest earned is added to my account at the end of the year, the value of the account at year-end will be Rs 2.00. What happens if the interest is computed and added more frequently during the year? If the interest is added twice in the year, the interest rate for each 6 months will be 50%, so the initial Rupee 1 is multiplied by 1.5 twice, yielding Re 1.00×1.5x1.5 = Rs 2.25 at the end of the year. This is called {\em compound interest}. Compounding quarterly yields Re 1.00×1.254 = Rs 2.4414..., and compounding monthly yields Rs 1.00×(1+1/12)12 = 2.613035... If there are n compounding intervals, the interest for each interval will be 100% / n and the value at the end of the year will be Rs 1.00×(1 + 1/n)n. As n gets larger and larger, something interesting happens, the compounding interval gets smaller. Compounding weekly (n = 52) yields Rs 2.692597..., while compounding daily (n = 365) yields Rs 2.714567..., just two paise more ! In fact this sequence approaches a {\em limit}. This is an observation made by Jacob Bernoulli around the year 1683. The limit as n grows large is the number that came to be known as e; with "continuous" compounding, the account value will reach Rs 2.7182818.... The Euler-Mascheroni constant, or simply the Euler's constant, denoted e, is one of the most important constants of mathematics, and plays an important role in the famous Euler's equation, considered by many to be the most beautiful equation in mathematics: e^{\pi \cdot \iota} + 1 = 0. The value of e is approximately 2.7182818284590452353602874... and it is the base of the {\em natural logarithm}. Here is another problem. n guests are invited to a wedding, and at the door each guest hands her/his handbag with a helper who then places them into n boxes, each labelled with the name of one guest. But the helper does not know the identities of the guests, and so he puts the bags into boxes selected at {\em random}. The problem, proposed de Montmort, is to find the chance (probability) that *none* of the bags gets put into the right box. The answer is: p_n = 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots+\frac{(-1)^n}{n!} = \sum_{k = 0}^n \frac{(-1)^k}{k!}. As the number n of guests tends to infinity, p_n approaches 1/e. Furthermore, the number of ways the bags can be placed into the boxes so that none of the bags is in the right box, called {\em derangements}, is n!/e rounded to the nearest integer, for every positive n. Once again, our friend e makes an appearance. In case you thought these were coincidences, wait until you get to college. It would not be an exaggeration to say that *any* mathematics at all you study, whether it is in engineering, physics, chemistry, biology, economics, computer science, or political science, let alone the study of mathematics itself, will see e appearing often and repeatedly. In mathematics, e appears in the study of analysis, number theory, geometry, combinatorics, .... indeed in too many places to list. Such a famous constant entered mathematics in a rather minor way. It first made an appearance in 1618 when, in an appendix to Napier's work on logarithms, a table appeared giving the natural logarithms of various numbers. That these were to the base e was not recognized then. Slowly logarithms were understood but it was Euler who introduced the notation e, and popularized it. In a letter to Goldbach in 1731, he discussed its properties, and in his famous book published in 1748, he gave the equation: e = 1 + 1/1! + 1/2! + 1/3! + ... and showed the limit (1 + 1/n)^n that we saw above to be e, computing the value to 18 decimal places. (By the way, Srinivasa Ramanujan discovered this identity independently by the age of 12, and computed the value to 15 decimal places by the age of 17.) Euler also gave the continued fraction expansions of e (figures cf1.gif, cf2.gif) and proved that it is irrational. But it was only in 1873 that e was also shown to be non-algebraic by Hermite: that is, it can be expressed only by infinite series like the above. In the Box, you see some equations that mathematicians call characterizations of e. (If you do not understand all the symbols yet, do not worry, you will learn all these before you leave school.) They show why mathematicians think of e as something elegant, even beautiful. BOX ------------------------------------------------ 1. The number e is the unique positive real number such that p1.png. 2. The number e is the unique positive real number such that p2.png. 3. The number e is the limit p3a.png. Similarly, p3b.png. 4. The number e is the sum of the infinite series p4.png. 5. The number e is the unique positive real number such that p5.png. ------------------------------------------------