Euler’s Number: What is “e” in Math?

Calculus Definitions >

Euler’s number, usually written as e, is a special number with a very important place in mathematics. The first few digits are:

2.7182818284590452353602874713527;

It’s an irrational number, which means you can’t write it as a fraction.

History of Euler’s Number

The number e was “‘discovered” in the 1720s by Leonard Euler as the solution to a problem set by Jacob Bernoulli. He studied it extensively and proved that it was irrational. He was also the first to use the letter e to refer to it, though it is probably coincidental that that was his own last initial.

The equation most commonly used to define it was described by Jacob Bernoulli in 1683:


The equation expresses compounding interest as the number of times compounding approaches infinity. With the binomial theorem, he proved this limit we would later call e.

We can actually follow the history of e even further back than Bernoulli. It turns out that e is the base for natural logarithms, and since these were studied extensively by John Napier one hundred years before Euler—in 1614—e is sometimes also called Napier’s constant. Napier published a table of natural logarithms, but didn’t include in his publication the constant they were calculated from.

Ways to Express Euler’s Number

Since Euler’s number is irrational, there is no way to express it as a fraction of integers, or as a finite or periodic decimal number. It comes up so often in both pure and applied math, however, there are many other ways it can be expressed. Some of these include:

In summation notation:



for any real number x, or.



Or using limits:



Or as a sum of trig functions:

Euler’s Number: The First 100 Digits

The first 100 digits of Euler’s number are:

2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274

References

McIntosh, Avery. On Euler’s Number e. Retrieved from http://people.bu.edu/aimcinto/number.e.pdf on May 27, 2018.

O’Connor & Robertson. The Number e. Retrieved from http://www-history.mcs.st-and.ac.uk/HistTopics/e.html on May 27, 2018.

NDT Resource Center. What is e? Retrieved from https://www.nde-ed.org/EducationResources/Math/Math-e.htm on May 27, 2018.

McCartin, Brian J. (2006). “e: The Master of All”. The Mathematical Intelligencer. 28 (2): 10–21 doi:10.1007/bf02987150. Retrieved from https://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/mccartin.pdf on May 27, 2018.

Sandifer, Ed (Feb 2006). “How Euler Did It: Who proved e is Irrational?”. MAA Online. Archived from the original (PDF) on 2014-02-23. Retrieved https://web.archive.org/web/20140223072640/http://vanilla47.com/PDFs/Leonhard%20Euler/How%20Euler%20Did%20It%20by%20Ed%20Sandifer/Who%20proved%20e%20is%20irrational.pdf from May 27, 2018.

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