Value of e

Value of e in mathematics is approximately 2.71828. “e” is an irrational number i.e., a non-repeating non-repeating number; meaning “e” cannot be expressed as a simple fraction. Euler’s Number or “e” is the base of the natural logarithm and has many applications in various mathematical and scientific contexts. Euler’s Number is similar to Pi (π).

You have heard about ‘e’ before but in this article, we’ll learn the significance of ‘e’ or Euler number in mathematics. We will find the value of ‘e’ stepwise and also practice some cool examples.

What is ‘e’ or Euler Number?

Euler number, denoted as “e,” is a mathematical constant that is approximately equal to 2.71828. Euler Number is named after the Swiss mathematician Leonhard Euler, who made important contributions to the understanding of this number.

The Euler number is an irrational number, which means that its decimal representation goes on forever without repeating. Like the more well-known irrational number π (pi), e has many interesting mathematical properties. One of its key features is its significance in calculus, particularly in the study of exponential growth and decay.

Euler Number (e) Definition

Euler number (e) is an irrational number which means it can’t be written as a simple fraction and is a non-terminating and non-repeating decimal which is approximately equal to 2.71828.

Limit Definition of Euler Number

The Euler number (often denoted as “e”) can be defined using a limit. The most common definition is based on the limit of the following expression as n approaches infinity:

Where,

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

Symbol for ‘Euler number’

In mathematical notation the Euler number is defined by a lowercase alphabet ‘e’. It is also called ‘Euler constant’. The approximate value of Euler constant is 2.71828.

Symbol for Euler Number is ‘e‘.

Approximate value of ‘e’

As we have read that euler number is an irrational number and it never ends. So, the exact value of ‘e’ cannot be determined and that’s why we take the approximate value of ‘e’ which is 2.71828.

Value of ‘e’ ≈ 2.71828

Full Value of ‘e’

Euler number or ‘e’ is a non-repeating decimal which is infinite. It never ends. This is the reason we can not find the full and exact value of ‘e’. We can write some significan

We can write the full value of e as 2.7182818284590452353602874713527 . . .

Where . . . implies the endless nature of the value.

Formula for e

The formula for ‘e’ is quite interesting and unique. 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’.

How to Calculate the Value of e

Now, we have understood what exactly ‘e’ is. It’s high time to know how we can determine the value of ‘e’.

Calculating the formula of ‘e’ can be a bit challenging but it’s doable. 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.

Let’s determine the approximate value of ‘e’. Put all the values in formula:

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

⇒ e = 2.71828 . . .

This infinite series is a great way to calculate the value of ‘e’.

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.

Read More,

• Value of Pi
• Value of Log e

Solved Examples on Value of e

Example 1: If function f(x)= e^x then what will be the value of f(3).

Solution:

Given that,

f(x)= e^x

Now, to find f(3). we substitute the value x=3.

f(3)= e³

We know that

e= 2.71828

Then,

f(3)= 2.71828³

f(3)= 20.0855

Therefore, the value of f(3) is 20.0855.

Example 2: Given function is f(x)= e^x then determine f(7).

Solution:

Given that,

f(x)= e^x

Now, to find f(7). we substitute the value x=7.

f(7)= e⁷

We know that

e= 2.71828

Then,

f(7)= 2.71828⁷

f(7)= 1096.63

Therefore, the value of f(7) is 1096.63.

Practice Problems on Value of e

Probelm 1: Evaluate the limit as x approaches infinity: .

Probelm 2: Calculate the sum of the infinite series: 1+ 1/1! + 1/2! + 1/3! + 1/4! . . .

Probelm 3: Find the derivative of the function f(x) = e^2x.

Probelm 4: Solve the differential equation: dy/dx = y with the initial condition y(0) = 1.

Probelm 5: Compute the integral: ∫e^x dx .

FAQs on value of ‘e’

1. What is e?

‘e’ is an irrational number which is non-terminating and cannot be expressed in finite form.

2. Why is e called an Euler Number or Euler constant?

A scientist whose name is Leonhard Euler has discovered ‘e’ while performing some logarithmic functions. That is why, ‘e’ is known as Euler number or Euler constant.

3. What is the Exact Value of e?

The exact value of ‘e’ cannot be determined because euler numbers are infinite and they keep going on forever without stopping.

4. Where do we use e in Mathematics?

‘e’ is used in various fields of mathematics. The prominent ones are calculus, probability, exponential growth and radioactive decay.

5. How can we find the Value of e?

The value of ‘e’ can be calculated by the following formula:

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

Keep adding further terms and you will get the approximate value of ‘e’.

6. What is the Approximate Value of e?

The approximate value of ‘e’ is 2.71828.

7. Can we express e as a Fraction?

No, ‘e’ is an irrational number that means it can’t be written in p/q form or the ratio of integers.