Euler’s Number (e)
In short: Euler’s number, written e and about 2.71828, is the irrational constant that expressions like (1 + 1/n)^n approach as n grows, and it is the natural base for continuous growth.
Algebra 2Functions and GraphsF-LE.A.4
What is Euler's Number?
Euler’s number is a special constant, roughly 2.71828, that turns up whenever growth happens continuously rather than in steps. Banks that add interest every instant, cultures of bacteria and cooling objects all lead to it.
Its definition is a limit: \(e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n\). The number is irrational, so its decimals never end or repeat. The function \(f(x) = e^x\) has the neat property that its height and its steepness are equal at every point.
How to spot it
- A formula contains the letter e raised to a power, such as A = Pe^{rt} for continuous compounding.
- A calculator key labeled e^x or ln is needed to finish a growth or decay problem.
- A story says the interest is compounded continuously or the population grows at a rate proportional to its size.
- The problem compares (1 + 1/n)^n for larger and larger n, which is how Euler’s number is defined.
The method
- Identify the model that uses Euler’s number: \(A = Pe^{rt}\) for growth, or \(A = Pe^{-rt}\) for decay.
- Write the rate \(r\) as a decimal and multiply it by the time \(t\) to get the exponent.
- Evaluate \(e\) raised to that exponent with a calculator, keeping at least five decimal places.
- Multiply by the starting amount \(P\) and round only at the end.
Euler's Number: solved example
Problem
A savings fund holds $2,500 and earns 3.6% interest compounded continuously. How much is in the account after 7 years?
Solution
Use \(A = Pe^{rt}\) with \(P = 2500\), \(r = 0.036\) and \(t = 7\). The exponent is \(0.036 \times 7 = 0.252\).
Then \(e^{0.252} \approx 1.28660\), so \(A \approx 2500 \times 1.28660 = \)\(\$3{,}216.49\) (about).
Try it
Question 1
Compute \(\left(1 + \frac{1}{100}\right)^{100}\) and compare it with e.
Show the answer
It is about 2.7048, slightly below \(e \approx 2.71828\); larger n would come closer.
Question 2
Approximate \(e^3\).
Show the answer
\(e^3 \approx 20.0855\), so about 20.09.
Euler's Number vs. Natural Logarithm
The constant e is a number, about 2.718, and \(e^x\) is a growth function built on it. The natural logarithm \(\ln x\) is the opposite operation: it asks which power of e gives \(x\). They undo each other, so \(\ln(e^5) = 5\) and \(e^{\ln 5} = 5\).
Keep going
Lessons
Worksheets and quizzes
- Exponential Functions and the Number e – Worksheet
- Exponential Functions and the Number e – Quiz
- Exponential Growth and Decay – Worksheet
- Exponential Growth and Decay – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Functions and Graphs