Euler's Number

Pronunciation: /ˈɔɪ.lərz ˈnʌm.bər/ 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:
• The base of growth functions:
• Euler's equation that relates exponents and trigonometric functions:

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

• 
• 

Calculation of the Value of e

This calculation uses the infinite sum
n	1/n!	Sum (approx)
0	1/0! = 1/1 = 1	1
1	1/1! = 1/1 = 1	2
2	1/2! = 1/2 = 0.5	2.5
3	1/3! = 1/6 ≈ 0.16666667	2.66666667
4	1/4! = 1/24 ≈ 0.04166667	2.70833333
5	1/5! = 1/120 ≈ 0.00833333	2.71666667
6	1/6! = 1/720 ≈ 0.00138889	2.71805556
7	1/7! = 1/5040 ≈ 0.00019841	2.71825397
8	1/8! = 1/40320 ≈ 0.00002480	2.71827877
9	1/9! = 1/362880 ≈ 0.00000275	2.71828152
10	1/10! = 1/3628800 ≈ 0.00000027	2.71828179
11	1/11! = 1/39916800 ≈ 0.00000003	2.71828181
...
Table 1

References

1. McAdams, David E.. All Math Words Dictionary, euler's number. 2nd Classroom edition 20150108-4799968. pg 73. Life is a Story Problem LLC. January 8, 2015. Buy the book

Cite this article as:

McAdams, David E. Euler's Number. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/e/e.html.

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Revision History

12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
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.)