What Decays, and How Fast

Last timeWhy Every Score Is a Log Probability

A quantity that multiplies by less than one each step is the same mathematics upside down. Half life and continuous rate are two names for the same number, and every schedule is built from them.

Nothing new happens in this lesson. A decaying quantity multiplies by a fixed

factor each step exactly as a growing one does; the factor is simply below one.

Every result from the first lesson applies, with the logarithms coming out

negative.

What is new is the vocabulary. Decay gets described in at least three different

ways depending on the field, and a reader who cannot convert between them will

meet the same curve repeatedly without recognising it.

FIG 1Two descriptions of the same fall

half life of 3 minutescontinuous rate of 0.231 per minute

These are not two similar curves, they are one curve written twice, and plotted together they lie exactly on top of each other. The first says the amount halves every three minutes. The second says it falls at a continuous rate whose time constant is about four point three minutes. Neither description is more correct and both appear in the literature, so the conversion between them is a skill rather than a convenience. Notice also that the curve never touches the axis, which is a property of every decay and the reason any decaying process needs a stated threshold before it can be called finished.

Half life is a complete description

Because the process is multiplicative, the time to fall by half does not depend

on when you start measuring. It is the same from the beginning as it is from a

thousandth of the way down, and that single number determines everything else.

Counting half lives is counting factors of two, so the question of how much is

left after a given time is a base two logarithm question, which is why the

fourth lesson is a prerequisite for this one.

FIG 2Counting half lives

step	half lives elapsed	fraction remaining	worth saying out loud
1	1	one half	the definition
2	2	one quarter	halving twice, not halving by two halves
3	3	one eighth	about twelve per cent
4	7	about one per cent	seven halvings is the rule of thumb for one per cent
5	10	about one thousandth	ten halvings is a thousandfold, since two to the ten is 1024
6	20	about one millionth	and the pattern continues: every ten halvings is another factor of a thousand
7	33	about one in ten thousand million	below any threshold anybody sets, which is what practically gone means

7 steps

The two rows worth keeping are the fourth and the fifth. Seven halvings takes something to about one per cent and ten takes it to about a thousandth, which together cover almost every question about how long until this has effectively gone. The last row makes the earlier point concrete: thirty three half lives is a factor of ten thousand million and the quantity is still not nought, so an engineer who needs a time when the process has finished must choose a level and state it rather than wait for the curve to arrive.

The continuous form

Physics, finance and statistics usually prefer a continuous rate in base $e$ ,

for the reason the fourth lesson gave: the calculus produces that base whether

or not anybody wanted it.

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 plan The contents

This 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

What Decays, and How Fast

Half life is a complete description

The continuous form

3 more paragraphs to go

This is the reading half

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