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# e (mathematical constant)

**e** is a number. It is the base of natural logarithm and is about 2.71828.^{[1]}^{[2]} It is an important mathematical constant. The number e is occasionally called **Euler's number** after the Swiss mathematician Leonhard Euler, or **Napier's constant** in honor of the Scottish mathematician John Napier who introduced logarithms. It is equally important in mathematics as π and i. e is an irrational number, and Euler himself gave the first 23 digits of e.^{[3]}

The number e has great importance in mathematics,^{[4]} as do 0, 1, π, and i. All five of these numbers are important and occur again and again in mathematics. The five constants appear in one formulation of Euler's identity. Like the constant π, e is also irrational (it cannot be represented as a ratio of integers)^{[5]} and transcendental (it is not a root of any non-zero polynomial with rational coefficients).^{[2]}

The number e is very important for exponential functions. For example, the exponential function applied to the number one, has a value of e.

e was discovered in 1683 by the Swiss mathematician Jacob Bernoulli, while he was studying compound interest.^{[6]} The numerical value of e truncated to 20 decimal places is:^{[5]}

- 2.71828 18284 59045 23536...

## Alternate definitions of **e**

There are many different ways to define e. Jacob Bernoulli, who discovered e, was trying to solve the problem:

- [math]\lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n.[/math]

In other words, there is a number that the expression [math]\left(1+\tfrac{1}{n}\right)^n[/math] approaches as *n* becomes larger. This number is e.

Another definition is to find the solution of the following formula:

- [math]2+\cfrac{2}{2+\cfrac{3}{3+\cfrac{4}{4+\cfrac{5}{5+\cfrac{6}{\ddots\,}}}}}[/math]

## Related pages

## References

- ↑ "Compendium of Mathematical Symbols" (in en-US). 2020-03-01. https://mathvault.ca/hub/higher-math/math-symbols/.
- ↑
^{2.0}^{2.1}Weisstein, Eric W.. "e" (in en). https://mathworld.wolfram.com/e.html. - ↑ Euler, Leonhard (1748).
*Introductio in analysin infinitorum*. M. M. Bousquet. p. 90. https://books.google.com/?id=jQ1bAAAAQAAJ&pg=PA90. - ↑ Howard Whitley Eves (1969).
*An Introduction to the History of Mathematics*. Holt, Rinehart & Winston. . https://archive.org/details/introductiontohi00eves_0. - ↑
^{5.0}^{5.1}"e - Euler's number". https://www.mathsisfun.com/numbers/e-eulers-number.html. - ↑ "The number e". St Andrews University. http://www-history.mcs.st-and.ac.uk/HistTopics/e.html.