Beyond Interest Rates:
The Number That Defines Growth
This tutorial is designed to take you from the familiar concept of earning interest in a bank account to one of the most important discoveries in mathematics: Euler's number, $e$, and its function $e^x$.
We will follow the logical progression found in classic texts like Teach Yourself Calculus.
The Banker's Problem (Realistic Growth)
Let's start with a realistic scenario. Imagine you invest $100.00 (Principal, $P$) at a bank that offers a generic interest rate of 10% per annum ($r = 0.10$).
We want to see how much money you have after 1 year depending on how often the bank calculates (compounds) the interest.
Case A: Annual Compounding ($n=1$)
If the bank calculates interest once at the end of the year, you get your $100 plus 10% of it.
$$\text{Total} = 100(1 + 0.10) = \$110.00$$
Case B: Semi-Annual Compounding ($n=2$)
Now, assume the bank compounds every 6 months. They split the 10% annual rate into two chunks of 5% ($0.05$), but they apply it twice.
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First 6 months: You earn 5% on $100.$$100(1.05) = \$105.00$$
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Second 6 months: You earn 5% on the new total ($105.00).$$105(1.05) = \$110.25$$
Notice you earned an extra $0.25. This is "interest on interest."
$$A = 100\left(1 + \frac{0.10}{2}\right)^2 = 110.25$$
Case C: Generalizing the Formula
If we compound the interest $n$ times per year, we slice the 10% rate into $\frac{0.10}{n}$ pieces, but we apply it $n$ times.
$$A = 100\left(1 + \frac{0.10}{n}\right)^n$$
Let's see what happens to our money as we compound more and more frequently:
| Frequency | n | Calculation | Result (Balance) |
|---|---|---|---|
| Annual | 1 | $100(1 + 0.1/1)^1$ | $110.00 |
| Semi-Annual | 2 | $100(1 + 0.1/2)^2$ | $110.25 |
| Quarterly | 4 | $100(1 + 0.1/4)^4$ | $110.38 |
| Monthly | 12 | $100(1 + 0.1/12)^{12}$ | $110.47 |
| Daily | 365 | $100(1 + 0.1/365)^{365}$ | $110.51 |
| Every Second | 31M+ | $100(1 + \dots)^{\dots}$ | $110.517... |
Visualizing Compound Interest Growth
The Insight: Finding the Base
The Surprising Ceiling
You might hope that if you compounded every microsecond, you would become a millionaire. But as the table shows, the growth hits a "ceiling" around $110.52.
To understand why this specific number appears, mathematicians simplify the problem to its purest form. They ask: "What if we had $1 at 100% interest?"
This simplifies the formula $\left(1 + \frac{r}{n}\right)^n$ into the fundamental limit definition:
$$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.71828...$$
Convergence Chart: Here is a chart showing how the pure value approaches $e$ as $n$ increases:
Convergence to e
| n (Frequency) | Formula: $(1+1/n)^n$ | Approximation of e |
|---|---|---|
| 1 | $(1 + 1/1)^1$ | 2.00000 |
| 2 | $(1 + 1/2)^2$ | 2.25000 |
| 5 | $(1 + 1/5)^5$ | 2.48832 |
| 10 | $(1 + 1/10)^{10}$ | 2.59374 |
| 100 | $(1 + 1/100)^{100}$ | 2.70481 |
| 10,000 | $(1 + 1/10000)^{10000}$ | 2.71815 |
| 1,000,000 | $(1 + \dots)^{\dots}$ | 2.71828 |
| $\infty$ | Limit | 2.71828... |
Visualizing Convergence to e
Our specific bank account result ($110.517...) is actually calculated using this constant:
$$\text{Result} = 100 \times e^{0.10} \approx 110.51709$$
The Algebra of Infinity (The Binomial Theorem)
We know what $e$ is (a limit), but how do we calculate it easily without a calculator? We use algebra to expand that bracket.
Before we expand, a quick note on notation: The exclamation mark $n!$ (factorial) means multiplying a number by all integers below it (e.g., $3! = 3 \times 2 \times 1 = 6$). Recall the Binomial Theorem for expanding $(a+b)^n$:
$$(a+b)^n = a^n + n a^{n-1}b + \frac{n(n-1)}{2!} a^{n-2}b^2 + \frac{n(n-1)(n-2)}{3!} a^{n-3}b^3 + \dots$$
Let's apply this to the pure definition of $e$, where $a=1$ and $b=\frac{1}{n}$:
$$\left(1 + \frac{1}{n}\right)^n = 1^n + n(1)^{n-1}\left(\frac{1}{n}\right) + \frac{n(n-1)}{2!}(1)^{n-2}\left(\frac{1}{n}\right)^2 + \frac{n(n-1)(n-2)}{3!}(1)^{n-3}\left(\frac{1}{n}\right)^3 + \dots$$
Since $1$ to any power is just $1$, this simplifies to:
$$= 1 + n\left(\frac{1}{n}\right) + \frac{n(n-1)}{2! n^2} + \frac{n(n-1)(n-2)}{3! n^3} + \dots$$
The Simplification Step
Why This Step Confuses Students
This step often confuses students. How do we get from messy fractions like $\frac{n(n-1)}{n^2}$ to something clean? Let's break it down.
