Question

How do you calculate Euler’s Number?

Answer 1

Hint: Here, we have to calculate the value of Euler’s Number i.e., $e$. It is mentioned under the log function also known as the base of the logarithmic function and written as ${\log _e}$. We will use the law of exponents of an infinite series to calculate the value of $e$ which is ${e^x} = 1 + \dfrac{x}{{1!}} + \dfrac{{{x^2}}}{{2!}} + \dfrac{{{x^3}}}{{3!}} + \dfrac{{{x^4}}}{{4!}} + \ldots + \infty $

Complete answer:
The value of $e$ is also known as Napier’s constant. While studying compound interest, Jacob Bernoulli discovered the value of $e$ which is similar to other certain mathematical concepts, equations and problems.
Now, we will calculate the value of $e$.
According to the definition of exponential function ${e^x}$ is an infinite series, which is
${e^x} = 1 + \dfrac{x}{{1!}} + \dfrac{{{x^2}}}{{2!}} + \dfrac{{{x^3}}}{{3!}} + \dfrac{{{x^4}}}{{4!}} + \ldots + \infty $
Where the symbol $!$ refers to the factorial.
So, the value of $e$ can be calculated as
$ \Rightarrow e = 1 + \dfrac{1}{{1!}} + \dfrac{{{1^2}}}{{2!}} + \dfrac{{{1^3}}}{{3!}} + \dfrac{{{1^4}}}{{4!}} + \dfrac{{{1^5}}}{{5!}} + \ldots + \infty $
$ \Rightarrow e = 1 + \dfrac{1}{1} + \dfrac{1}{{1 \times 2}} + \dfrac{1}{{1 \times 2 \times 3}} + \dfrac{1}{{1 \times 2 \times 3 \times 4}} + \dfrac{1}{{1 \times 2 \times 3 \times 4 \times 5}} + \ldots + \infty $
On multiplying the numbers in denominators. We get,
$ \Rightarrow e = 1 + \dfrac{1}{1} + \dfrac{1}{2} + \dfrac{1}{6} + \dfrac{1}{{24}} + \dfrac{1}{{120}} + \ldots + \infty $
Now, let us assume the first few terms
$ \Rightarrow e = 1 + \dfrac{1}{1} + \dfrac{1}{2} + \dfrac{1}{6} + \dfrac{1}{{24}} + \dfrac{1}{{120}}$
Solving the above equation by taking least common factor and adding the numbers in denominators. We get,
$ \Rightarrow e = 2.7182818 \ldots $
The value of $e$ found by calculation is $2.7182818 \ldots $ which is an irrational number and also a real number as all irrational numbers are real numbers.
The approximate value of $e$ is $2.718$ which is used for calculation.

Note: The value of $e$ is an irrational number as there are infinite numbers after the decimal and such numbers are called an irrational number. Therefore, $e$ is an irrational number. The value of $e$ is special when it acts as the base of logarithmic function, its value is $1$ i.e., ${\log _e} = 1$. The limit of the Euler’s number is ${\left( {1 + \dfrac{1}{n}} \right)^n}$ where the value of $n$ addresses the infinity.