The ladder and the paper

The idea

A fixed increase at each step is linear growth; linear growth is additive whereas exponential growth is multiplicative, so exponential growth reaches very large values in far fewer steps.

A ladder to the Moon

Roxie imagines a ladder to the Moon, with a rung every 20 cm. The ladder gains 20 cm at every rung. Estu has a sheet of paper 0.001 cm thick, and every fold doubles it.

Both take 30 steps: 30 rungs and 30 folds. Which is taller?

• The ladder
• The paper
• About the same

Race them step by step

Press + to take one step. The ladder and the paper take the same number of steps. Find the first step where the paper is taller.

One adds, one multiplies

A fixed increase at each step is linear growth. Linear growth is additive, whereas exponential growth is multiplicative, so exponential growth reaches very large values in far fewer steps.

To reach the Moon, 3,84,400 km away: Ladder: 20 + 20 + 20 + … (1,92,20,00,000 times) Paper: 0.001 × 2 × 2 × … (46 times)

The ladder leads at first. The paper passes it at step 19 and stays ahead.

• Steps | Ladder | Paper
• 10 | 200 cm | 1.024 cm
• 18 | 360 cm | about 262 cm
• 19 | 380 cm | about 524 cm
• 30 | 600 cm | about 10.7 km
• 46 | 9.2 m | about 7,03,687 km

Notes

A fixed increase at each step is linear growth. Linear growth is additive, whereas exponential growth is multiplicative, so exponential growth reaches very large values in far fewer steps.

Check yourself

Which one shows linear growth?

A ladder gains 20 cm at every rung. How tall is it, in cm, after 12 rungs?

Answer: 240

20 + 20 + … (12 times) = 12 × 20 = 240 cm.

Rani saves ₹10 more every week: ₹10, 20, 30, … Ravi's savings double every week: ₹10, 20, 40, … After week 6, how many rupees does Ravi have?

Answer: 320

10 → 20 → 40 → 80 → 160 → 320. Ravi's savings are multiplied by 2 each week, so he has ₹320, while Rani has only ₹60.

The paper's first fold adds only 0.001 cm, but a rung adds 20 cm. Why does the paper reach the Moon in only 46 steps?

• Bacteria in a dish double every hour. The number is multiplied by 2 every hour. That is exponential growth.
• A sheet of paper folded again and again. Each fold multiplies the thickness by 2. That is exponential growth.
• A ladder where each rung is 20 cm higher than the one below — correct. Yes! The same amount, 20 cm, is added at every rung.
• Because a fold adds more than 20 cm right from the start. The first fold adds only 0.001 cm. The paper wins later.
• Because the paper gains the same amount at every fold. No. The ladder gains the same 20 cm at every step. The paper's gain doubles at every fold.
• Because the thickness is multiplied, so each gain is bigger than the one before — correct. Yes! Each fold doubles the whole height so far, so the gains keep growing. The ladder always adds the same 20 cm.