TwiddleLab / Math / Growth: Exponents and Logs
Half-Life: Growth in Reverse
What you will learn
- Exponential decay multiplies by the same factor below 1 every step, so the amount keeps halving.
- The half-life h is the time to lose half: y = a · (1/2)^(t/h).
- After n half-lives, (1/2)ⁿ is left: 1/8 means three half-lives.
How the lesson goes
Your tea is too hot to drink. When will it be just right?
First, a faster experiment with coins. Bet how many shakes it takes.
Shake the coins.
Each shake, every coin that lands heads leaves. Keep shaking until about one is left.
Now the tea: 90 °C in a 20 °C room.
The gap to room temperature halves every 10 minutes. Find when the tea is 37.5 °C, then Check.
Only the gap to the room halves, not the temperature. 70 halved twice is 17.5, and 20 + 17.5 = 37.5 °C.
Halving on a timer has a name.
Exponential decay: multiply by the same factor below 1 every step. Coins: ×0.5 per shake.
The half-life h is how long it takes to lose half: y = a · (1/2)^(t/h).
Tea: the gap halves every 10 minutes, so its half-life is 10 minutes.
Measure the half-life of dice.
600 dice, and every die that rolls a six leaves. Drag the purple handle until the curve fits the dots.
Each roll keeps 5/6 of the dice. Halving takes as many rolls as it needs for (5/6)ⁿ to reach 1/2: about 3.8.
Game: keep the dose working.
A medicine has a half-life of 6 hours. Tap Dose to add 200 mg. Keep the level in the green band for 48 hours, below the red line.
AI training halves its step size too.
The step starts at 0.08 and is halved every 10 epochs. Drag the flag to the first epoch where it is below 0.01, then Check.
Each halving takes 10 epochs. 0.08 needs four halvings to drop below 0.01 (to 0.005), so 4 × 10 = 40 epochs.
Carbon-14 has a half-life of 5,730 years. A bone has 1/8 of its carbon-14 left. How old is it?
1/8 is (1/2)³, so three half-lives passed: 3 × 5,730 = 17,190 years.
Your dice curve: when are about 150 of the 600 left?
Tap the roll, then Check.
150 is a quarter of 600, and a quarter is (1/2)². So it takes two half-lives: about 2 × 3.8 ≈ 7.6 rolls.
Previous lesson: Growth Curves and Percent Growth · Next lesson: Logs Count Doublings · Part of the course: Growth: Exponents and Logs
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