Euler’s Number
2026-02-28 08:25 Diff
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Last updated on December 10, 2025

Euler’s number, denoted by 𝑒, is a fundamental mathematical constant introduced by Jacob Bernoulli in 1683. Later, Leonhard Euler studied it further, which is why it is named after him.

What is Euler's Number?

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Euler's number, written as e, is an irrational and transcendental constant, with its value being approximately 2.71828. This value is approximate because the decimal expansion goes infinitely without repeating.

We see this value appear naturally in many areas of math and science while describing continuous growth or decay. The Euler's number helps describe any process that requires continuous and smooth change. 

Formula of Euler's Number

Euler’s number, e is defined by the following equation: 

Euler’s formula for compound interest,

\(A = P^{rt}\)

Where, 

FV stands for future value

PV represents present value of balance or sum

e is the mathematical constant

r  is Interest rate being compounded, and 

t is time in years

Euler's Formula for Complex Analysis

Euler’s form connects trigonometry and exponential functions helping in complex analysis. it provides an efficient framework unifying exponential and trigonometric expressions helping simplify mathematical computations. 

For any value of x, the formula is given by:

\(e^ix = cos x + isin x\)

Here,
cos and sin represent the trigonometric ratios functions.
i is the imaginary unit, and
e is the base of the natural logarithm.

Geometrically, this formula can be visualized on a complex plane where \(e^{i𝜃} \) traces a unit circle as the angle θ is measured in radians.

Let us look at the approximate proof of this formula for better understanding.

Let’s start with the taylor series expansion of the exponential function \(e^x\):

\(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots \)

This expansion  holds true for real as well as complex numbers.

Substituting x = iθ, where i is the imaginary unit (i2 = -1)

\(e^{i\theta} = 1 + i\theta + \frac{(i\theta)^2}{2!} + \frac{(i\theta)^3}{3!} + \frac{(i\theta)^4}{4!} + \cdots \)

Now, we group real and imaginary terms separately:

\(e^{i\theta} = \left(1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots \right) + i\left(\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots \right) \)
 

The trigonometric series is 

\(\cos \theta = 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots \)

\(\sin \theta = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots \)

Now, substituting these expansions back to our expression for \(e^iθ\), we get

\(e^iθ = cosθ + isinθ\)

Hence, proved \(e^iθ = cosθ + isinθ\)

Euler's identity: From the above formula, we get \(e^ix = cosx + isinx\)

when x =  π, this formula give the identity 

\(e^{i π} = cos π + isin π\)

\(e^{i π} = -1 + i (0) \), because cos π = -1 and sin π = 0

\(e^{i π} = -1\) or  \(e^{iπ} + 1 = 0\)

This is Euler's identity.

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Euler's Formula for Polyhedra

A polyhedron is a three-dimensional solid shape. It consists of flat faces and straight edges. Cubes, cuboids, prisms and pyramids are some examples of polyhedra. If a polyhedron does not intersect itself, its vertices, faces and edges follow a specific relationship.

According to Euler's formula, the sum of the number of vertices and faces is exactly two more than the number of edges. Mathematically, it can be expressed as:

\(F + V - E = 2\)

Here, F, V and E represent the number of faces, vertices and edges.

Why is Euler’s Number (e) Important?

Euler’s number (e) often appears in situations involving growth and decay, where the rate of change depends on the current amount. This idea is closely connected to Euler’s formula, sometimes referred to as Euler’s formulas or Euler’s rule, which helps describe continuous change. Understanding what Euler’s number is used for in mathematics and science.

For example, in biology, Euler’s number is observed when bacterial populations grow continuously and double at predictable intervals. Another real-life use of Euler’s number formula is in radiometric dating, where the number of radioactive atoms decreases steadily according to the element’s half-life.
The Euler’s number symbol is e, and although it has infinitely many digits, the Euler’s number begins as 2.71828. It plays a crucial role in explaining natural growth patterns, decay processes, and many continuous change phenomena.

How to Use Euler’s Number

The value of e is approximately 2.718. Euler's number is mostly used to calculate the rate of change or growth, such as in finance, radioactive decay, and so on. Here are some examples


Example 1: Calculate the final amount when $100 is invested for 5 years at a 4% interest rate compounded continuously.

Solution: Euler's formula for compounding interest is
A = Pert

Given, \(P = 100\)
\( r = 0.04\)
\(t = 5\)

\(A = 100e^{0.04 \times 5} \)
\(= 100 × 1.2214\)
\(  = 122.14\)

Therefore, the money in the account after 5 years is $122.14.

Example 2: Find the value of e when n = 3 

Solution: Given n = 3,

\(\lim\limits_{x \to \infty} ( 1 + \frac1n)^n \)

\(e =\) \((1 + \frac13)^3 \) \(= 2.37037\)

This is an approximation; Euler's number e ≈ 2.71828.

Important Notes

  • The value of e is approximately 2.718.
     
  • The number e can be defined mathematically as:

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

  • Euler introduced the constant e for use in various mathematical applications. For example, the compound interest formula uses e and is written as,

        \(A = P e^{rt} \)

  • e is the base of natural logarithms, and it plays a key role in solving equations involving continuous growth or decay.
     
  • Euler’s number appears in many real-life applications, such as population growth, radioactive decay, and continuously compounded interest, making it one of the most important constants in mathematics.

