One Grain of Rice
There's a legend — possibly apocryphal, certainly instructive — about the inventor of chess presenting his game to an Indian king. The king, delighted, offered any reward. The inventor's request was modest: one grain of rice on the first square, two on the second, four on the third, and so on, doubling each square across all sixty-four.
The king laughed. That's it? A few handfuls of rice?
Let's count. The first row of the chessboard is easy: 1, 2, 4, 8, 16, 32, 64, 128. That's 255 grains total — about a pinch. By the end of the second row, you're at 65,535 grains. Getting there. A decent bowl of rice.
But math has a wicked sense of humor. By the halfway mark — square 32 — you're at about 4.3 billion grains. That's roughly 100 metric tons, enough to fill a couple of Olympic swimming pools. The king is starting to sweat.
By square 64, the total is 264 − 1 = 18,446,744,073,709,551,615 grains. That's 18.4 quintillion. More rice than has ever been harvested in all of human history.1 The king, depending on which version you prefer, either beheaded the inventor or appointed him chief advisor. The math doesn't change either way.
This is the story of compound interest — the most powerful force in finance, and possibly the most misunderstood force in human cognition. Albert Einstein allegedly called it "the eighth wonder of the world," adding: "He who understands it, earns it; he who doesn't, pays it."2 Whether Einstein actually said this is unclear. But whoever did was right.
The Rule of 72
Here's the most useful piece of mental arithmetic you'll ever learn. Want to know how long it takes your money to double? Divide 72 by the interest rate.
The Rule of 72
Doubling Time ≈ 72 ÷ Interest Rate (%)
A quick approximation accurate to within a year for rates between 2% and 72%.
At 7% annual return — roughly the long-run stock market average after inflation — your money doubles every 10.3 years. A 25-year-old who invests $10,000 and never touches it will have $80,000 by retirement. Not from cleverness. Not from trading tips. Just from sitting still and letting mathematics do its thing.
But here's where Jordan Ellenberg would lean forward in his chair and say: wait. The Rule of 72 is symmetric. It works in both directions.
Credit card debt at 24% APR? That doubles in 3 years. A $5,000 balance you "forget about" becomes $10,000, then $20,000, then $40,000. The same exponential curve that builds generational wealth — pointed at you.
Payday loans at 400% APR? Your debt doubles in 66 days.
The math doesn't have a moral compass. It compounds whatever you give it.
This is the fundamental asymmetry of modern finance: the people who most need to understand compound interest — borrowers drowning in high-rate debt — are the ones least likely to have been taught it. Meanwhile, the people who benefit most from compound growth — investors with capital — learned about it early enough to put it to work.3
Student loan debt in the United States surpassed $1.7 trillion in 2023. Much of that growth isn't new borrowing — it's old borrowing, compounding. A $30,000 loan at 6.8% that you defer for a decade doesn't stay at $30,000. It becomes $58,000. The principal hasn't changed. The interest has been having babies.
Franklin's Gamble
Benjamin Franklin understood all of this, of course. When he died in 1790, he left £1,000 (about $4,400 at the time) each to the cities of Boston and Philadelphia. But Franklin being Franklin, the gift came with conditions: the cities couldn't touch most of the money for 200 years.4
His will specified that the funds should be lent to young tradesmen at 5% interest. After 100 years, each city could withdraw a portion for public works. After 200 years — in 1990 — the remainder would be released.
The result? Boston's fund had grown to approximately $5 million. Philadelphia's reached about $2 million (they'd managed it less aggressively). Combined: roughly $7 million from an initial $8,800.5
Now, $7 million across 200 years isn't going to make anyone a billionaire. The average annual return was modest — around 3.5%. But that's the point. Even mediocre compound growth, given enough time, produces extraordinary results. The chess inventor didn't need a high rate of return. He just needed 64 squares.
Where e Comes From
There's a deeper mathematical story hiding inside compound interest, and it leads to one of the most important numbers in all of mathematics.
Suppose you invest $1 at 100% annual interest. If it compounds once a year, you get $2. Compound twice a year (50% every six months), and you get $2.25. Four times a year? $2.44. Monthly? $2.61.
