Euler's Number

Pronunciation: /ˈɔɪlərz ˈnʌmbər/ Explain

In mathematics, e represents a constant, sometimes called Euler's number after Leonhard Euler (1707-1783). The importance of this constant comes from its use for:

  • the base of natural logarithms: log base e of x = ln x
  • the base of growth functions: f(x) = a*e^(bx)
  • Euler's equation that relates exponents and trigonometric functions: e^(i theta) = cos(theta) + i*sin(theta)

The value of e is approximately 2.71828. A more exact value of e can be calculated by one of several methods:

  • e = limit as n approaches infinity of (1 + 1/n) ^ n
  • e = sum from n = 1 to infinity of 1 divided by n factorial.

Calculation of the Value of e
This calculation uses the infinite sum e = sum from n = 1 to infinity of 1 divided by n factorial.
n1/n!Sum (approx)
01/0! = 1/1 = 11
11/1! = 1/1 = 12
21/2! = 1/2 = 0.52.5
31/3! = 1/6 ≈ 0.166666672.66666667
41/4! = 1/24 ≈ 0.041666672.70833333
51/5! = 1/120 ≈ 0.008333332.71666667
61/6! = 1/720 ≈ 0.001388892.71805556
71/7! = 1/5040 ≈ 0.000198412.71825397
81/8! = 1/40320 ≈ 0.000024802.71827877
91/9! = 1/362880 ≈ 0.000002752.71828152
101/10! = 1/3628800 ≈ 0.000000272.71828179
111/11! = 1/39916800 ≈ 0.000000032.71828181
...
Table 1

Cite this article as:

McAdams, David E. Euler's Number. 7/9/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/e/e.html.

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Revision History

7/5/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
1/31/2010: Added "References". (McAdams, David E.)
11/25/2008: Changed equations to images. (McAdams, David E.)
7/14/2008: Added table on the calculation of e. Expanded text on natural logarithms and growth functions. Changed equations from images to Hot_Eqn. (McAdams, David E.)
7/7/2008: Corrected spelling. (McAdams, David E.)
5/8/2008: Initial version. (McAdams, David E.)

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