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What is e? Who first used e? How do you find it? How many digits does it have?



e = 2.71828..., the Base of Natural Logarithms

e is a real number constant that appears in some kinds of mathematics problems. Examples of such problems are those involving growth or decay (including compound interest), the statistical "bell curve," the shape of a hanging cable (or the Gateway Arch in St. Louis), some problems of probability, some counting problems, and even the study of the distribution of prime numbers. It appears in Stirling's Formula for approximating factorials. It also shows up in calculus quite often, wherever you are dealing with either logarithmic or exponential functions. There is also a connection between e and complex numbers, via Euler's Equation.

e is usually defined by the following equation:

e = limn->infinity (1 + 1/n)n.

Its value is approximately 2.718281828459045... and has been calculated to 869,894,101 decimal places by Sebastian Wedeniwski (you'll find the first 50 digits in a Table of constants with 50 decimal places, from the Numbers, constants and computation site, by Xavier Gourdon and Pascal Sebah).

The number e was first studied by the Swiss mathematician Leonhard Euler in the 1720s, although its existence was more or less implied in the work of John Napier, the inventor of logarithms, in 1614. Euler was also the first to use the letter e for it in 1727 (the fact that it is the first letter of his surname is coincidental). As a result, sometimes e is called the Euler Number, the Eulerian Number, or Napier's Constant (but not Euler's Constant).

An effective way to calculate the value of e is not to use the defining equation above, but to use the following infinite sum:

e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ...

If you need K decimal places, compute each term to K+3 decimal places and add them up. You can stop adding after the term 1/n! where n! > 10K+3, because, to K+3 decimal places, the rest of the terms are all zero. Even though there are infinitely many of them, they will not change the decimal places you have already calculated. Now the last decimal place or two of the resulting sum may be off due to truncation or rounding of each term, but the first K places should be correct. That is why the computation uses extra decimal places.

As an example, here is the computation of e to 22 decimal places:

   1/0!  = 1/1       = 1.0000000000000000000000000
   1/1!  = 1/1       = 1.0000000000000000000000000
   1/2!  = 1/2       = 0.5000000000000000000000000
   1/3!  = 1/6       = 0.1666666666666666666666667
   1/4!  = 1/24      = 0.0416666666666666666666667
   1/5!  = 1/120     = 0.0083333333333333333333333
   1/6!  = 1/720     = 0.0013888888888888888888889
   1/7!  = 1/5040    = 0.0001984126984126984126984
   1/8!  = 1/40320   = 0.0000248015873015873015873
   1/9!  = 1/362880  = 0.0000027557319223985890653
   1/10! = 1/3628800 = 0.0000002755731922398589065
   1/11!             = 0.0000000250521083854417188
   1/12!             = 0.0000000020876756987868099
   1/13!             = 0.0000000001605904383682161
   1/14!             = 0.0000000000114707455977297
   1/15!             = 0.0000000000007647163731820
   1/16!             = 0.0000000000000477947733239
   1/17!             = 0.0000000000000028114572543
   1/18!             = 0.0000000000000001561920697
   1/19!             = 0.0000000000000000082206352
   1/20!             = 0.0000000000000000004110318
   1/21!             = 0.0000000000000000000195729
   1/22!             = 0.0000000000000000000008897
   1/23!             = 0.0000000000000000000000387
   1/24!             = 0.0000000000000000000000016
   1/25!             = 0.0000000000000000000000001
                      -----------------------------
                       2.7182818284590452353602875
Then to 22 decimal places, e = 2.7182818284590452353603, which is correct. (Actually,it's correct to 25 places, but that was luck!).

There have been recent discoveries of even more efficient ways of computing e, one of which was used for setting the record mentioned above.

It is a fact (proved by Euler) that e is an irrational number, so its decimal expansion never terminates, nor is it eventually periodic. Thus no matter how many digits in the expansion of e you know, the only way to predict the next one is to compute e using the method above using more accuracy.

It is also true that e is a transcendental number (a fact first proved in 1873 by the French mathematician Charles Hermite), which means that e is not the root of any polynomial with rational number coefficients. These are properties that e shares with pi. The Dr. Math archives contain one proof of The Irrationality of e, and on the Web is another by Kevin Brown.

e is also the base of natural logarithms. The natural logarithm function ln(x) is defined that way: ln(x) = loge(x). This is "natural" for several reasons. One is the following limit:

ln(x) = limk->0 (xk-1)/k.

Another example from calculus is that if y = ln(x) + c, for c constant, then dy/dx = 1/x, and these are the only functions for which this is true. Another is that the curve y = ln(x) has a tangent at (1,0) with slope 1, and among all logarithmic functions, it is the only one that does.



Note: The term
Euler's Constant is usually reserved for another number also first studied by Euler, 0.5772156649... = limn->infinity [1/1 + 1/2 + 1/3 + ... + 1/n - ln(n)]. [Return to text.]



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