Growth & decay

You take a tablet, and the active ingredient does not vanish all at once: it drops by the same share hour after hour. Your savings do the opposite, they gain the same share year after year. Both follow the same rule. It is not the same amount that gets added each time, it is the same number that multiplies. This single idea sits behind interest, bacterial cultures and the drug level in your blood.

One factor, again and again

If something grows by 3 % per year, the old 100 % stays and 3 % is added. Together that is 103 %, written as a factor: 1.03. So 500 euros become 515 euros after one year. In the second year you multiply by 1.03 again, but this time starting from the 515 euros, which gives 530.45 euros. After two years the account holds $500 \cdot 1.03 \cdot 1.03$, and after ten years the factor has acted ten times. That is exactly what a power is for: the exponent simply counts how often the factor took its turn.

Decay: the factor stays below 1

A factor smaller than 1 makes the amount smaller at every step, without any minus sign. If the drug in the blood drops by 30 % per hour, then 70 % remains, so the factor is 0.7. From 400 mg you get 280 mg after one hour and 196 mg after two hours. Notice how the loss shrinks: first 120 mg, then only 84 mg. It always refers to whatever is left right now, and that is why a decay never mathematically reaches zero.

Doubling time and half life

With growth you ask for the doubling time, with decay for the half life. The doubling time is the time until twice as much is there, the half life the time until only half is left. Both depend only on the factor, not on how much is there right now. A medicine with a half life of 6 hours has dropped to half after 6 hours, to a quarter after 12 hours and to an eighth after 18 hours. So 600 mg becomes 300 mg, then 150 mg, then 75 mg. Saving works as a mirror image: at roughly 7 % growth per year an amount doubles about every 10 years, whether it is 200 euros or 5000 euros.