One Rule Inside Another
Last timeHow Fast, Exactly Here
Most rules worth asking about are built out of simpler ones. When one rule is fed into another the rates multiply, and the reason is that a small nudge is scaled twice on its way through.
The previous lesson worked out one rate by hand. It took a paragraph of algebra for
the squaring rule, which is about the simplest rule there is. Doing that from scratch
for something like the square of three times the input plus one would be unpleasant,
and for anything genuinely complicated it would be impossible.
It is also unnecessary. That rule is not one rule, it is two rules stacked: triple and
add one, then square. This lesson is about assembling the rate of a stack from the
rates of its parts, and the answer is the most useful single fact in the subject.
- the inner rule, applied first. It is the one that touches the input
- the outer rule, applied to whatever the inner rule produced. It never sees the original input at all
- the change in the middle value. It is caused by the change in the input and is generally a different size from it
- how much the outer rule magnifies a change, measured at the middle value currently in play. Read somewhere else, it is simply the wrong number
Watching a nudge go through
Take the rule that triples the input, adds one, and squares the result. At an input of
two the middle value is seven and the end value is forty nine. Now push the input by a
small amount and look at what arrives.
| step | nudge at the input | change in the middle | change at the end | end change per middle change | end change per nudge | what happened |
|---|---|---|---|---|---|---|
| 1 | 0.1 | 0.3 | 4.29 | 14.3 | 42.9 | A tenth at the input becomes three tenths at the middle: the inner rule tripled it, as it must. |
| 2 | 0.05 | 0.15 | 2.1225 | 14.15 | 42.45 | Halve the nudge and every change halves with it, but the two ratios barely move. |
| 3 | 0.01 | 0.03 | 0.4209 | 14.03 | 42.09 | The fourth column is heading for fourteen, which is twice the middle value of seven. |
| 4 | 0.001 | 0.003 | 0.042 | 14.003 | 42.009 | And the last column is heading for forty two, which is fourteen times three. |
The number fourteen is worth pausing on. The outer rule is squaring, and the rate of
squaring at an input of two would be four. Four is nowhere in this table. The outer
rule is not being applied at two, it is being applied at seven, and twice seven is
fourteen. That single point is where almost all the mistakes live.
The lesson stops here
4 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