First — talk for 2 minHas anyone in your family had a savings account or invested money? What grew the most? What didn't grow at all? If no one, Mr. Bennett will share.
How fast can money grow?
In 1683, a Swiss mathematician named Jakob Bernoulli asked a strange question. Imagine a bank that pays you 100% interest each year on a $1 deposit. At year's end, you would have $2. But what if the bank added the interest twice a year — 50% halfway, then another 50% on the new total? You would do a little better. Every month? Better still. Every second?
Bernoulli could not write the answer down as a fraction or a square root. Fifty years later, the mathematician Euler gave the limit a name: the letter e. Like π, it is irrational — its decimal never repeats or ends. And like π, it shows up everywhere — in finance, in biology, even in how a hot drink cools.
Today we will use e to look at two families. Both invest at the same rate. Both wait the same time. Family A starts with $10,000; Family B starts with $1,000,000. The dollar gap between them at the start is $990,000. What happens when interest compounds not annually, not monthly, not daily, but continuously — at every instant — and what does that limit do to the gap after forty years?
Worked Example #1 — Bernoulli's compounding question (build the limit)
ProblemA bank pays 100% interest per year on $1. If it compounds n times per year, after 1 year you have \(P\!\left(1 + \dfrac{r}{n}\right)^{n t}\). With \(P=1\), \(r=1\), \(t=1\), this becomes \(\left(1 + \tfrac{1}{n}\right)^{n}\). What does the value approach as n grows very large?
DataCompute the value for several n. As n grows, the base \(1 + \tfrac{1}{n}\) shrinks toward 1, but the exponent n grows without bound.
Word Bank
exponent
base
level off
irrational
limit
infinity
round up
subtract
Number Bank
2.0000
2.2500
2.7146
2.71828…
3.1416
2.5000
| # | Reasoning (fill in) | Math |
|---|
| 1 |
For \(n = 1\) (annually), the is \(1 + 1 = 2\). Raise to the power 1. |
\(\left(1+\tfrac{1}{1}\right)^{1} = \) |
| 2 |
For \(n = 2\) (semi-annually), the base is \(1.5\). Raise to the 2. |
\(\left(1+\tfrac{1}{2}\right)^{2} = 1.5^{2} = \) |
| 3 |
For \(n = 365\) (daily), use a calculator. The value is just over 2.7. |
\(\left(1+\tfrac{1}{365}\right)^{365} \approx \) |
| 4 |
The values — they do not blow up to infinity. The ceiling is a special number. |
\(\displaystyle \lim_{n \to \infty}\left(1+\tfrac{1}{n}\right)^{n} = \) \(e\) |
| 5 |
The number e is — its decimal never repeats. Write its first few digits. |
\(e \approx \) |
Page 1 of 3 · The Number e & the Math of Wealth Gaps · Algebra 2 · Unit 5 · Lesson 12
Worked Example #2 — Family A: continuous compounding for 40 years
ProblemFamily A invests a starting principal at a continuous rate. Use the formula \(A = P \cdot e^{r t}\) to find the amount after 40 years.
DataFamily A starts with P = $10,000. The S&P 500 has returned about r = 0.07 per year (7%, after inflation) over the last 50 years. The horizon is t = 40 years (one career to retirement).
Word Bank
multiply
exponent
principal
continuous
substitute
divide
add up
Number Bank
2.8
16.4446
164,446
280,000
10,000
2.7183
47
| # | Reasoning (fill in) | Math |
|---|
| 1 |
Identify the P, the rate r, and the time t. |
\(P = \) , \(r = 0.07\), \(t = 40\) |
| 2 |
Compute the \(r \cdot t\) first. |
\(r \cdot t = 0.07 \cdot 40 = \) |
| 3 |
Use the calculator's \(e^{x}\) key to find \(e^{x}\) at \(x = 2.8\). |
\(e^{2.8} \approx \) |
| 4 |
the principal P by the growth factor \(e^{2.8}\). |
\(A = 10{,}000 \cdot 16.4446\) |
| 5 |
Round to the nearest dollar. Family A's $10,000 grows to about $. |
\(A \approx \$164{,}446\) |
Activity 2 — Family B & the gap (fill in from your work)
Family B: P = $1,000,000, r = 0.07, t = 40. Same multiplier \(e^{2.8} \approx\) . So \(A_B = 1{,}000{,}000 \cdot e^{2.8} \approx\) $.
Dollar gap (B − A): at year 0 the gap is $. At year 40 the gap is about $.
Ratio B/A: at year 0 it is ×. At year 40 it is ×. The ratio is (the same / different) — circle one.
Conclude
The math is "fair." The dollars are not. What lever would you pull?
Same rate. Same time. Same \(e^{2.8}\). The ratio is preserved at 100× — that is just algebra: the factor cancels. But the dollar gap grew by the same multiplier as everyone's wealth: from $990,000 at year 0 to about $16.28 million at year 40.
Look at \(A = P \cdot e^{r t}\). Three letters can move: P (who has a starting principal?), r (fees, redlining, predatory lending bend it), t (an earlier start = more years of compounding). Which lever does each policy pull?
Reflect: Pick one intervention from the slide deck (Baby Bonds, expanded CTC, estate-tax restoration, or your own). Which letter in the formula does it change — and why does that move the outcome more than changing the rate?
Page 2 of 3 · The Number e & the Math of Wealth Gaps · Algebra 2 · Unit 5 · Lesson 12
Tier 2 · General Academic Words
- approach — to get closer and closer to a value without quite reaching it.
- ceiling — the highest a value can reach; an upper limit.
- compound — to grow on top of itself, so each gain is added to the next round.
- continuous — happening at every moment, with no pauses or steps.
- credit — to add money to an account.
- factor — a number you multiply by.
- gap — the difference between two amounts.
- horizon — how far ahead in time you are looking (here, 40 years).
- interest — extra money a bank pays you for keeping your money there.
- irrational — cannot be written as a fraction; the decimal never repeats and never ends.
- level off — to stop growing as fast and settle near a value.
- preserve — to keep the same.
- principal — the starting amount of money you invest or borrow.
- ratio — how many times bigger one number is than another (B ÷ A).
Tier 3 · Math & Finance Words
- e (Euler's number) — an irrational constant, about 2.71828… Like π, its decimal never repeats. It is the limit of \(\left(1 + \tfrac{1}{n}\right)^{n}\) as n grows.
- ex (the natural exponential function) — the function with base e. Read aloud as "e to the x." Its key on a calculator is usually labeled \(e^{x}\).
- limit — the value an expression gets closer and closer to as the input gets very large (or very close to something).
- compound interest — interest that gets added to the principal, so next time you earn interest on a bigger pile.
- continuous compounding — compounding at every instant, with no waiting between credits. The math uses e.
- P(1 + r/n)nt — the general compounding formula. P = principal, r = annual rate, n = times compounded per year, t = years.
- P · ert — the continuous compounding formula. Read aloud as "P times e to the r-t." Use this when the bank credits interest at every moment.
- interest rate (r) — the percent of the principal added per year, written as a decimal (7% = 0.07).
- multiplier — the growth factor \(e^{rt}\) that you multiply the principal by. Same rate and time gives the same multiplier for everyone.
- baby bonds — a policy that gives every newborn a starting principal in an account that compounds until age 18.
- estate tax — a tax on very large amounts of money passed from one generation to the next.
- wealth gap — the difference in total wealth between two groups (often by race, region, or income bracket).
Page 3 of 3 · The Number e & the Math of Wealth Gaps · Algebra 2 · Unit 5 · Lesson 12