What if you compound continuously — every instant, every infinitesimal sliver of time? Jacob Bernoulli asked this question in 1683, and the answer turns out to be a specific, irrational number:6
The Birth of e
e = lim (1 + 1/n)n = 2.71828...
As compounding frequency approaches infinity, growth converges to Euler's number.
This number — e, approximately 2.71828 — shows up everywhere: radioactive decay, population growth, the bell curve, the distribution of primes, even the optimal strategy for choosing a spouse (the famous secretary problem). It's the universe's growth constant, and it was born from a banker's question about interest rates.
The formula for continuous compounding is elegant:
Continuous Compounding
A = P · ert
- A
- Final amount
- P
- Principal (starting amount)
- r
- Annual interest rate (as decimal)
- t
- Time in years
The practical difference between daily compounding and continuous compounding is negligible — a few cents on a thousand dollars. But the conceptual difference is enormous. Continuous compounding tells us that exponential growth isn't a staircase of discrete steps. It's a smooth, relentless curve. It never pauses, never sleeps, never takes a day off.
Try It Yourself
Play with the numbers. Toggle between wealth mode and debt mode to see how the same mathematics creates fortunes and traps.
Compound Growth Calculator
Watch exponential growth in action — for better or worse.
Final Value
$76,123
From $10,000 over 30 years
Rule of 72
At 7%, your money doubles every ~10.3 years.
Exact doubling time: 10.24 years
The Linear Bias
Here's the uncomfortable truth about human cognition: we are, at our core, linear thinkers trapped in an exponential world.
When we see something growing, our brains instinctively project a straight line. If a pandemic case count went from 100 to 200 in a week, we think: in ten weeks, it'll be around 1,100. The actual answer, at a weekly doubling rate, is 102,400.7
This isn't stupidity. It's biology. For most of human evolution, the things that mattered — the distance to the next village, the number of predators, the amount of food — changed linearly, if they changed at all. Our brains evolved to think in straight lines because, for 200,000 years, straight lines were good enough.
They're not good enough anymore.
Why This Matters
Climate feedback loops, viral spread, technological acceleration, compound debt — the forces shaping the 21st century are all exponential. And every time you underestimate an exponential curve (which your brain will do, automatically, without asking), you're the king laughing at a few grains of rice.
During the early weeks of COVID-19, epidemiologists were screaming about exponential growth while politicians pointed at the "small" case counts. When Italy had 322 confirmed cases on February 25, 2020, the doubling time was roughly 2.5 days. Within a month, they had over 100,000. The math was never a secret. The problem was that human brains are constitutionally incapable of feeling exponential growth until it's too late.
Let's test your own linear bias:
Linear vs Exponential Guesser
I'll show you a growing quantity. You guess where it ends up. Most people are wildly off.
Both Sides of the Curve
The real lesson of compound interest isn't "invest early" — though you should — or "avoid debt" — though you should do that too. The real lesson is about the shape of exponential curves and why they surprise us every single time.
An exponential curve looks linear at the beginning. This is the dangerous part. When your student loan balance ticks up by a few hundred dollars in interest, it feels manageable. When your retirement account grows by $500 in a year, it feels pointless. Both feelings are wrong, and they're wrong for the same reason: you're standing at the left edge of the curve, where everything looks flat.
The policy implications are staggering. A credit card company doesn't need you to borrow a lot. It just needs you to borrow a little, at a high rate, for a long time. The compounding does the rest. Meanwhile, a 22-year-old putting $200 a month into an index fund doesn't feel like they're building wealth. They are. They just can't see it yet because they're stuck on the flat part of the curve.8
Ellenberg's entire project in How Not to Be Wrong is about mathematical thinking as a superpower — a way to see structure where others see chaos. Compound interest is maybe the purest example. The structure is right there, encoded in a formula anyone can learn in five minutes. But feeling it — really internalizing what exponential growth means — that takes deliberate effort against our own cognitive wiring.
The king learned this lesson the hard way. Most of us do too. But you don't have to. You have the formula. You have the Rule of 72. And now, maybe, you have the gut-level sense that when something doubles, and doubles again, the world has changed before you've noticed.
Pay attention to the flat part of the curve. That's where your future lives.