**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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