Home - Education Resources

 

What is e?

"e" is a numerical constant that is equal to 2.71828. Just as pi (3.14159) is a numerical constant that occurs whenever the circumference of a circle is divided by its diameter. The value of "e" is found in many mathematical formulas such as those describing a nonlinear increase or decrease such as growth or decay (including compound interest), the statistical "bell curve," the shape of a hanging cable or a standing arch. "e" also shows up in some problems of probability, some counting problems, and even the study of the distribution of prime numbers. In the field of nondestructive evaluation it is found in formulas such as those used to describe ultrasound attenuation in a material. The sound energy decays as it moves away from the sound source by a factor that is relative to "e." Because it occurs naturally with some frequency in the world, "e" is used as the base of natural logarithms.

e is usually defined by the following equation:

Its value is approximately 2.718 and has been calculated to 869,894,101 decimal places by Sebastian Wedeniwski. 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. It was proven by Euler that "e" is an irrational number, so its decimal expansion never terminates, nor is it ever periodic.

An effective way to calculate the value of e is not to use the defining equation above, but to use
the following infinite sum of factorials. Factorials are just products of numbers indicated by an exclamation mark. For instance, "four factorial" is written as "4!" and means 1×2×3×4 = 24.

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

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! = 1/39916800
= 0.0000000250521083854417188
1/12! = 1/479001600
= 0.0000000020876756987868099
1/13! = 1/6227020800
= 0.0000000001605904383682161
1/14! = 1/87178291200
= 0.0000000000114707455977297
1/15! = 1/1307674368000
= 0.0000000000007647163731820
1/16! = 1/20922789887989
= 0.0000000000000477947733239
1/17! = 1/355687428101759
= 0.0000000000000028114572543
1/18! = 1/6402373705148490
= 0.0000000000000001561920697
1/19! = 1/121645101098757000
= 0.0000000000000000082206352
1/20! = 1/2432901785214670000
= 0.0000000000000000004110318
1/21! = 1/51091049359062800000
= 0.0000000000000000000195729
1/22! = 1/1123974373384290000000
= 0.0000000000000000000008897
1/23! = 1/25839793281653700000000
= 0.0000000000000000000000387
1/24! = 1/625000000000000000000000
= 0.0000000000000000000000016
1/25! = 1/10000000000000000000000000
= 0.0000000000000000000000001


The sum of the values in the right column is 2.7182818284590452353602875 which is "e."

For more information on e, visit the the math forum at mathforum.org

Reference: The mathforum.org