Value of e

The value of e in mathematics is approximately 2.71828. The number (e) is an irrational number, which means its decimal expansion is non-terminating and non-repeating. Similar to (π), the value of (e) continues forever without forming a repeating pattern.

The constant (e) is also known as Euler’s number and serves as the base of the natural logarithm.

The number (e) is extremely important in mathematics because it naturally appears in problems involving:

• exponential growth,
• continuous compounding,
• calculus,
• probability,
• and differential equations.

The formula for e is:

e = 1+ 1/1! + 1/2! + 1/3! + 1/4! . . .

This is an infinite series which when taken till any finite number of terms gives the approximate value of e, and the more numbers you add, the closer you get towards the value of 'e'.

Limit Definition of Euler Number

The Euler number can be defined using a limit. The most common definition is based on the limit of the following expression as n approaches infinity:

\bold{e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n}

Where,

• n is a positive integer,
• 1/n represents the reciprocal of n.

Calculating the Value of e

We know that the formula of 'e' is expressed as:

e = 1+ 1/1! + 1/2! + 1/3! + 1/4! . . .....

Let's breakdown it stepwise.

• Start with the first term which is 1.
• Now, add second term which is 1/1!. The factorial of 1! is equal to 1. So, here we get 1/1! = 1.
• Add third term which is 1/2!. The factorial of 1/2! is equal to 1/4. So, here we get 1/2! = 0.5.
• Moving to the fourth term which is 1/3!. The factorial of 1/3! is equal to 1/6. So, here we get 1/3! = 0.1667.
• Coming to fifth term which is 1/4!. The factorial of 1/4! is equal to 1/24. So, here we get 1/4! = 0.04167.
• Keep adding the next terms to find the approximate value. The more numbers you add, the more accurate approximate value you will get.
• Let's make it more simple by adding some more terms: the sixth term 1/5! = 0.008333 and the seventh term 1/6! will be 0.001389.

Put all the values in formula:

e = 1 + 1 + 0.5 + 0.1667 + 0.04167 + 0.008333 + 0.001389 + . . .

e = 2.71828 . . .

Properties of Euler's Number

There are some of the most common properteis of Euler's Number:

• Euler number plays a crucial role in various topics of math such as calculus, probability and complex analysis.
• The Euler number or 'e' makes it easy for us to understand the phenomena of exponential decay, radioactive decay etc.
• It is used as an important parameter to create an exponential function.
• In real life also, it plays a vital role in calculations. Suppose, you have deposited money in your bank account and you get a certain amount of interest rate which keeps compounding, 'e' is the magical element that makes it happen.

Derivatives of e^x

Whenever you have dealt with functions including 'e^x' (where 'x' is a variable), the derivative of 'e^x' is itself the function. In simple words, the rate of change of 'e^x' is 'e^x.' This characteristics of e makes it special because the derivative of 'e' is as similar as the function itself.

For example: if f(x) = e^x, then the derivative f'(x) = e^x which is same as function.

Integrals of e^x

While in the case of integration of 'e^x' with respect to 'x,' the integral of 'e^x' is 'e^x' and a constant (C). The integral of 'e^x' also remains 'e^x' when you perform integration with respect to 'x.'

For example: if f(x) = e^x , then the integration f'(x) = e^x + C.