Euler’s number, usually written as e, is a special number with a very important place in mathematics. The number’s first few digits are 2.7182818284590452353602874713527; it’s an irrational number, which means that 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 compounded 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:
for any real number x.
Euler’s Number: The First 100 Digits
The first 100 digits of Euler’s number are:
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.
Stephanie Glen. "Euler’s Number: What is “e” in Math?" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/eulers-number/
Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!