TwiddleLab / Math / Growth: Exponents and Logs
Doubling Is Wild
What you will learn
- Doubling again and again is a power: after n folds the stack is 0.1 mm × 2ⁿ.
- In 2ⁿ, 2 is the base and n is the exponent: how many times you multiply.
- Exponential growth multiplies each step, so it starts slow and then leaves any adding behind.
How the lesson goes
How many folds to reach the Moon?
A sheet of paper is 0.1 mm thick. Each fold doubles the stack. Drag the lever to your bet.
Fold it 10 times.
Tap Fold and watch the stack.
10 folds made a book. What do 20 folds make?
Tap the landmark you think it reaches, then Check.
Each fold doubles. Ten doublings multiply by 2 × 2 × ... × 2 = 1024. So 20 folds is 10 cm × 1024, about 105 m.
Doubling n times is a power.
After n folds the stack is 0.1 mm × 2ⁿ. 2ⁿ means 2 × 2 × ... × 2, with n twos.
2 is the base: what you multiply by. n is the exponent: how many times.
This is exponential growth: every step multiplies. Adding 1 mm a fold is linear growth.
Game: fold to the Moon.
You are at 20 folds. Keep folding until the stack passes the Moon. The yellow stack adds 1 mm a fold.
Build your own grower.
Pick a start and a multiplier. Make it reach the Moon in exactly 12 steps.
Each step multiplies by your number, so the dots climb the same height on this ladder every step. Twelve equal climbs reach the Moon.
Your grower against a rocket.
The rocket adds 10,000 km a step. Find the first step where your grower is ahead.
The rocket adds 10,000 km every step. Your grower multiplies, so each step it adds more than the step before. Sooner or later it adds more than 10,000 km a step and passes.
Next lesson: Exponent Rules by Counting · Part of the course: Growth: Exponents and Logs
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