Exponential Growth: Doubling, Rule of 72 and Half-Life

Adding against doubling

Two plants start with one leaf each. The first plant grows 2 leaves every day. The second plant doubles its leaves every day. After a week the gap is already clear: the first has 1 + 2 × 7 = 15 leaves, the second 2⁷ = 128. After 10 days it is 21 against 1 024, and after 30 days it is 61 against 1 073 741 824. Adding the same amount each time is linear growth, and its graph is a straight line. Multiplying by the same factor each time is exponential growth, and its graph bends upward faster and faster. Watch the two curves part in the drawing.

The chessboard legend

A famous legend tells of the inventor of chess who asked a king for a reward: one grain of wheat on the first square, two on the second, four on the third, doubling on every square. The king laughed at such a small request. It is only a story, and the details change from teller to teller. But the arithmetic is real. Square 10 holds 2⁹ = 512 grains, square 11 holds 1 024, and by square 21 there are over a million. Square 64 alone holds 2⁶³ = 9 223 372 036 854 775 808 grains, about 9.2 quintillion.

Adding up the whole board

How many grains does the whole board need? Add the squares: 1 + 2 + 4 + 8 and so on. For the first three squares, 1 + 2 + 4 = 7, which is 2³ − 1. This pattern always works: the first n squares together hold 2ⁿ − 1 grains. For all 64 squares the total is 2⁶⁴ − 1 = 18 446 744 073 709 551 615, about 1.8 × 10¹⁹. By a rough estimate, that is many hundreds of times the wheat the whole world harvests in a year. In the sandbox set the rate to 100 % per step, so the amount doubles each time, and flip the axis to a log scale: the curve straightens into a line.

Compound interest

Put 1000 in an account that pays 10 % a year, and leave the interest in. After one year you have 1000 × 1.10 = 1100. After two years the interest is paid on 1100, not on 1000, so you have 1100 × 1.10 = 1210, and after three years 1331. In general, A = P × (1 + r)ⁿ, where P is the start amount, r is the rate as a decimal and n is the number of years. Simple interest would add only 100 a year, giving 1300 after three years. The extra 31 is interest on interest, and it is why exponential growth starts slowly and then runs away.

The rule of 72

How long does money take to double? Solve 1.10ⁿ = 2 and you get n = ln 2 / ln 1.10 ≈ 7.27 years. You can skip the logarithms with the rule of 72: divide 72 by the rate in percent. At 10 %, 72 / 10 = 7.2 years, very close to 7.27. At 6 % the rule says 12 years, and the exact answer is 11.9. At 8 % it says 9 years, and the exact one is 9.01. A check with the table: 1000 becomes 1949 after 7 years and 2144 after 8 years, so the doubling happens during year 8. The rule is at its best for rates between about 6 % and 10 %.

Halving and carbon-14

Growth has a mirror twin: halving. A radioactive atom decays at random, but a large crowd of them obeys a clear rule: after a fixed time, called the half-life, half remain. For carbon-14 the half-life is 5 730 years. Living things take in carbon-14 from the air and food. When they die the supply stops, and the amount halves every 5 730 years. A bone with a quarter of the original amount is two half-lives old: 2 × 5 730 = 11 460 years. With one eighth left it is 3 × 5 730 = 17 190 years old. Willard Libby invented radiocarbon dating in 1949. Beyond about 50 000 years too little is left to measure.

The number e

Return to the bank. Suppose a bank pays 100 % a year. Pay the interest once and 1 becomes 2. Pay it half-yearly, 50 % twice, and you get 1.5 × 1.5 = 2.25. Pay monthly: (1 + 1/12)¹² ≈ 2.613. Pay daily: (1 + 1/365)³⁶⁵ ≈ 2.7146. Do it more often still and the total creeps up but never passes a limit: e ≈ 2.71828. Jacob Bernoulli met this limit in 1683 while studying interest, and Leonhard Euler later gave it the letter e. Like π, e is irrational, so its digits never end.

Continuous growth

When growth compounds all the time, the amount after time t is A = P × e^(r t), where r is the rate as a decimal. Put 1000 into an account with 10 % growth compounded continuously for 1 year: 1000 × e^0.1 ≈ 1105.17, slightly more than the 1100 of a yearly payment. The doubling time is ln 2 / r = 0.693 / 0.10 ≈ 6.93 years. Bacteria in a dish, a spreading rumour and a discharging capacitor follow this same curve, with a negative r for decay. Try different rates in the sandbox and compare the curves.

Growth, doubling and halving