Thought Toys · Exhibit 17

How money doubles

Savings don't grow in a straight line — they grow by doubling, and the doublings stack on top of each other. Drag the interest rate and watch how long one doubling takes, and how that doubling time slides across a human lifetime. There's a shortcut hiding in here so simple you can do it in your head.

Balance of $1,000 over time — compounding vs. a straight line
compound — interest on interest simple — a straight line each doubling

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0% · never doubles15% · doubles in ~5 yr

What you're seeing

Put $1,000 in an account that pays interest once a year. The blue line is what most people picture: simple interest, the same few dollars added every year — a straight ramp. The amber curve is what actually happens. Each year's interest is paid on everything you have, including last year's interest, so the gains feed on themselves and the curve bends upward, leaving the straight line far behind.

The clean way to feel exponential growth is to stop watching dollars and start watching doublings. The faint gold rungs mark where your money has doubled — to $2,000, then $4,000, then $8,000. Notice they're spaced evenly in time: every doubling takes the same number of years, no matter how big the pile already is. That even spacing is the signature of exponential growth, and it's why a steady rate eventually runs away.

Now drag the rate. At 0% the money never doubles at all — the curve is flat and the gold rungs vanish. Nudge it up and the first doubling marches in from the right edge; raise it more and a second and third rung appear inside your lifetime. A small change in rate doesn't add a little — it changes how many times your money doubles, and each extra doubling is another ×2. That's why 2% and 7% aren't close: over forty years they end nearly seven times apart.

The rule, exactly. With annual compounding the balance is A(t) = P(1 + r)t so the time to double solves (1+r)T = 2, giving Tdouble = ln 2 ⁄ ln(1 + r) The famous shortcut: for the ordinary single-digit rates, ln(1+r) ≈ r and 100·ln 2 ≈ 69, nudged to 72 because it divides cleanly and corrects for annual (not continuous) compounding — so years to double ≈ 72 ⁄ rate%. At 8% that's 9.00 years against an exact 9.01. Counter-example, verified in node: at r = 0 the doubling time is infinite — the balance never doubles and compound equals simple (the negative control: doubling needs a rate above zero, and the gap over 2% vs 7% compounds to 6.8× over 40 years).

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