Value of e

e Value of Math

In Mathematics, 2.71828182845 and so on is the value of e. If you have carefully noticed, it has been mentioned under the log function also known as the base of the logarithmic function. It looks something like - loge. This mathematical constant is not only used in Mathematics, but it is also applied in Physics. Here are a few values of e when they are raised to a certain power:


  • \[{e^1}\]= e [ value of e power one (1) or value of e1 will remain e]

  • \[{e^o}\]= 1 [ value of e power zero (0) will remain 1]

  • \[{e^\infty }\]= 0 [ value of e power infinity (\[\infty \]) will remain 0]


The value of e is special and when it acts as the base of a logarithmic function, its value is 1 i.e., loge= 1.


Euler’s Number (e)

(1+1/n)n is the limit of the Euler’s number. In (1+1/n)n, when the value n addresses infinity, this expression arises while studying compound interest. It’s also represented as the sum of infinite numbers.


\[e = \sum {_{n = 0}^\infty } \frac{1}{{n!}} = \frac{1}{1} + \frac{1}{1} + \frac{1}{{1.2}} + \frac{1}{{1.2.3}} + .......\]

With the value of e, the above equation can be solved. Once it is solved, the resultant will be an irrational number that is usually used in multiple Mathematical calculations and concepts. The value e is named after a great mathematician called Leonhard Euler. 


The other name of the value of e is known as Napier’s constant. While studying Compound Interest, Jacob Bernoulli discovered the value of e. Similar to other 

certain mathematical concepts, equations, and problems. The properties of the value of e are also similar to the other constant that has been previously mentioned.


What is the Value of e?

As mentioned above, Jacob Bernoulli was the mathematician who discovered the value of constant e. The Euler’s constant expression by the sum of infinite numbers can also be expressed in multiple different terms, like:

\[e = \mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{1}{n}} \right)^n}\]


Hence, this proves that the value (1 + 1/n)n will reach e only when the value of n reaches infinity. To calculate the approximate value, let’s substitute the value of n and calculate the value of e. Let us start with n = 1 and increment the value. 



Value of n

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

Value of e

1

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

2.000000

2

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

2.25000

5

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

2.48832

10

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

2.59374

100

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

2.70481

1000

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

2.71693

10000

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

2.71815

100000

\[{\left( {1 + \frac{1}{{100000}}} \right)^{100000}}\]

2.71827


Value of Exponential Constant

The exponential constant plays a vital role in Maths. In addition to that, it is denoted by the letter “e”. The approximate value of e is 2.718. But, the value of e is much larger than that. MAthematically, the complete value of e can go on for thousands of digits.


e =2.718281828459045235360287471352662497757247093699959574966967627724076630353


How to Calculate the Value of e?

Until now, we studied the value of e and its vital role and importance in Mathematics and other subjects. It is also known as the Euler’s constant, the base of a logarithmic function, etc. To calculate the value of e, you can use the formula or the expression given below: 


\[e = \sum {_{n = 0}^\infty } \frac{1}{{n!}} = \frac{1}{1} + \frac{1}{1} + \frac{1}{{1.2}} + \frac{1}{{1.2.3}} + \frac{1}{{1.2.3.4}} + \frac{1}{{1.2.3.4.5}} + \frac{1}{{1.2.3.4.5.6}} + ....\]


When the above expression is solved, the approx value of the constant e can be found. 


\[e = \frac{1}{1} + \frac{1}{1} + \frac{1}{{1.2}} + \frac{1}{{1.2.3}} + \frac{1}{{1.2.3.4}} + ..................or\]

\[e = \frac{1}{1} + \frac{1}{{1!}} + \frac{1}{{2!}} + \frac{1}{{3!}} + \frac{1}{{4!}} + \frac{1}{{5!}} + ................or\]

\[e = \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{6} + \frac{1}{{24}} + \frac{1}{{120}} + .............\]

Now, let’s just assume the first few terms . . . .  


\[e = \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{6} + \frac{1}{{24}} + \frac{1}{{120}}\]

e = 2.718


Hence, the exponential constant value e = 2.72.

FAQ (Frequently Asked Questions)

1. What is Euler’s number?

(1+1/n)n is the limit of the Euler’s number. In (1+1/n)n, when the value n addresses infinity, this expression arises while studying compound interest. It’s also represented as the sum of infinite numbers.


                                     


With the value of e, the above equation can be solved. Once it is solved, the resultant will be an irrational number that is usually used in multiple Mathematical calculations and concepts. The value e is named after a great mathematician called Leonhard Euler. The other name of the value of e is known as Napier’s constant. While studying Compound Interest, Jacob Bernoulli discovered it. Similar to other mathematical constants like , , , and others, the value of e also plays a vital role in certain mathematical concepts, equations, and problems. The properties of the value of e are also similar to the other constant that has been previously mentioned.


2. What is the value of e?

As mentioned above, Jacob Bernoulli was the mathematician who discovered the value of constant e. The Euler’s constant expression by the sum of infinite numbers can also be expressed as:


                           


reaches infinity. To calculate the approximate value, let’s substitute the value of n and calculate the value of e. Let us start with n = 1 and increment the value.

3. How to calculate the value of e?

Until now, we studied the value of e and its vital role and importance in Mathematics and other subjects. It is also known as the Euler’s constant, the base of a logarithmic function, etc. To calculate the value of e, you can use the formula or the expression given below: 


            


Now, let’s just assume the first few terms…


             


e = 2.718


Hence, the exponential constant value e = 2.72.