We can split the $n^2$ in the denominator into $n \times n$:
$$\frac{n(n-1)}{n^2} = \frac{n}{n} \times \frac{n-1}{n}$$
Since $\frac{n}{n} = 1$ and $\frac{n-1}{n} = 1 - \frac{1}{n}$, this becomes:
$$1 \times \left(1 - \frac{1}{n}\right)$$
Using this logic for all terms, our series becomes:
$$= 1 + 1 + \frac{1(1 - 1/n)}{2!} + \frac{1(1 - 1/n)(1 - 2/n)}{3!} + \dots$$
Taking the Limit
Now, let $n$ go to infinity ($n \to \infty$). As $n$ becomes huge, fractions like $1/n$ and $2/n$ approach zero. This means factors like $(1 - 1/n)$ effectively become $1$. The expression cleans up beautifully:
$$e = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \dots$$
This is the infinite series definition of $e$. It converges very fast:
- • $1$
- • $+ 1 = 2$
- • $+ 0.5 = 2.5$
- • $+ 0.166 = 2.66...$
- • $+ 0.0416 = 2.708...$
The Exponential Function ($e^x$)
In calculus, we rarely deal with just the number $e$. We deal with the function $y = e^x$. This represents continuous growth over a time period $x$.
Using a similar logic with the Binomial Theorem (replacing $1/n$ with $x/n$), we get the Exponential Series:
$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \dots$$
This gives us the famous Continuous Compound Interest Formula: $A = P e^{rt}$.
Remember our bank account with rate $r=0.10$? We can find the continuous compounding factor just by plugging $x=0.10$ into this series:
$$e^{0.1} = 1 + 0.1 + \frac{0.01}{2} + \frac{0.001}{6} + \dots \approx 1.10517$$
Multiply that by your $100 principal, and you get $110.517, matching our table from Part 1 exactly.
Generalizing to Finance ($A = Pe^{rt}$)
In our specific example, we had a rate of 10% ($x=0.10$) and a time of 1 year. In the wider world of finance, this leads to the formula for Continuous Compound Interest:
$$A = P e^{rt}$$
Where:
- • $P$ is the Principal (starting money).
- • $r$ is the annual interest rate.
- • $t$ is the time in years.
- • $e$ is our constant (2.718...).
This formula is derived directly from the limit we just found. It calculates the maximum possible interest you can earn if the bank compounded your money every instant.
The Calculus Connection (The Derivative)
This is likely the content of your final screenshot: applying calculus to this series.
Why the Derivative Matters
Instantaneous vs. Average Growth
In algebra, we calculate average growth over a year. In calculus, we want to know the instantaneous speed of growth at any specific moment. If you freeze time, how fast is your money growing right now? To find that, we need the derivative (slope).
Notation: $\frac{d}{dx}$ means "take the derivative with respect to x".
Recall the Power Rule for derivatives: $\frac{d}{dx}(x^n) = nx^{n-1}$.
- • The derivative of $x$ is $1$.
- • The derivative of $x^2$ is $2x$.
- • The derivative of a constant ($1$) is $0$.
Let's differentiate the infinite series of $e^x$ term by term:
$$y = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \dots$$
$$\frac{dy}{dx} = 0 + 1 + \frac{2x}{2!} + \frac{3x^2}{3!} + \frac{4x^3}{4!} + \dots$$
Look closely at the fractions.
- • $\frac{2x}{2!} = \frac{2x}{2 \times 1} = x$
- • $\frac{3x^2}{3!} = \frac{3x^2}{3 \times 2 \times 1} = \frac{x^2}{2!} = \frac{x^2}{2}$
- • $\frac{4x^3}{4!} = \frac{4x^3}{4 \times 3 \times 2 \times 1} = \frac{x^3}{3!} = \frac{x^3}{6}$
So, the derivative simplifies to:
$$\frac{dy}{dx} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$$
Do you recognize this? It is the exact same series we started with!
$$\frac{d}{dx}(e^x) = e^x$$
The Physical Meaning: Bacteria & Money
Unique in All of Mathematics
This result is unique in all of mathematics. The function $y = e^x$ is the only function where the rate of growth (slope) is exactly equal to the current size (value).
- 1. Money: If you have $e^x$ dollars, your wealth is growing at a rate of exactly $e^x$ dollars per year. The more you have, the faster you get rich.
- 2. Biology: Imagine a colony of bacteria. If the size of the colony is $y$, and they reproduce freely, the rate at which new bacteria are born ($\frac{dy}{dx}$) is directly proportional to the current number of bacteria ($y$).
This is why $e$ appears everywhere in nature—from radioactive decay to population growth—wherever change is proportional to size.
Summary Flow
- 1. Algebra: We start with a bank account ($100(1 + 0.1/n)^n$).
- 2. Limits: We let $n \to \infty$ and find that the interest doesn't explode; it settles on a multiple of $e$.
- 3. Expansion: We use the Binomial Theorem to write $e$ and $e^x$ as sums of simple factorials.
- 4. Calculus: We differentiate that sum to prove that the slope of $e^x$ is $e^x$ itself.
The Big Picture
From a simple question about bank accounts, we've discovered one of mathematics' most fundamental constants. The number $e$ connects finance, algebra, calculus, and natural phenomena in a way that no other number does. It's not just a number—it's the language of growth itself.