Question

What is the approximate value of \[1/e\] ?

Answer 1

Hint: In this type of problem we have to understand the concept of logarithmic and exponential function. After understanding the concept we have to find the value of \[e\] . When the value of \[e\] is known, simply substitute it in the given \[1/e\] and you will get your required answer.

Complete step by step answer:
\[e\] is a real number which is called Euler's number (Other name of \[e\] is also known as Napier’s constant) . The value of \[e\] is named after a great mathematician called Leonhard Euler. As if you have noticed it earlier, then it has been mentioned under the log function and it is also known as the base of the logarithmic function. This constant is not only used in Mathematics but also in Physics.
The limit of the Euler’s number is: \[{{(1+1/n)}^{n}}\] . It is also represented as the sum of infinite numbers. This expression arises in the study of compound interest.
\[e=\sum\limits_{n=0}^{\infty }{\dfrac{1}{n!}=\dfrac{1}{1}+\dfrac{1}{1}+\dfrac{1}{1.2}+\dfrac{1}{1.2.3}+..........}\]
To calculate the value of \[e\] , we can use the above formula or expression as:
\[e=\sum\limits_{n=0}^{\infty }{\dfrac{1}{n!}=\dfrac{1}{1}+\dfrac{1}{1}+\dfrac{1}{1.2}+\dfrac{1}{1.2.3}+\dfrac{1}{1.2.3.4}+\dfrac{1}{1.2.3.4.5}+\dfrac{1}{1.2.3.4.5.6}..........}\]
The approx value of the constant \[e\] can be found as:
\[\Rightarrow e=\dfrac{1}{1}+\dfrac{1}{1!}+\dfrac{1}{2!}+\dfrac{1}{3!}+\dfrac{1}{4!}+\dfrac{1}{5!}+...........\]
\[\Rightarrow e=\dfrac{1}{1}+\dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{24}+\dfrac{1}{120}+.........\]
Now, let’s just assume the first few terms:
\[\Rightarrow e=\dfrac{1}{1}+\dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{24}+\dfrac{1}{120}\]
After simplification we get,
\[\Rightarrow e=2.718\]
So, the value of exponential constant \[e=2.72\]
Mathematically, the complete value of \[e\] can go on for thousands of digits. So, we can say that it is an irrational number which is a real number.
Now for calculating value of \[1/e\]
The value of constant is equal to \[2.72\]
So, \[\dfrac{1}{e}=\dfrac{1}{2.72}\]
\[\Rightarrow \dfrac{1}{e}=0.3676\]
\[\Rightarrow \dfrac{1}{e}=0.37\]
\[e\] is irrational (i.e. it cannot be written as a ratio of integers) and transcendental (i.e. it is not a root of any non zero polynomial with rational coefficients). The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change in the dependent variable. If the change is positive it is called exponential growth and if it is negative, it is also called exponential decay.

Note: Euler’s number is used in the most practical sense for working with radioactive decay. In mathematics, it is a crucially important tool that allows mathematicians to convert complex numbers to more usable form. It is also used in growth problems, such as population models.