Doing It With Arithmetic, and the Error You Buy
Last timeThe Straight Line That Fits, Then the Parabola
When nothing can be written down, a known rate can be marched forward in small steps using only arithmetic, and the error that buys is predictable and controllable.
The previous lessons left a gap on purpose. A total is a difference of values of
anything with the right rate, which is a magnificent shortcut and sometimes simply
unavailable: for a great many rules nothing with the right rate can be written down,
and for a rate that came from a measurement rather than a formula there is nothing to
write down in the first place.
In that situation nothing is lost except the shortcut. A rate is an instruction as much
as a description, and following the instruction with arithmetic works every time.
- the value of the quantity after k steps. The march only ever knows where it is now, which is why it works without a closed form
- the step size. Smaller is more accurate and more arithmetic, and the exchange rate between the two is the central question
- the rate, evaluated where the march currently stands. It is allowed to depend on the clock, on the quantity itself, or on both
- the step's whole contribution: a rate times a duration, exactly one term of the sums in the lesson on slices
The march, running
Here is the simplest interesting case: a quantity whose rate equals its current amount,
which is money at compound interest, a population with unlimited food, or a decaying
isotope with the sign flipped. The march multiplies by one plus the step every time.
The error, measured
Now march to a fixed finishing time with different step sizes and compare against the
true answer, which for this quantity is known independently.
The lesson stops here
3 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents