Summarize this article:
139 LearnersLast updated on 17 September 2025
Euler’s Number
Euler’s number, written as 𝑒, is a 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?
- Euler's number is an irrational and transcendental mathematical constant. Its value is approximately equal to 2.71828 and is denoted as ‘e’.
- e = 2.71828
- The decimal expansion of Euler's number’s goes on forever without repeating.
- Euler’s number is used in math and science to describe things that grow or decay over time.
Formula of Euler's Number
e here represents Euler’s number, and it is defined by the following equation:
Euler’s formula for compound interest,
A = Pert
Where
FV = Future value
PV = Present value of balance or sum
e = mathematical constant
r = Interest rate being compounded
t = Time in years
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 = 100e0.04 × 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.
Real-Life Applications of Euler’s Number
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.
Common Mistakes and How to Avoid Them in Euler’s Number
Mistake 1
Confusing e as a variable
Students might misunderstand e for a variable like m or n. Remember that e is a constant and its value is approximately 2.718
Mistake 2
Not dividing the interest rate by 100.
Children might use the interest rate as a whole number instead of converting it into a decimal by dividing it by 100 while solving a compound interest problem.
For example, they can use the interest rate 5 as it is, by not converting it into 0.05 (5/100). By doing so, the result will be incorrect.
Mistake 3
Getting confused between Euler’s number and Euler’s constant
Note that the value of Euler's number e is ≈ 2.71828, and Euler's constant is ≈ 0.57721
Mistake 4
Not using proper exponents correctly. e4x when it is actually e4x
Make sure to write the exponents correctly.
Not writing exponents correctly, e.g., writing e4x instead of e4x.
Mistake 5
Assuming that e is a whole number
The value of e is always a decimal number, which is approximately equal to 2.718. Students might incorrectly approximate e as 2 or 3.
Solved Examples for Euler's Number
Problem 1
Calculate the final amount when $1000 is invested for 4 years at a 6% interest rate compounded continuously.
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
Problem 2
Find the value of e when n = 5
2.48832
Explanation
Given n = 5,
\(\lim\limits_{x \to \infty} ( 1 + \frac1n)^n\)
\((1 + \frac15)^5\) = 2.48832
Problem 3
Evaluate lim(n→∞) (1+3/n)^n
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
Problem 4
Calculate the final amount when $800 is invested for 9 years at a 6% interest rate compounded continuously.
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
FAQs on Euler's Number
1.What is the formula for Euler’s number?
2.What is the Euler’s formula for compound interest?
3.What is the value of e?
4.Are Euler's number and Euler's constant the same?
5.Is Euler's number irrational?
6.How can children in Indonesia use numbers in everyday life to understand Euler’s Number?
7.What are some fun ways kids in Indonesia can practice Euler’s Number with numbers?
8.What role do numbers and Euler’s Number play in helping children in Indonesia develop problem-solving skills?
9.How can families in Indonesia create number-rich environments to improve Euler’s Number skills?
Explore More numbers
![Important Math Links Icon]()
Previous to Euler’s Number
![Important Math Links Icon]()
Next to Euler’s Number
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.
