# Carbon dating and logarithms

However, this new, green money is stagnant — it can’t grow! Only the original 0 can do “work” to generate money.Simple interest has a simple formula: Every period you earn P * r (principal * interest rate).When you have a growing thing, which creates more growing things, which creates more growing things… The most basic type is period-over-period return, which usually means “year over year”.

There’s no trickery because there’s no compounding — interest can’t grow.

Simple interest is useful when: In practice, simple interest is fairly rare because most types of earnings can be reinvested. You can drive “only” 30 minutes and go 25 miles (50 mph * .5 hours). An interest rate gives you a “trajectory” or “pace” to follow.

If you grew for 6 months, you should be entitled to $25. Each year that blue contributes $50 (in green) to our total amount.

Of course, with simple interest our earnings are based on our original amount, not the “new total”.

There really isn’t an APR vs APY distinction, since your earnings can’t change: you always earn the same amount per year. I think of it as a type of “speed”: But both types of speed have a subtlety: we don’t have to wait the full time period! You could drive 15 minutes and go 12.5 miles (50 mph * .25 hours). If you have $100 at a 50% simple interest rate, your pace is $50/year.

Most interest explanations stop there: here’s the formula, now get on your merry way. But you don’t need to follow that pace for a full year!

But at any instant, there’s a single speed, a single trajectory.

(The math gurus will call this trajectory a “derivative” or “gradient”.

By that logic, do 100 and 200 earn the same amount, too? This issue didn’t seem to bother the ancient Egyptians, but did raise questions in the 1600s and led to Bernoulli’s discovery of e (sorry math fans, e wasn’t discovered via some hunch that a strange limit would have useful properties).