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Euler's Number

Euler's Number

In mathematics, we deal with several terminologies with fixed numeric values. These are called constants, and they help in solving mathematical problems with ease.

For example, \(\pi =3.14\)

Likewise, there is a constant e, known as Euler’s number. It was discovered in the 18th century. It is quite interesting to learn about the history of Euler’s number. It is the base of the natural logarithm. The 

In math, the term e is called Euler's number after the Swiss mathematician Leonhard Euler.

It is mathematically known as Euler's (pronounced as oiler) constant or Napier's constant.

In this chapter, you will learn about

  • Euler's number value
  • How to calculate e without using Euler's number calculator
  • Euler's number proof
  • Euler's number digit

Lesson Plan 

What is Euler's Number?

Euler's number (e) play's a crucial role in mathematics. It represents the basis of the exponential functions and logarithms.

euler’s number value,euler’s number calculator,euler’s number proof,euler’s number digit

Euler's number digit are\[e = 2.717281728459045\]

The value of e is approximately equal to
\[e = 2.718\]

The number was first coined by Swiss mathematician Leonhard Euler in the 1720s. John Napier, the inventor of logarithms, made significant contributions to developing the number, while Sebastian Wedeniwski calculated it to 869,894,101 decimal places.

\(e\) is called the Euler Number, the Eulerian number, or Napier's Constant. Euler's number proof was first given as that \(e\) is an irrational number, so its decimal expansion never terminates.

This mathematical constant is not only used in math but equally important and applicable in physics.

General Formula of Euler's Number

\(e\) is mathematically represented and defined by the following equation:

\[\begin{align} e=\lim_{n\rightarrow\infty}\left( 1+\frac{1}{n}\right)^n\end{align}\]

Let us have some practical experience of putting the value n to calculate the value of Euler's number digit:

Value of n

Putting the value of n in the equation

Value of e

1 \[\ e_1=\left( 1+\frac{1}{1}\right)^{1}\]


2 \[\ e_2=\left( 1+\frac{1}{2}\right)^{2}\]


5 \[\ e_5=\left( 1+\frac{1}{5}\right)^{5}\]


10 \[\ e_{10}=\left( 1+\frac{1}{10}\right)^{10}\]


100 \[\ e_{100}=\left( 1+\frac{1}{100}\right)^{100}\]


1000 \[\ e_{1000}=\left( 1+\frac{1}{1000}\right)^{1000}\]


10000 \[\ e_{10000}=\left( 1+\frac{1}{10000}\right)^{10000}\]



important notes to remember
Important Notes
1. The value of e is approximately equal to\(e = 2.718\)
2.\(e\) is mathematically represented as  \(\begin{align} e=\lim_{n\rightarrow\infty}\left( 1+\frac{1}{n}\right)^n\end{align}\)
3. Euler introduced the number e with respect to many mathematical calculations. For example, to calculate compound interest we use \(\ A= Pe^{rt}\).

Solved Examples

Let us observe a practical approach to know more about Euler's number.

Example 1



The number of starfish in the sea varies throughout the year. The Euler’s constant also varies. The value of n is given in the table; find the value of \(e.\)

Value of n

Value of e


\[\ e_{1}=\left( 1+\frac{1}{1}\right)^{1}\]








To find the value of e, substituting the value of n in the equation we get, 

\[\ e_{1}=\left( 1+\frac{1}{1}\right)^{1}\\\ e_{2}=\left( 1+\frac{1}{2}\right)^{2}\\\ e_{5}=\left( 1+\frac{1}{5}\right)^{5}\\\ e_{10}=\left( 1+\frac{1}{10}\right)^{10}\]

\(\therefore\) Value of \[\begin{align} e_1 &=2.00000\\ e_2 &=2.25000\\ e_5 &=2.48832\\ e_{10} &=2.59374\end{align}\]

Example 2


John will deposit $180 in an account at a rate of 2% that is compounded continuously. How much money will be in the account after 4 years?



Using formula \[\begin{align} A &= Pe^{rt}\\\ &= 180 e^{0.02\times 4}\\\ &= 180\times 1.0832\\\ &= \ $194.99\end{align}\]

\(\therefore\) Money in the account after 4 years is \[\ $194.99\]

Thinking out of the box
Think Tank

In calculus, the Euler's number is derived from the natural logarithm. Can you think about such logarithm equations that relate directly with \(e\)?


Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.


Let's Summarize

We hope you enjoyed learning about Euler's Number with the simulations and practice questions. Now you will be able to easily solve problems on Euler's number value, Euler's number calculator, Euler's number proof and Euler's number digit.

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Frequently Asked Questions (FAQs)

1. What is special about Euler's Number?

Euler's number \(e\) represents the basis of the exponential functions and logarithm. It can be used in Mathematics as well as in Physics.

2. What is Euler's Number equal to?

Euler’s number is a mathematical constant used as the base of the natural logarithm. It is denoted by \(e\) and is also represented by the general formula of cube  \[F+V-E= \chi\] Where \(\chi\)  is called the "Euler Characteristic." The constant value of Euler's number digit is \(= 2.718\)

3. What is the value of \(e\)?

The constant value of \(e = 2.718\)

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