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Lesson ID: 13744
Discover how the number e plays a crucial role in exponential growth, from continuous compound interest to population models. Learn the formulas, solve problems, and explore real-world applications.
Imagine you have a magic bank account. Every time you check it, there's a little more money inside.
No one put it there, and it didn't come from thin air—it just keeps growing on its own.
Well, it’s not magic—it’s math!
At the heart of this growth is a special number: e. And if you understand how e works, you can use it to maximize your savings, predict population growth, and even understand how things like bacteria multiply.
Dig into why this number is so powerful.
You may already be familiar with compound interest—the idea that money can grow over time when interest is added to the principal repeatedly.
The more often interest is added, the faster the money grows.
That’s where e comes in.
Mathematicians Jacob Bernoulli and Leonhard Euler studied this idea in the 1600s and 1700s. Bernoulli discovered that if interest is compounded more and more frequently—hourly, by the second, or even at every tiny moment—the formula for compound interest approaches a limit.
Euler later proved that this limit is an irrational number called e, approximately equal to 2.718.
Break it down step by step.
The Compound Interest Formula
The formula for compound interest is:
A = P(1 + r/n)nt
Where:
A = final amount in the account
P = initial principal (starting amount)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
The more frequently interest is added (larger n), the greater the final amount.
Replace the formula with this version, using e:
A = Pe(rt)
This is the formula for continuous compounding—where n becomes infinitely large.
Example: Growing Your Money
Say you invest $1,000 at an annual interest rate of 5%, compounded continuously.
A = 1000e(0.05 × 10)
A ≈ 1000e0.5
A ≈ 1000 × 1.6487
A ≈ $1,648.72
With continuous compounding, your money grows a little faster than with daily or monthly interest. That’s why banks, businesses, and scientists rely on this formula for precise calculations.
Now, practice using the formula yourself in the Got It? section!
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