Tips and Tricks to Master Euler's Number

Given below are a few tips and tricks that help students better understand and apply the Euler number.
 

  • The approximate value of Euler's number is 2.718. For most calculations, you can use 2.72 as a quick estimate.
     
  • e naturally appears in situations involving growth or decay, so understanding its connection to compound interest is essential.
     
  • You can define e using the limit. This helps you remember that e is linked to repeated growth.
     
  • The function f(x) = ex is unique because its derivative and its integral are both equal to the original function. This makes many calculus problems more straightforward to handle.
     
  • Parents can encourage children to say the number aloud. Repeating helps them remember better.
     
  • Teachers can use real-life examples, such as population growth or simple interest, to show why 2.718 keeps appearing in math.
     
  • Children should imagine adding tiny pieces again and again until they reach a special number, e.

Common Mistakes and How to Avoid Them in Euler’s Number

It’s easy to get confused when working with Euler’s number because it behaves differently from regular numbers. Spotting the mistakes most people make can help you use e correctly and with confidence.

Real-Life Applications of Euler’s Number

The number e quietly shapes many processes happening around us every day. From interest on savings to the way living things grow, it helps explain continuous change.

1. Biology: It is used to calculate the exponential growth and decay of organisms

2.  Physics: Radioactive decay follows an exponential pattern modeled using Euler’s number. 

3. Finance: Compound interest calculations in finance reveal growth and decline patterns, which support better risk management

4. Computer Science: It helps study complex algorithms in fields such as machine learning, computer graphics, optimization, and many more.

5. Weather: Euler’s number is used in studying weather changes, such as temperature changes over time, which involves exponential functions.

Problem 1

Calculate the final amount when $1000 is invested for 4 years at a 6% interest rate compounded continuously.

Okay, lets begin

A = 1271.24

Explanation

Using the formula A = Pert

A = Total money with interest

P = 1000 

r = 0.06

t = 4 

A = 1000 e0.06 × 4

A = 1000 e0.24

A = 1000 × 1.271249

A = 1271.24

Well explained 👍

Problem 2

Find the value of e when n = 5

Okay, lets begin

2.48832

Explanation

Given n = 5,
 

\(\lim\limits_{x \to \infty} ( 1 + \frac1n)^n\)
 

\((1 + \frac15)^5\) = 2.48832

Well explained 👍

Problem 3

Evaluate lim(n→∞) (1+3/n)^n

Okay, lets begin

20.0855

Explanation

We know that,
 

\(\lim\limits_{x \to \infty} ( 1 + \frac1n)^n\) = e, 
 

Given \((1 + \frac3n)^n\)which is equivalent to,
 

\((1 + \frac3n)^n\) = \(1 + \frac1 {n/3}^{n/3 \times 3}\)= \(1 + \frac1 {n/3}^{n/3 }\)


The limit approaches,
 

\(\lim\limits_{x \to \infty} ( 1 + \frac3n)^n\) = e3
 

We know e = 2.71828, then, e3 =  2.718283 = 20.0855
 

Therefore, (1+3n)n = 20.08553

Well explained 👍

Problem 4

Calculate the final amount when $800 is invested for 9 years at a 6% interest rate compounded continuously.

Okay, lets begin

1372.80

Explanation

 Using the formula A = Pert

A = Total money with interest

P = 800 

r = 0.06

t = 9

A = 800 × e0.06 ×  9

A = 800 ×  e0.54

A = 800 × 1.7160068

A = 1372.80

Well explained 👍

Problem 5

Sonia invests $2,000 in a savings account that earns 5% annual interest, compounded continuously. How much money will she have in her account after 6 years?

Okay, lets begin

$2,699.72

Explanation

Sonia’s investment grows with continuous compounding, which uses the formula: \(A = P e^{rt} \)

Where:
A = amount after t years
P = principal = 2000
r = annual interest rate in decimal = 0.05
t = time in years = 6
\(​​​​​​​e ≈ 2.71828\)

Substituting the values, we get

\(A = 2000 \cdot e^{0.05 \cdot 6} = 2000 \cdot e^{0.3} \approx 2000 \cdot 1.34986 \approx 2699.72 \)

So, after 6 years, Sonia will have $2,699.72 in her account. Continuous compounding grows the money slightly faster than regular compounding because interest is added constantly.

Well explained 👍

FAQs on Euler's Number

1.What is the formula for Euler’s number?

The formula for Euler's number is \(\lim\limits_{x \to \infty} ( 1 + \frac1n)^n\)

2.What is the Euler’s formula for compound interest?

Euler’s formula for compound interest,

A = Pert

3.What is the value of e?

The value of e is 2.71828

4.Are Euler's number and Euler's constant the same?

No. Euler's number ‘e’ is used in exponential function and calculus, while Euler's constant. Both represent different values.

5.Is Euler's number irrational?

Yes, Euler's number is irrational, and it has a non-repeating decimal.

6.Can Euler’s number be visualized, and how can I explain it to my child?

Yes! Euler’s number (e ≈ 2.71828) can be visualized using the idea of continuous growth.

7.Is it necessary for children to memorize the value of e, or is conceptual understanding enough?

Memorizing e isn’t necessary for children. Understanding that e ≈ 2.718 represents continuous growth or decay in real-life situations is enough to build a strong foundation. 

Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

Fun Fact

: She loves to read number jokes and games.