The paper that reaches the Moon
Doubling again and again grows faster than you can imagine.
Fold it again and again
A sheet of paper is 0.001 cm thick. Imagine you could fold it in half as many times as you wish. Each fold doubles the thickness.
How thick will the paper be after 30 folds?
What this lesson covers
The idea
A quantity multiplied by the same number at every step shows multiplicative, or exponential, growth; a quantity that doubles each step grows, from any point, by 2 multiplied by itself as many times as there are steps.
Fold it again and again
A sheet of paper is 0.001 cm thick. Imagine you could fold it in half as many times as you wish. Each fold doubles the thickness.
How thick will the paper be after 30 folds?
- A few centimetres
- As tall as a building
- Higher than an aeroplane flies
Slide the folds
Slide the number of folds. Watch each bar become twice the one before it.
Multiply, multiply, multiply
A quantity multiplied by the same number at every step shows multiplicative, or exponential, growth. A quantity that doubles each step grows, from any point, by 2 multiplied by itself as many times as there are steps.
After any 3 folds the thickness grows 8 times (2 × 2 × 2). From any point, after 10 folds it grows 1024 times (2 multiplied by itself 10 times).
- Folds | Thickness
- 0 | 0.001 cm
- 10 | 1.024 cm
- 17 | about 131 cm
- 26 | about 670 m
- 30 | about 10.7 km
- 46 | more than 7,00,000 km
Notes
A quantity multiplied by the same number at every step grows exponentially. A quantity that doubles each step grows, from any point, by 2 multiplied by itself as many times as there are steps.
Check yourself
The thickness doubles at every fold. Which shows the thickness after 4 folds?
A stack has been folded 5 times. After 3 more folds, it is how many times thicker than it is now?
Answer: 8
From any point, 3 folds make the thickness 2 × 2 × 2 = 8 times as much. It does not matter how many folds came before.
After 10 folds, how many times thicker is the paper than the sheet you started with?
Answer: 1024
2 multiplied by itself 10 times = 1024. So 10 folds make the paper 1024 times as thick.
Which one shows exponential growth?
- 0.001 × 4. That multiplies by 4 only once. Each of the 4 folds multiplies by 2.
- 0.001 × 2 × 2 × 2 × 2 — correct. Yes! Four folds, so ×2 four times: 0.001 × 16 = 0.016 cm.
- 0.001 + 2 + 2 + 2 + 2. That adds 2 each time. A fold doubles the thickness: it multiplies by 2.
- A plant grows 2 cm every day. It adds the same amount each day. That is not multiplying.
- Lotuses in a pond double every day — correct. Yes! The number is multiplied by the same number (2) each day.
- A bus travels 5 km every minute. The distance goes up by the same 5 km each minute. It adds. It does not multiply.